# A Survey on Uncertainty Quantification of Large Language Models: Taxonomy, Open Research Challenges, and Future Directions
**Authors**: Ola Shorinwa, Zhiting Mei, Justin Lidard, Allen Z. Ren, AnirudhaMajumdar
> shoa@princeton.edu
> maymei@princeton.edu
> jlidard@princeton.edu
> allen.ren@princeton.edu
> ani.majumdar@princeton.eduPrincetonUniversityPrincetonNJUSA
Abstract.
The remarkable performance of large language models (LLMs) in content generation, coding, and common-sense reasoning has spurred widespread integration into many facets of society. However, integration of LLMs raises valid questions on their reliability and trustworthiness, given their propensity to generate hallucinations: plausible, factually-incorrect responses, which are expressed with striking confidence. Previous work has shown that hallucinations and other non-factual responses generated by LLMs can be detected by examining the uncertainty of the LLM in its response to the pertinent prompt, driving significant research efforts devoted to quantifying the uncertainty of LLMs. This survey seeks to provide an extensive review of existing uncertainty quantification methods for LLMs, identifying their salient features, along with their strengths and weaknesses. We present existing methods within a relevant taxonomy, unifying ostensibly disparate methods to aid understanding of the state of the art. Furthermore, we highlight applications of uncertainty quantification methods for LLMs, spanning chatbot and textual applications to embodied artificial intelligence applications in robotics. We conclude with open research challenges in uncertainty quantification of LLMs, seeking to motivate future research.
Uncertainty Quantification; Large Language Models (LLMs); Confidence Estimation. doi: 1111111.1111111 ccs: Computing methodologies ccs: Computing methodologies Artificial intelligence ccs: Computing methodologies Natural language processing ccs: Computing methodologies Natural language generation
1. Introduction
Large language models have demonstrated remarkable language generation capabilities, surpassing average human performance on many benchmarks including math, reasoning, and coding (Achiam et al., 2023; Anthropic, 2024; Brown, 2020; Touvron et al., 2023; Dubey et al., 2024; Chiang et al., 2023). For example, recent (multi-modal) large language models were shown to achieve impressive scores, e.g., in the $90\%$ percentile, on simulated Law School Admission Test (LSAT) exams, the American Mathematics Competition (AMC) contests, the Multistate Bar Exam, and the Graduate Record Exam (GRE) General Test, outperforming a majority of test takers (Katz et al., 2024; Achiam et al., 2023; Anthropic, 2024). Likewise, LLMs have advanced the state of the art in machine translation, text summarization, and question-and-answer tasks. However, LLMs also tend to produce plausible, factually-incorrect responses to their input prompts, termed hallucinations (Lee et al., 2018). In some scenarios, the hallucinated response is overtly incorrect; however, in many cases, the factuality of the LLM response is harder to discern, posing significant risk as a user might falsely assume factuality of the response, which can result in devastating consequences, especially when safety is of paramount importance. As a result, hallucinations pose a notable danger to the safe, widespread adoption of LLMs.
<details>
<summary>extracted/6588127/images/teaser/uq_for_llms.png Details</summary>
### Visual Description
# Technical Document Extraction: Chat Interface Analysis
## 1. Image Overview
The image depicts a digital chat interface between a user and an AI assistant, followed by a verification footer. The layout follows a standard messaging format with right-aligned user bubbles and left-aligned assistant bubbles.
## 2. Component Isolation
### Region A: User Input (Top Right)
* **Icon:** Black silhouette of a person's head and shoulders.
* **Bubble Color:** Blue.
* **Text Transcription:** "What is the lowest-ever temperature recorded in Antarctica?"
### Region B: AI Response (Middle Left)
* **Icon:** A circular green logo with a black interlocking knot/geometric pattern.
* **Bubble Color:** Royal Blue.
* **Text Transcription:** "The lowest-ever temperature recorded in Antarctica was a staggering -128.6ÂșF (-89.2ÂșC)."
### Region C: User Follow-up (Middle Right)
* **Icon:** Black silhouette of a person's head and shoulders.
* **Bubble Color:** Blue.
* **Text Transcription:** "How confident are you?"
### Region D: AI Confirmation (Lower Left)
* **Icon:** A circular green logo with a black interlocking knot/geometric pattern.
* **Bubble Color:** Royal Blue.
* **Text Transcription:** "I am 100% confident."
### Region E: Footer / Fact-Check Status (Bottom Center)
* **Visual Indicator:** A large, dark red "X" mark (cross).
* **Text Transcription:** "Fact-Check: False"
* **Note:** The word "False" is highlighted in a reddish-brown color.
## 3. Data and Information Summary
| Field | Content |
| :--- | :--- |
| **Subject Matter** | Record low temperatures in Antarctica. |
| **Claimed Value (Imperial)** | -128.6ÂșF |
| **Claimed Value (Metric)** | -89.2ÂșC |
| **AI Confidence Level** | 100% |
| **Verification Status** | **False** |
## 4. Technical Analysis of Content
The image serves as a demonstration of AI "hallucination" or overconfidence in incorrect data.
* **The Fact:** While -89.2°C (-128.6°F) was the record set at Vostok Station in 1983, more recent satellite data (2010/2013) has suggested lower temperatures (approx. -93.2°C or -135.8°F), though these are surface skin temperatures rather than air temperatures.
* **The Discrepancy:** The "Fact-Check: False" label at the bottom indicates that the AI's claim of 100% confidence in its specific data point is being flagged as incorrect or outdated by an external verification layer.
</details>
Figure 1. A user asks an LLM the question: What is the lowest-ever temperature recorded in Antarctica?; in response, the LLM answers definitively. Afterwards, the user asks the LLM how confident the LLM is. Although the LLM states that it is â100% confident,â the LLMâs response fails to pass a fact-check test. Confidence scores provided by LLMs are generally miscalibrated. UQ methods seek to provide calibrated estimates of the confidence of LLMs in their interaction with users.
\Description
[A user asks an LLM the question: What is the lowest-ever temperature recorded in Antarctica?; in response, the LLM answers definitively. Afterwards, the user asks the LLM how confident the LLM is. Although the LLM states that it is â100% confident,â the LLMâs response fails to pass a fact-check test. Confidence scores provided by LLMs are generally miscalibrated. UQ methods seek to provide calibrated estimates of the confidence of LLMs in their interaction with users.]A user asks an LLM the question: What is the lowest-ever temperature recorded in Antarctica?; in response, the LLM answers definitively. Afterwards, the user asks the LLM how confident the LLM is. Although the LLM states that it is â100% confident,â the LLMâs response fails to pass a fact-check test. Confidence scores provided by LLMs are generally miscalibrated. UQ methods seek to provide calibrated estimates of the confidence of LLMs in their interaction with users.
To ensure the trustworthiness of LLMs, substantial research has been devoted to examining the mechanisms behind hallucinations in LLMs (Lee et al., 2018; Chen et al., 2023; Azamfirei et al., 2023; Xu et al., 2024a; Ji et al., 2023), detecting its occurrence, identifying potential causes, and proposing mitigating actions. However, even in the absence of hallucinations, LLMs are susceptible to doubt when given prompts at the boundary of their knowledge base. In these situations, prior work has shown that LLMs fail to accurately convey their uncertainty to a user, either implicitly or explicitly, unlike typical humans (Liu et al., 2023c; Alkaissi and McFarlane, 2023). In fact, LLMs tend to be overconfident even when they should be uncertain about the factuality of their response (Xiong et al., 2023; Groot and Valdenegro-Toro, 2024). We provide an example in Figure 1, where an LLM is asked: âWhat is the lowest-ever temperature recorded in Antarctica?â, to which the LLM responds definitively. Even when prompted for its confidence in its answer, the LLM claims that it is â100% confident.â However, the LLMâs answer fails to pass a fact-check test. Knowing how much to trust an LLM-generated response is critical for users (Kim et al., 2024a), helping inform the development of contingency strategies commensurate with the degree of uncertainty of the LLM in its response. For example, in applications such as robotics, an LLM-equipped robot could seek human guidance (Ren et al., 2023a) or necessitate further review in the judicial practice (Delacroix, 2024). Uncertainty quantification (UQ) methods for LLMs seek to address this challenge by providing users with an estimate of an LLMâs confidence in its response to a given prompt. Indeed, uncertainty quantification can be important in factuality analysis (Huang et al., 2023a).
The rapid adoption of LLMs in many applications has contributed to the fast-pace development of UQ methods for LLMs to promote their safe integration into a wide range of applications. However, the huge volume of UQ methods for LLMs has made it particularly challenging to ascertain the research scope and guarantees provided by existing UQ methods, complicating the identification of useful UQ methods for practitioners seeking to leverage them in their application areas, as well as the identification of impactful future directions for research. We claim that this challenge arises from the lack of a taxonomy that unifies related existing methods and presents an organized view of existing work in this research area.
Through this survey, we seek not only to enumerate existing work in UQ for LLMs, but also to provide a useful taxonomy of UQ methods for LLMs to aid understanding the state of the art in this research area. We reiterate that the introduction of an effective taxonomy for these methods can facilitate their adoption in wide-ranging applications, such as in factuality analysis, hallucination detection, and robotics. We categorize existing uncertainty quantification methods for LLMs into four main classes: (1) token-level uncertainty quantification methods; (2) self-verbalized uncertainty quantification methods; (3) semantic-similarity uncertainty quantification methods; and (4) mechanistic interpretability methods. These categories encompass uncertainty quantification of multi-claim, multi-sentence LLM responses. We elaborate on each category in this survey, identifying the key features shared by methods within each category. Moreover, we identify open research challenges and provide directions for future research, hoping to inspire future effort in advancing the state of the art.
Comparison to other Surveys
A number of surveys on hallucinations in LLMs exists, e.g., (Rawte et al., 2023; Huang et al., 2023b; Tonmoy et al., 2024; Liu et al., 2024e; Bai et al., 2024). These surveys discuss hallucinations in detail, introducing the notion of hallucinations (Rawte et al., 2023), identifying its types and potential causes (Huang et al., 2023b), and presenting mitigation techniques (Tonmoy et al., 2024). However, these papers provide little to no discussion on uncertainty quantification methods for LLMs, as this research area lies outside the scope of these surveys. In contrast, only two surveys on uncertainty quantification methods for LLMs exist, to the best of our knowledge. The first survey (Geng et al., 2024) categorizes confidence estimation and calibration methods into two broad classes: methods for generation tasks and methods for classification tasks, defined by the application domain. The survey in (Geng et al., 2024) focuses more heavily on calibration methods, with a less extensive discussion on confidence estimation methods. In contrast, our paper provides an extensive survey of uncertainty quantification methods with a brief discussion on calibration of uncertainty estimates. For example, whereas (Geng et al., 2024) lacks a detailed discussion on the emerging field of mechanistic interpretability, our survey presents this field in detail, along with potential applications to uncertainty quantification. Moreover, our survey discusses a broad range of applications of uncertainty quantification methods for LLMs, e.g., embodied applications such as in robotics, beyond those discussed in (Geng et al., 2024). A concurrent survey (Huang et al., 2024) on uncertainty quantification of LLMs categorizes existing uncertainty quantification methods within more traditional classes, which do not consider the unique architecture and characteristics of LLMs. In contrast, our survey categorizes existing work within the lens of LLMs, considering the underlying transformer architecture of LLMs and the autoregressive token-based procedure utilized in language generation.
Organization
In Section 2, we begin with a review of essential concepts that are necessary for understanding the salient components of uncertainty quantification of LLMs. We discuss the general notion of uncertainty and introduce the main categories of uncertainty quantification methods within the broader field of deep learning. Subsequently, we identify the relevant metrics utilized by a majority of uncertainty quantification methods for LLMs. In Sections 3, 4, 5, and 6, we discuss the four main categories of uncertainty quantification methods for LLMs, highlighting the key ideas leveraged by the methods in each category. In Section 7, we provide a brief discussion of calibration techniques for uncertainty quantification, with applications to uncertainty quantification of LLMs. In Section 8, we summarize the existing datasets and benchmarks for uncertainty quantification of LLMs and present applications of uncertainty quantification methods for LLMs in Section 9. We highlight open challenges in Section 10 and suggest directions for future research. Lastly, we provide concluding remarks in Section 11. Figure 2 summarizes the organization of this survey, highlighting the key details presented therein.
Uncertainty Quantification for LLMs
Taxonomy
Datasets and Benchmarks
Applications
Open Challenges and Future Directions
Token-Level UQ
Self-Verbalized UQ
Semantic-Similarity UQ
Mechanistic Interpretability
Reading Comprehension
Mathematics
Multi-Hop Reasoning
Factuality Analysis
Chatbot and Textual
Robotics
Consistency and Factuality
Entropy and Factuality
Multi-Episode UQ for Interactive Agents
Mechanistic Interpretability and UQ
Datasets and Benchmarks
(Xiao and Wang, 2021; Kadavath et al., 2022; Bakman et al., 2024; Ling et al., 2024; Vazhentsev et al., 2024; Fadeeva et al., 2024; Ren et al., 2023b)
(Mielke et al., 2022; Lin et al., 2022; Stengel-Eskin et al., 2024; Yang et al., 2024b; Xu et al., 2024b; Tao et al., 2024; Band et al., 2024)
(Kuhn et al., 2023; Chen and Mueller, 2023; Lin et al., 2023; Kossen et al., 2024; Wang et al., 2024a; Qiu and Miikkulainen, 2024; Ao et al., 2024)
(Ahdritz et al., 2024)
(Joshi et al., 2017; Reddy et al., 2019; Lebret et al., 2016)
(Lin et al., 2022)
(Yang et al., 2018; Geva et al., 2021)
(Lin et al., 2021; Li et al., 2023; Thorne et al., 2018)
(Tsai et al., 2024; Ren et al., 2023a; Wang et al., 2023b; Liang et al., 2024; Mullen Jr and Manocha, 2024; Wang et al., 2024c; Zheng et al., 2024)
(Zhang et al., 2023a; Yadkori et al., 2024; Mohri and Hashimoto, 2024; Pacchiardi et al., 2023; Tai et al., 2024; Kolagar and Zarcone, 2024; Steindl et al., 2024)
Figure 2. The overview of this survey, including a taxonomy of uncertainty quantification methods for LLMs, relevant datasets and benchmarks, applications, and open challenges and directions for future research.
2. Background
We review fundamental concepts that are crucial to understanding uncertainty quantification of LLMs. We assume basic familiarity with deep learning and build upon this foundation to introduce more specific concepts, describing the notion of uncertainty, the inner workings of LLMs, and the development of metrics and probes to illuminate the uncertainty of LLMs in their response to a userâs prompt.
2.1. Uncertainty
Uncertainty is a widely-known, yet vaguely-defined concept. For example, people generally associate uncertainty with doubt or a lack of understanding, knowledge, or control, but cannot generally provide a precise definition, especially a mathematical one. This general ambiguity applies to the field of LLMs (Keeling and Street, 2024). For example, a subset of the LLM research field considers the uncertainty of a model to be distinct from its level of confidence in a response generated by the model (Lin et al., 2023), stating that confidence scores are associated with a prompt (input) and a prediction by the model, whereas uncertainty is independent of the modelâs prediction. However, a large subset of the field considers uncertainty and the lack of confidence to be mostly-related, generally-interchangeable concepts. In this section, for simplicity, we consider uncertainty and confidence to be mostly interchangeable.
When prompted, LLMs tend to hallucinate when uncertainty about the correct answer exists, e.g., when a lack of understanding or a lack of knowledge exists (see Figures 4 and 4). In Figures 4 and 4, we ask GPT-4o mini to name the best cooking book written by a (likely) fictional person Jamie Feldman. GPT-4o mini provides a confident response: âThe Ultimate Guide to Cooking for One.â However, based on an internet search, this cookbook does not exist (although many similar ones do). Moreover, when prompted about its confidence, GPT-4o mini apologizes before providing yet another confident, but factually-incorrect response: âThe Jewish Cookbook.â This book is authored by Leah Koenig, not Jamie Feldman. Uncertainty quantification (UQ) methods aim to provide a more rigorous estimate of the modelâs confidence in its response, e.g., from the entropy of the distribution from which the tokens are sampled. Before discussing UQ techniques for LLMs, we identify the types of uncertainty and the methods suitable for characterizing uncertainty in deep-learned models, more broadly.
<details>
<summary>extracted/6588127/images/background/uncertainty/hallucination_chatgpt_4.png Details</summary>
### Visual Description
# Technical Document Extraction: Chat Interface Screenshot
## 1. Image Overview
The image is a digital screenshot of a mobile or web-based chat interface. It depicts a two-part exchange between a user and an AI assistant. The interface uses a light gray background with blue message bubbles.
## 2. Component Isolation
### Region A: User Input (Top Right)
* **Visual Description:** A blue speech bubble aligned to the right side of the screen. To the right of the bubble is a black silhouette icon representing a person (User).
* **Transcription:**
> "What is the best cooking book written by Jaime Feldman?"
### Region B: AI Response (Bottom Left)
* **Visual Description:** A blue speech bubble aligned to the left side of the screen. To the left of the bubble is a circular green icon containing a black stylized "X" or interlocking loop symbol (AI Assistant Logo).
* **Transcription:**
> "Jamie Feldman is known for her book \"The Ultimate Guide to Cooking for One.\" This book features recipes designed for solo cooks, focusing on simple, delicious meals that minimize waste and maximize flavor. It's a great resource for anyone cooking for themselves and looking for practical and tasty ideas!"
## 3. Technical Details & Observations
* **Language:** The text is entirely in English.
* **Typographical Note:** There is a slight spelling discrepancy between the user's query and the AI's response. The user spells the name "Jaime Feldman," while the AI response spells it "Jamie Feldman."
* **Content Summary:** The AI identifies a specific literary work, "The Ultimate Guide to Cooking for One," attributed to Jamie Feldman, and provides a brief summary of the book's value proposition (solo cooking, waste minimization, and simplicity).
* **UI/UX Elements:**
* **Background Color:** Light Gray (#E6E6E6 approx).
* **Bubble Color:** Royal Blue (#3B6DED approx).
* **Text Color:** White (#FFFFFF).
* **Alignment:** Standard chat "right-for-user, left-for-system" layout.
## 4. Data Summary
| Field | Value |
| :--- | :--- |
| **Subject Name (Query)** | Jaime Feldman |
| **Subject Name (Response)** | Jamie Feldman |
| **Primary Work Cited** | The Ultimate Guide to Cooking for One |
| **Key Themes** | Solo cooking, waste reduction, flavor maximization, practical recipes |
</details>
Figure 3. Hallucination in LLMs: When asked for information about a possibly fictional person, LLMs tend to fabricate a response that sounds coherent but is entirely false.
\Description
[Hallucination in LLMs: When asked for information about a possibly fictional person, LLMs tend to fabricate a response that sounds coherent but is entirely false.]Hallucination in LLMs: When asked for information about a possibly fictional person, LLMs tend to fabricate a response that sounds coherent but is entirely false.
<details>
<summary>extracted/6588127/images/background/uncertainty/hallucination_chatgpt_4_followup.png Details</summary>
### Visual Description
# Technical Document Extraction: Chat Interface Screenshot
## 1. Image Overview
The image is a digital screenshot of a conversational user interface (chat bubble format). It depicts a two-part exchange between a human user and an automated assistant/AI.
## 2. Component Isolation
### Region A: User Input (Top Right)
* **Visual Description:** A blue rounded rectangular speech bubble aligned to the right side of the frame. To the right of the bubble is a black silhouette icon representing a person.
* **Text Content:** "Are you confident about your response on Jamie Feldman?"
### Region B: Assistant Response (Bottom Left)
* **Visual Description:** A blue rounded rectangular speech bubble aligned to the left side of the frame. To the left of the bubble is a circular green icon containing a black stylized "X" or interlocking knot symbol.
* **Text Content:** "I apologize for the confusion. Jamie Feldman is best known for âThe Jewish Cookbook,â which features a wide range of traditional and modern Jewish recipes. If youâre looking for a comprehensive guide to Jewish cooking, that would be the book to check out! Thank you for your patience."
## 3. Detailed Text Transcription
| Speaker | Text Content |
| :--- | :--- |
| **User** | Are you confident about your response on Jamie Feldman? |
| **Assistant** | I apologize for the confusion. Jamie Feldman is best known for âThe Jewish Cookbook,â which features a wide range of traditional and modern Jewish recipes. If youâre looking for a comprehensive guide to Jewish cooking, that would be the book to check out! Thank you for your patience. |
## 4. Technical Analysis & Observations
* **Language:** The text is entirely in English.
* **Context:** The assistant is providing a corrective response (indicated by the apology) regarding the identity or work of "Jamie Feldman."
* **Fact Check Note:** The assistant attributes "The Jewish Cookbook" to Jamie Feldman. (Note: In external reality, *The Jewish Cookbook* is widely attributed to Leah Koenig; this screenshot may demonstrate a "hallucination" or error in the AI's knowledge base, which is the subject of the user's skeptical inquiry).
* **Formatting:** The assistant's text uses standard punctuation and curly quotes for the book title.
## 5. Visual Specifications
* **Background Color:** Light grey/silver.
* **Bubble Color:** Royal Blue (#4169E1 or similar).
* **Text Color:** White.
* **Font Style:** Sans-serif.
</details>
Figure 4. Hallucination in LLMs: When asked about its confidence, the LLM apologizes before hallucinating another response. The Jewish Cookbook is authored by Leah Koenig, not Jaime Feldman.
\Description
[Hallucination in LLMs: When asked about its confidence, the LLM apologizes before hallucinating another response. The Jewish Cookbook is authored by Leah Koenig, not Jaime Feldman.]Hallucination in LLMs: When asked about its confidence, the LLM apologizes before hallucinating another response. The Jewish Cookbook is authored by Leah Koenig, not Jaime Feldman.
2.2. Types of Uncertainty
Uncertainty can be broadly categorized into two classes, namely: aleatoric uncertainty and epistemic uncertainty. When considered collectively, the resulting uncertainty is referred to as predictive uncertainty, without a distinction between the two components.
2.2.1. Aleatoric Uncertainty
Aleatoric uncertainty encompasses the lack of definiteness of the outcome of an event due to the inherent randomness in the process which determines the outcome of the event. For example, a model cannot predict with certainty the outcome of an unbiased coin toss due to the random effects in the coin toss, regardless of the complexity of the model or the size of the training dataset used in training the model. This irreducible uncertainty is referred to as aleatoric uncertainty. For example, in the case of LLMs, aleatoric uncertainty can arise when there is inherent randomness in the ground-truth response, e.g., when prompted with âWhat will the temperature be tomorrow?â, the uncertainty associated with the LLMâs output can be characterized as aleatoric uncertainty, which is entirely due to the random effects associated with daily weather conditions. In essence, daily weather conditions cannot be predicted with absolute certainty, irrespective of the amount of training data available.
2.2.2. Epistemic Uncertainty
In contrast to aleatoric uncertainty, epistemic uncertainty characterizes the doubt associated with a certain outcome (prediction) due to a lack of knowledge or âignoranceâ by a model, often due to limited training data. For example, when prompted to provide the digit in the $7$ th decimal place in the square-root of $2$ , GPT-4o mini responds with the answer $6$ . However, this answer is wrong: the digit in the $7$ th decimal place is $5$ . The uncertainty in the LLMâs output can be characterized as epistemic uncertainty, which can be eliminated by training the LLM on more data specific to this prompt. In other words, epistemic uncertainty describes reducible uncertainty, i.e., epistemic uncertainty should reduce when there is more knowledge about the state on which the decision is being made, e.g., via choosing the right model to use for learning, using more training data, or by incorporating any prior knowledge. The uncertainty associated with the response in Figure 4 is entirely epistemic and stems from missing training data. If we train the LLM on more data, including the fact that Jamie Feldman did not write a cookbook, we can eliminate the uncertainty associated with the modelâs response. Before concluding, we note that prior work has explored decomposing predictive uncertainty into epistemic and aleatoric components (Hou et al., 2023).
2.3. Uncertainty Quantification in Deep Learning
Broadly, uncertainty quantification for deep learning lies along a spectrum between two extremes: training-based and training-free methods, illustrated in Figure 5. Whereas training-based methods assume partial or complete visibility and access to the internal structure of the neural network, modifying it to probe its uncertainty, training-free methods use auxiliary models or additional data to quantify the uncertainty of the model post-hoc.
Training-based Training-free BNNs (Jospin et al., 2022) MCMC (Hastings, 1970) Variational Inference (Posch et al., 2019) MC-Dropout (Gal and Ghahramani, 2016; Gal et al., 2017)
| Deep Ensemble |
| --- |
| (Lakshminarayanan et al., 2017; Guo et al., 2018; Cavalcanti et al., 2016; Martinez-Munoz et al., 2008; BuciluÇ et al., 2006; Hinton, 2015) |
| ENNs |
| --- |
| (Osband et al., 2023; Wang and Ji, 2024) |
| Test-time Data Augmentation |
| --- |
| (Lee and AlRegib, 2020; Ayhan and Berens, 2018; Wu and Williamson, 2024; Bahat and Shakhnarovich, 2020) |
Dropout Injection (Loquercio et al., 2020; Ledda et al., 2023) Gradient-based (Lee and AlRegib, 2020; Huang et al., 2021; Igoe et al., 2022)
Figure 5. Uncertainty quantification methods in deep learning span the spectrum from training-based methods to training-free methods.
2.3.1. Training-Based Methods
Training-based uncertainty quantification methods span Bayesian Neural Networks, Monte Carlo Dropout methods, and Deep Ensembles, which we review in the subsequent discussion. Instead of training a set of parameters to predict a single outcome, a Bayesian neural network (BNN) (Jospin et al., 2022) learns a distribution over the modelâs weights $\theta$ . Specifically, a BNN learns a distribution over the parameters, $p(\theta|D)$ , with dataset $D$ , with its prediction consisting of two parts: a maximum a posteriori estimation component $\hat{y}$ , and the uncertainty associated with it, defined by the covariance of the prediction $\Sigma_{\hat{y}|x,D}$ .
Despite being statistically principled, the prohibitive computational costs associated with BNNs prevent them from being directly employed. In order to train BNNs, a variety of methods have been proposed, among which the most popular ones are Markov Chain Monte-Carlo (MCMC) (Hastings, 1970) and variational inference (Posch et al., 2019). The former samples from the exact posterior distribution, while the latter learns to approximate the posterior with a variational distribution, $q_{\varphi}$ . Due to the relaxed requirement of access to large amounts of samples, the variational inference method has been more widely used, with Monte-Carlo dropout (Gal and Ghahramani, 2016; Gal et al., 2017) and Deep ensemble (Lakshminarayanan et al., 2017) being representative methods. More recently, epistemic neural networks (ENNs) (Osband et al., 2023; Wang and Ji, 2024) have been introduced to reduce the computational challenges associated with BNNs. To make ensemble methods more efficient, e.g., in out-of-distribution detection (Vyas et al., 2018), pruning methods (Guo et al., 2018; Cavalcanti et al., 2016; Martinez-Munoz et al., 2008), which reduce redundancy among ensemble members, and distillation methods (BuciluÇ et al., 2006; Hinton, 2015), which reduce the number of networks to one, teaching it to represent the knowledge of a group of networks, have been introduced. While these methods are easy to implement and require much less computation compared to regular BNNs or MCMC, they do suffer from being an approximation of the true posterior distribution. In fact, the modelâs uncertainty predictions could be worse when data augmentation, ensembling, and post-processing calibration are used together (Rahaman et al., 2021).
2.3.2. Training-Free Methods
Training-free methods for estimating uncertainty have become popular due to their ease of implementation. Since neither the network architecture nor the training process need to be revised, training-free methods work well with large-scale foundation models that are costly to train or fine-tune. In (Ayhan and Berens, 2018; Lee and AlRegib, 2020; Wu and Williamson, 2024; Bahat and Shakhnarovich, 2020), the authors perform data augmentation at test time to generate a predictive distribution, quantifying the modelâs uncertainty. Similarly, dropout injection (Loquercio et al., 2020; Ledda et al., 2023) extends MC-dropout to the training-free domain by only performing dropout at inference time to estimate epistemic uncertainty. In (Mi et al., 2022), the authors estimate uncertainty for regression using similar perturbation techniques. Lastly, gradient-based uncertainty quantification methods (Lee and AlRegib, 2020) generate gradients at test-time, which provide an signal for epistemic uncertainty and for OOD detection in (Huang et al., 2021; Igoe et al., 2022), by constructing confounding labels.
2.4. Uncertainty Quantification for LLMs
The introduction of the transformer (Vaswani, 2017) for sequence-to-sequence machine translation tasks spurred the development of large language models. However, as noted in the preceding discussion, LLMs have gained some notoriety for their tendency to hallucinate when uncertain about a response to a specified prompt. Here, we review the general architecture of LLMs and provide some motivation for the development of LLM-specific metrics for quantifying uncertainty.
2.4.1. LLM Architecture
LLMs use the transformer architecture to provide free-form responses to input prompts specified in natural language. The transformer architecture consists of an encoder, which processes the input to the model, and a decoder, which generates the modelâs outputs auto-regressively, where the previous outputs of the model are passed into the model to generate the future outputs. Given an input prompt, the words (elements) of the prompt are tokenized (i.e., the sentences/phrases in natural-language are decomposed into simple units referred to as tokens) and transformed to input embeddings using a learned model. The encoder takes in the input embeddings augmented with positional encodings to incorporate positional context and generates a sequence of latent embeddings, which serves as an input to the decoder, using a stack of $N$ multi-head attention sub-blocks and fully-connected feedforward networks. The decoder takes in the embeddings associated with the previous outputs of the decoder, preceded by a start token, and computes an output embedding using a similar stack of multi-head attention heads and feedforward networks as the encoder. The resulting output embeddings are passed into a linear layer prior to a softmax output layer, which converts the decoder embeddings to a probability distribution over the tokens in the dictionary of the model. In subsequent discussion, we denote the probability of the $j$ âth token in the $i$ âth sentence of an LLMâs output as $p_{ij}$ . The output token is selected from this probability distribution: e.g., by greedily taking the token associated with the maximum probability mass. The resulting output is passed into the decoder for auto-regressive generation of text.
<details>
<summary>extracted/6588127/images/background/architecture/llm_architecture.png Details</summary>
### Visual Description
# Technical Document Extraction: Transformer Decoder Architecture Diagram
## 1. Overview
This image is a technical flow diagram illustrating the architecture of a neural network, specifically a Transformer-based decoder or generative model. It details the sequence of operations from input embedding to the final probability distribution output.
## 2. Component Segmentation
### Region A: Input Processing (Left)
* **Sine Wave Icon:** Located at the far left, representing the periodic nature of positional signals.
* **Positional Encoding (Green Block):** Provides information about the relative or absolute position of tokens in the sequence.
* **Embedding (Light Blue Block):** Converts input tokens into dense vectors.
* **Summation Operator (Circle with +):** Combines the output of the Embedding and Positional Encoding blocks.
### Region B: Core Processing Block (Center - Dashed Box)
This region is enclosed in a dashed rectangle labeled **"$N$ multi-head attention sub-blocks"**, indicating that the internal sequence repeats $N$ times.
1. **Residual Connection 1:** A path that bypasses the first sub-layers, connecting the input of the block to the first internal summation operator.
2. **Norm (Blue Block):** Layer normalization applied to the input.
3. **Masked Multi-Head Attention (Purple Block):** The primary attention mechanism, "masked" to prevent the model from attending to future tokens.
4. **Summation Operator (Circle with +):** Adds the residual connection to the output of the Masked Multi-Head Attention.
5. **Residual Connection 2:** A path that bypasses the second sub-layers, connecting the output of the first summation to the final summation of the block.
6. **Norm (Blue Block):** Second layer normalization.
7. **Feed-Forward (Orange Block):** A point-wise fully connected neural network.
8. **Summation Operator (Circle with +):** Adds the second residual connection to the output of the Feed-Forward block.
### Region C: Output Head (Right)
* **Norm (Blue Block):** A final layer normalization applied after the $N$ blocks.
* **Linear (Magenta Block):** A fully connected layer that projects the vector to the vocabulary size.
* **Softmax (Red Block):** Converts the linear output into a probability distribution over the vocabulary.
---
## 3. Data Flow and Logic
The diagram follows a linear left-to-right flow with internal loops (residual connections).
| Step | Component | Action/Trend |
| :--- | :--- | :--- |
| 1 | **Input** | Embedding and Positional Encoding are summed together. |
| 2 | **Entry to $N$ Blocks** | The combined vector enters the repeating sub-block structure. |
| 3 | **Attention Sub-layer** | Data is normalized, processed via Masked Multi-Head Attention, and then added back to the original input (Residual). |
| 4 | **Feed-Forward Sub-layer** | The result is normalized, processed via a Feed-Forward network, and added back to the sub-layer input (Residual). |
| 5 | **Repetition** | Steps 3 and 4 repeat $N$ times. |
| 6 | **Final Output** | The data undergoes a final Normalization, a Linear transformation, and a Softmax activation to produce the final result. |
---
## 4. Textual Transcriptions
### Labels and Blocks
* **Positional Encoding** (Green)
* **Embedding** (Light Blue)
* **Norm** (Blue - appears 3 times)
* **Masked Multi-Head Attention** (Purple)
* **Feed-Forward** (Orange)
* **Linear** (Magenta)
* **Softmax** (Red)
### Annotations
* **$N$ multi-head attention sub-blocks**: Text located at the top center of the dashed bounding box.
### Symbols
* **$\oplus$ (Summation)**: Represented by a circle containing a plus sign, appearing 3 times in the main flow.
* **$\sim$ (Sine Wave)**: Icon representing the mathematical basis for positional encoding.
* **$\rightarrow$ (Arrows)**: Indicate the directional flow of data.
</details>
Figure 6. Many state-of-the-art LLMs are decoder-only transformers, with $N$ multi-head attention sub-blocks, for auto-regressive output generation.
\Description
[Many state-of-the-art LLMs are decoder-only transformers, with $N$ multi-head attention sub-blocks, for auto-regressive output generation.]Many state-of-the-art LLMs are decoder-only transformers, with $N$ multi-head attention sub-blocks, for auto-regressive output generation.
While early LLM models utilized encoder-only or encoder-decoder transformer architectures, state-of-the-art LLMs now generally utilize a decoder-only architecture. For example, the GPT family of models, such as GPT-4 (Achiam et al., 2023), and the Llama family of models, such as Llama 3 (Dubey et al., 2024), are decoder-only transformers. In Figure 6, we show a decoder-only transformer model. These state-of-the-art models leverage advances in transformers to improve computational efficiency, given the huge size of these models: Llama $3$ has $8$ B parameters for the small variant and $70$ B parameters for the large variant, while GPT- $4$ is rumored to have over one trillion parameters. Llama $3$ uses rotary positional embeddings (RoPE) (Su et al., 2024a) instead of absolute positional embeddings, which have been shown to be more effective than alternative embedding schemes. For a more detailed review of LLMs, we refer readers to (Minaee et al., 2024). Before presenting the metrics utilized by UQ methods for LLMs, we discuss natural-language inference, which is an important component of many UQ methods for LLMs.
2.4.2. Natural-Language Inference
Natural-language inference (NLI) refers to the task of characterizing the relationship between two text fragments, where one text fragment represents a premise (i.e., a statement that is believed to be true) while the other fragment represents a hypothesis (i.e., a statement whose veracity we seek to evaluate based on the premise) (Williams et al., 2017; Dagan et al., 2005; Fyodorov et al., 2000). Given a premise and a hypothesis, we can classify the relation between the text pair as: an entailment, if one can infer that the hypothesis is most likely true given the premise; a contradiction, if one can infer that the hypothesis is most likely false given the premise; or a neutral label, if one cannot infer the truthfulness of the hypothesis from the premise (MacCartney and Manning, 2008; Condoravdi et al., 2003; Monz and de Rijke, 2001). In Figure 7, we provide some examples of text pairs that exhibit entailment, contradiction, or neutrality. In the first example, the premise indicates that the student presented a research paper at a conference (i.e., the student did not skip the conference), hence, the contradiction. In the second example, the premise indicates that the orchestra enjoyed the concert, but does not state whether the orchestra performed at the concert (or just observed the event), hence the neutral label. In the third example, we can infer that the hypothesis is true, since the premise indicates that the team was on vacation, hence, not in the office.
<details>
<summary>extracted/6588127/images/background/nli/nli.png Details</summary>
### Visual Description
# Technical Document Extraction: Natural Language Inference (NLI) Examples
## 1. Image Overview
This image is a conceptual diagram illustrating the three primary categories of Natural Language Inference (NLI): **Contradiction**, **Neutral**, and **Entailment**. The diagram uses a horizontal layout for each example, consisting of a premise (left), a relationship indicator (center), and a hypothesis (right).
## 2. Component Segmentation
### Region A: Left Column (Premises)
* **Visual Style:** Light blue rounded rectangular blocks with black text.
* **Content:**
1. "A student presented a research paper at the conference."
2. "The orchestra enjoyed the concert."
3. "The team was out on vacation yesterday."
### Region B: Center Column (Relationship Indicators)
* **Visual Style:** Each row contains a colored circle flanked by black chain-link icons, symbolizing a logical connection or disconnection. A text label is positioned directly above each circle.
* **Row 1 (Top):** Red circle. Label: "**Contradiction**"
* **Row 2 (Middle):** Yellow circle. Label: "**Neutral**"
* **Row 3 (Bottom):** Green circle. Label: "**Entailment**"
### Region C: Right Column (Hypotheses)
* **Visual Style:** Light purple rounded rectangular blocks with black text.
* **Content:**
1. "The student skipped the conference."
2. "The orchestra performed at the concert."
3. "The team was not in the office yesterday."
---
## 3. Data Table Extraction
The following table reconstructs the logical relationships presented in the diagram:
| Premise (Blue Block) | Relationship (Center Label/Color) | Hypothesis (Purple Block) |
| :--- | :--- | :--- |
| A student presented a research paper at the conference. | **Contradiction** (Red) | The student skipped the conference. |
| The orchestra enjoyed the concert. | **Neutral** (Yellow) | The orchestra performed at the concert. |
| The team was out on vacation yesterday. | **Entailment** (Green) | The team was not in the office yesterday. |
---
## 4. Logical Flow and Trend Analysis
### Row 1: Contradiction (Red)
* **Logic:** The premise states the student was at the conference presenting. The hypothesis states they skipped it. These two statements cannot both be true simultaneously.
* **Visual Indicator:** The red color typically signifies a "stop" or "conflict" in logical processing.
### Row 2: Neutral (Yellow)
* **Logic:** The premise states the orchestra enjoyed the concert. While it is likely they performed, they could have been in the audience. The premise does not provide enough information to prove or disprove the hypothesis.
* **Visual Indicator:** The yellow color signifies "caution" or "indeterminacy," where the truth value is unknown based solely on the provided text.
### Row 3: Entailment (Green)
* **Logic:** If the team was out on vacation, it is a logical necessity that they were not in the office. The truth of the premise guarantees the truth of the hypothesis.
* **Visual Indicator:** The green color signifies a "go" or "positive match," indicating a direct logical follow-through.
## 5. Language Declaration
The text in this image is entirely in **English**. No other languages are present.
</details>
Figure 7. Natural-language inference models characterize the relationship between a pair of texts, namely: a premise and a hypothesis. The possible relations include: (1) an entailment where the hypothesis can be inferred from the premise; (2) a contradiction, where the hypothesis is more likely false given the premise; and (3) a neutral relation, where the veracity of the hypothesis cannot be determined from the premise.
\Description
[Natural-language inference models characterize the relationship between a pair of texts, namely: a premise and a hypothesis. The possible relations include: (1) an entailment where the hypothesis can be inferred from the premise; (2) a contradiction, where the hypothesis is more likely false given the premise; and (3) a neutral relation, where the veracity of the hypothesis cannot be determined from the premise.]Natural-language inference models characterize the relationship between a pair of texts, namely: a premise and a hypothesis. The possible relations include: (1) an entailment where the hypothesis can be inferred from the premise; (2) a contradiction, where the hypothesis is more likely false given the premise; and (3) a neutral relation, where the veracity of the hypothesis cannot be determined from the premise.
NLI methods play an important role in uncertainty quantification of LLMs. Many UQ methods for LLMs rely on characterization of the semantic relationship between multiple realizations of the LLMâs responses to a given input prompt to determine the confidence of the model. Many of these methods rely on learned models for natural-language inference, such as BERT (Devlin, 2018), which utilizes a transformer-based architecture to learn useful language representations that are crucial in natural-language tasks such as question answering and natural-language inference. Unlike many standard language models, e.g., Generative Pre-trained Transformer (GPT) (Radford and Narasimhan, 2018), which impose a unidirectionality constraint where every token can only attend to previous tokens, BERT employs a bidirectional approach where each token can attend to any token regardless of its relative position, using a masked language model, potentially enabling the model to capture broader context, especially in sentence-level tasks. In (Liu, 2019), the authors demonstrate that the performance of BERT is limited by inadequate pre-training and propose an improved model, named RoBERTa (Liu, 2019), which retains the same architecture as BERT but is trained for longer with larger mini-batches of data with longer sequences. DeBERTa (He et al., 2020) further improves upon the performance of RoBERTa by introducing a disentangled attention mechanism and an enhanced mask decoder.
2.4.3. Metrics for Uncertainty Quantification for LLMs.
Uncertainty quantification in the LLM community has largely eschewed traditional UQ methods for learned models due to the notable computation cost of running inference on LLMs (Balabanov and Linander, 2024), although, a few UQ methods for LLMs utilize deep ensembles, e.g., (Wang et al., 2023a; Balabanov and Linander, 2024; Zhang et al., 2024; Arteaga et al., 2024), generally based on low-rank adaptation (LoRA) (Hu et al., 2021). Consequently, many UQ methods in this space have introduced less computationally intensive approximate quantification methods that directly harness the unique architecture of LLM models to assess the uncertainty of these models. In some cases, these methods retain the high-level idea of ensemble methods, quantifying the uncertainty of the model on a given prompt using the outputs of a set of individual models or a collection of outputs from the same model, with a temperature parameter less than one to promote greater stochasticity in the tokens generated by the model. UQ methods for LLMs can be broadly categorized into white-box models and black-box models (Liu et al., 2024d; Vashurin et al., 2024), illustrated in Figure 9 and Figure 9, respectively.
White-Box UQ Methods
White-box UQ models assume that the underlying architecture of the model is partially or completely visible and accessibleâhence the term white-box âtaking advantage of access to the intermediate outputs of the underlying models, such as the probability distribution over the generated tokens or outputs at the inner layers of the model, to assess the uncertainty of the model (Kuhn et al., 2023; Azaria and Mitchell, 2023; Fadeeva et al., 2024). We provide some metrics utilized by white-box UQ methods for LLMs, where $p_{ij}$ denotes the conditional probability of token $j$ (conditioned on all preceding tokens) in sentence $i$ :
<details>
<summary>extracted/6588127/images/background/uq_techniques/white_box.png Details</summary>
### Visual Description
# Technical Document Extraction: Chat Interface Analysis
## 1. Image Overview
The image depicts a digital user interface (UI) representing a chat conversation between a human user and an AI assistant. The interface includes a query, a response with a highlighted keyword, and a technical pop-up box displaying probability or confidence scores for specific entities.
---
## 2. Component Isolation
### Region A: User Input (Header/Top Right)
* **Visual Description:** A blue speech bubble with white text, aligned to the right. A black silhouette icon of a person is positioned to the right of the bubble.
* **Transcribed Text:** "When did volleyball become an Olympic Sport?"
### Region B: AI Response (Main Body/Left)
* **Visual Description:** A blue speech bubble with white text, aligned to the left. A circular green icon containing a stylized "X" or knot symbol is positioned to the left of the bubble.
* **Transcribed Text:** "Volleyball debuted at the Tokyo Olympic Games in 1964."
* **Formatting Note:** The word **"Tokyo"** is highlighted in a pink/magenta font color within the blue bubble, indicating an interactive element or a specific entity extraction.
### Region C: Data Overlay/Pop-up (Center/Foreground)
* **Visual Description:** A white rectangular box with a black border, superimposed over the AI response bubble. It contains a list of geographic locations paired with percentage values.
* **Spatial Placement:** Centered horizontally, overlapping the bottom edge of the AI response bubble.
* **Text Color:** All text within this box is rendered in an orange/copper font.
---
## 3. Data Table Extraction (Pop-up Content)
The following table reconstructs the data found in the foreground pop-up box. This data appears to represent a "Top-K" prediction list or confidence scores for the entity "Tokyo" mentioned in the response.
| Entity (Location) | Value (Percentage) |
| :--- | :--- |
| Tokyo | 87.28% |
| Osaka | 7.49% |
| Kyoto | 4.16% |
| Beijing | 1.04% |
| Mumbai | 0.03% |
---
## 4. Technical Analysis & Trends
* **Primary Data Point:** "Tokyo" is the highest-ranked entity with a dominant confidence score of **87.28%**.
* **Trend Verification:** The values follow a sharp downward trend. There is a significant drop of **~79.79 percentage points** between the first result (Tokyo) and the second result (Osaka). The remaining values (Kyoto, Beijing, Mumbai) show a continuing exponential decay in probability.
* **Contextual Inference:** This image likely demonstrates an "Explainable AI" (XAI) or "Attribution" feature where the system shows the internal probability distribution for a specific fact or word choice generated in the response.
---
## 5. Language Declaration
* **Primary Language:** English (US).
* **Other Languages:** None detected. All geographic entities are written in English script.
</details>
Figure 8. White-box uncertainty quantification methods utilize an LLMâs internal information, e.g., the modelâs probabilities for the token associated with each output.
\Description
[White-box uncertainty quantification methods utilize an LLMâs internal information, e.g., the modelâs probabilities for the token associated with each output.]White-box uncertainty quantification methods utilize an LLMâs internal information, e.g., the modelâs probabilities for the token associated with each output.
<details>
<summary>extracted/6588127/images/background/uq_techniques/black_box.png Details</summary>
### Visual Description
# Technical Document Extraction: LLM Confidence Estimation Flowchart
## 1. Document Overview
This image is a technical flowchart illustrating a process for calculating a "Confidence Estimate" from a Large Language Model (LLM) based on the consistency of multiple generated responses to a single prompt.
## 2. Component Isolation and Analysis
### Region 1: Input (Header)
* **Visual Element:** A blue rounded rectangular box with a black silhouette icon of a person's head and shoulders to the top right.
* **Text Content:** "Who was Abraham Lincoln?"
* **Function:** Represents the user-provided query or prompt entering the system.
### Region 2: Processing (Middle-Top)
* **Visual Element:** A lime-green square containing a black stylized knot/infinity-like logo.
* **Label:** "LLM" (positioned to the left of the square).
* **Function:** Represents the Large Language Model processing the input.
* **Flow:** A downward-pointing arrow connects the Input box to this LLM component.
### Region 3: Output Generation (Middle-Bottom)
* **Visual Element:** A large light-cyan rectangular container labeled "Randomly-Generated Responses".
* **Internal Components:** Two distinct white boxes with black borders, separated by an ellipsis ("...").
* **Left Box Text:** "Abraham Lincoln was the fifteenth president of the U.S., serving from 1861 to 1865."
* **Right Box Text:** "Abraham Lincoln was the sixteenth president of the U.S., serving from 1861 to 1864."
* **Function:** Illustrates the model generating multiple variations of an answer. Note the factual discrepancies between the two samples (15th vs 16th president; 1865 vs 1864 end date).
* **Flow:** A downward-pointing arrow connects the LLM component to this container.
### Region 4: Result (Footer)
* **Visual Element:** A pink rounded rectangular box.
* **Text Content:** "Confidence Estimate: 75%"
* **Function:** The final output of the process, quantifying the reliability of the model's answers based on the variance in the generated responses.
* **Flow:** A downward-pointing arrow connects the "Randomly-Generated Responses" container to this final box.
## 3. Process Flow Summary
1. **Prompting:** A user asks a factual question.
2. **Inference:** The LLM processes the prompt.
3. **Sampling:** Instead of a single output, the system generates a set of "Randomly-Generated Responses."
4. **Evaluation:** The system compares these responses. Because the responses contain conflicting information (e.g., different ordinal numbers for the presidency and different end dates), the system calculates a numerical confidence score.
5. **Output:** The process concludes with a "Confidence Estimate" (in this example, 75%).
## 4. Text Transcription (Precise)
| Element Type | Text Content |
| :--- | :--- |
| **Input Prompt** | Who was Abraham Lincoln? |
| **Processor Label** | LLM |
| **Container Title** | Randomly-Generated Responses |
| **Response A** | Abraham Lincoln was the fifteenth president of the U.S., serving from 1861 to 1865. |
| **Separator** | ... |
| **Response B** | Abraham Lincoln was the sixteenth president of the U.S., serving from 1861 to 1864. |
| **Final Output** | Confidence Estimate: 75% |
## 5. Language Declaration
All text in this image is in **English**. No other languages are present.
</details>
Figure 9. Black-box uncertainty quantification methods do not access the internal states or probabilities computed by the model, quantifying the modelâs uncertainty entirely from its natural-language response.
\Description
[Black-box uncertainty quantification methods do not access the internal states or probabilities computed by the model, quantifying the modelâs uncertainty entirely from its natural-language response.]Black-box uncertainty quantification methods do not access the internal states or probabilities computed by the model, quantifying the modelâs uncertainty entirely from its natural-language response.
1. Average Token Log-Probability. The average of the negative log-probability of the tokens, which captures the average confidence of the model (Manakul et al., 2023), is given by: ${\mathrm{Average}(p)=-\frac{1}{L_{i}}\sum_{j}\log(p_{ij}),}$ where sentence $i$ consists of $L_{i}$ tokens. Note that the value of this metric increases as the conditional probability distribution of each token decreases, signifying an decrease in the modelâs confidence. The average token probability is related to the product of the token probabilities.
1. Perplexity. The perplexity of a modelâs prediction represents the exponential of the average of the negative log-probability of the tokens which comprise the sentence (response) generated by the LLM (Fadeeva et al., 2024). Perplexity is given by: ${\mathrm{Perplexity}(p)=\exp\left(-\frac{1}{L_{i}}\sum_{j}\log(p_{ij})\right).}$
1. Maximum Token Log-Probability. The maximum token log-probability captures the token with the lowest conditional probability, which is given by: ${\mathrm{Maximum}(p)=\max_{j}-\log(p_{ij}).}$
1. Response Improbability. This metric entails computing the probability of a given sentence given the conditional distribution for each token (Fadeeva et al., 2024), where the probability distribution is conditioned on preceding tokens, and subtracting the resulting value from one. The uncertainty metric is defined as: ${\mathrm{Improb.}=1-\prod_{j}p_{ij}.}$
1. Entropy. The maximum entropy of the probability distribution associated with each token can be utilized as a metric for UQ, given by: ${\mathrm{Entropy}=\max_{j}\mathcal{H}(p_{j}),}$ where $\mathcal{H}$ represents the entropy of the probability distribution $p_{j}$ of token $j$ . Some existing methods claim that this metric is better than the perplexity (Fadeeva et al., 2024). Similarly, the predictive entropy (Malinin and Gales, 2020) at input $x$ and output $y$ is defined as: ${\mathcal{H}(Y\mid x)=-ât p(y\mid x)\ln p(y\mid x)dy.}$ In the discrete case, the entropy associated with the output distribution of token $j$ in sentence $i$ is defined by: ${\mathcal{H}_{ij}=-\sum_{wâ\mathcal{D}}p_{ij}(w)\log p_{ij}(w),}$ where $\mathcal{D}$ denotes the dictionary containing all possible words in the model and $w$ represents a word in $\mathcal{D}$ .
Black-Box UQ Methods
In contrast, black-box methods assume that the modelâs internal outputs cannot be accessed externally (Manakul et al., 2023; Chen and Mueller, 2023). Hence, these methods quantify the uncertainty of the model entirely from the modelâs response to an input prompt. Prior work has discussed the pros and cons of both categories of UQ methods (Lin et al., 2023). Concisely, white-box methods generally require access to the underlying architecture and intermediate outputs of an LLM, which is increasingly difficult to obtain given that many LLMs have become closed-source models, posing a significant limitation. In contrast, black-box models enable UQ of closed-source models such as OpenAIâs GPT-4 (Achiam et al., 2023) and Anthropic Claude (Anthropic, 2024), which do not provide complete access to the model. In general, black-box UQ methods for LLMs require the evaluation of the similarity between multiple responses generated by an LLM or an ensemble of LLMs on the same or similar prompts to quantify the uncertainty of the LLM on a given input prompt. Other black-box UQ methods, such as self-verbalized UQ methods, train the model to directly provide a natural-language estimate of its confidence. Here, we identify some prominent techniques for measuring the similarity between a pair of text fragments:
1. NLI Scores. As described in Section 2.4.2, NLI models, such as RoBERTa (Liu, 2019) and DeBERTa (He et al., 2020), classify the relationship between a pair of text fragment as either an entailment, a contradiction, or a neutral relation. Many black-box methods utilize the probabilities (or logits) predicted by the NLI model for these three classes as a measure of the similarity between the two text fragments, which is ultimately used to quantify the uncertainty of the LLM. For example, given the probability $p_{\mathrm{entail}}$ predicted by an NLI model that a text fragment $t_{1}$ entails another text fragments $t_{2}$ , we can define the similarity between both text fragments as: ${\mathrm{sim}(t_{1},t_{2})=p_{\mathrm{entail}}}$ . Conversely, given the probability of contradiction $p_{\mathrm{contradict}}$ , we can define the similarity between $t_{1}$ and $t_{2}$ as: ${\mathrm{sim}(t_{1},t_{2})=1-p_{\mathrm{contradict}}}$ .
1. Jaccard Index. The Jaccard index, also referred to as Intersection-over-Union measures the similarity between two sets by computing the ratio of the intersection of both sets and the union of both sets. Hence, the Jaccard index $J$ between two sets $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ , where each set consists of the words that make up its associated text fragment, is given by: ${J(\mathcal{T}_{1},\mathcal{T}_{2})=\frac{|\mathcal{T}_{1}\cap\mathcal{T}_{2}|%
}{|\mathcal{T}_{1}\cup\mathcal{T}_{2}|}.}$ Although the Jaccard index always lies between $0 0$ and $1$ , making it a suitable metric (Pilehvar et al., 2013; Cronin et al., 2017; Qurashi et al., 2020), the Jaccard index does not consider the context of the text fragments, which is important in evaluating the similarity between both text fragments.
1. Sentence-Embedding-Based Similarity. The similarity between two text fragments can also be determined by computing the cosine-similarity between the sentence embeddings associated with each text fragment. Sentence-embedding models transform natural-language inputs (or tokens) into a vector space, enabling direct computation of the similarity between two sentences (phrases). For example, Sentence-BERT (SBERT) (Reimers, 2019) builds upon the pretrained BERT architecture to train a model that computes semantically-relevant sentence embeddings. Other similar models include LaBSE (Feng et al., 2020) and SONAR (Duquenne et al., 2023). Since the sentence embeddings capture the context of the text fragment, this approach is less susceptible to the challenges faced by the Jaccard index, such as those that arise with negated words.
1. BERTScore. The BERTScore (Zhang et al., 2019) measures the similarity between two sentences by computing the cosine-similarity between the contextual embedding of each token (word) in the reference sentence $t_{r}$ and the contextual embedding of the associated token in the candidate sentence $t_{c}$ . The token embeddings are generated from NLI models to capture the context of the sentence. As a result, a given word might have different embeddings, depending on the context of the sentence in which it is used, addressing the challenges faced by the Jaccard similarity metric and word-embedding-based metrics. The BERTScore is composed of a precision $P_{\mathrm{BS}}$ , recall $R_{\mathrm{BS}}$ , and F1 $F_{\mathrm{BS}}$ score, given by:
$$
P_{\mathrm{BS}}=\frac{1}{|t_{c}|}\sum_{\hat{w}_{j}\in t_{c}}\max_{w_{i}\in t_{%
r}}w_{i}^{\top}\hat{w}_{j},\enspace R_{\mathrm{BS}}=\frac{1}{|t_{r}|}\sum_{w_{%
i}\in t_{r}}\max_{\hat{w}_{j}\in t_{c}}w_{i}^{\top}\hat{w}_{j},\enspace F_{%
\mathrm{BS}}=2\frac{P_{\mathrm{BS}}\cdot R_{\mathrm{BS}}}{P_{\mathrm{BS}}+R_{%
\mathrm{BS}}}, \tag{1}
$$
where each token in the candidate sentence is matched to its most similar token in the reference sentence. The BERTScore is obtained by computing the cosine-similarity between matched pairs. Since each token embedding is normalized, the cosine-similarity between a pair of embeddings simplifies to the inner-product. The recall score is related to the ROUGE metric (Lin, 2004) used in evaluating text summaries and more broadly to the BARTScore (Yuan et al., 2021). However, the ROUGE metric utilizes human-provided summaries as the reference.
In the following sections, we describe the main categories of UQ methods for LLMs in detail, namely: (1) Token-Level UQ Methods; (2) Self-Verbalized UQ Methods; (3) Semantic-Similarity UQ Methods; and (4) Mechanistic Interpretability, outlined in Figure 2. Although mechanistic interpretability has not been widely applied to uncertainty quantification, we believe that insights from mechanistic interpretability can be more extensively applied to the uncertainty quantification of LLMs; hence, we include these methods within our taxonomy.
3. Token-Level UQ
We recall that the outputs of an LLM are generated by sampling from a probability distribution over the tokens that make up the outputs, conditioned on the preceding tokens in the outputs (see LABEL:{sec:background_uq_llm}). Token-level UQ methods leverage the probability distribution over each token to estimate the probability of generating a given response from an LLM. Although a high predicted probability in a particular generation may not be indicative of its correctness over another, direct quantification of the modelâs generating distribution may lead to better understanding of the stochasticity of generations. Token-level UQ methods utilize the white-box UQ metrics discussed in Section 2.4.3 to estimate the randomness in the probability distribution associated with an LLMâs response. For example, some token-level UQ methods compute the entropy of the underlying probability distribution over the tokens (Xiao and Wang, 2021; Ling et al., 2024) or semantic clusters (Kuhn et al., 2023) (referred to as semantic entropy) to estimate the LLMâs confidence. Likewise, a variant of SelfCheckGPT (Manakul et al., 2023) trains an $n$ -gram model using multiple samples of the response of an LLM to a given query including its main response. Subsequently, SelfCheckGPT estimates the LLMâs uncertainty by computing the average of the log-probabilities of the tokens generated by the $n$ -gram model, given the original response of the LLM. Moreover, SelfCheckGPT proposes using the maximum of the negative log-probability to estimate the LLMâs uncertainty.
Token-based UQ methods generally perform poorly with long-form responses, because the product of the token probabilities decrease with longer responses, even when the responses are semantically equivalent to a shorter response. To address this limitation, token-based UQ methods employ a length-normalized scoring function (Thomas and Joy, 2006; Malinin and Gales, 2020), to reduce the dependence of the UQ metrics on the length of the sequence, given by: ${\mathrm{Product(p)}=\prod_{j=1}^{L_{i}}p_{ij}^{\frac{1}{L_{i}}}}$ , where $L_{i}$ denotes the length of sentence $i$ , and $p_{ij}$ is the conditional probability of token $j$ , given all preceding tokens, in sentence $i$ . The work in (Bakman et al., 2024) introduces Meaning-Aware Response Scoring (MARS) as an alternative to length-normalized scoring. MARS utilizes an importance function to assign weights to each token based on its contribution to the meaning of the sentence. The contribution of each token to the meaning of the sentence is determined using BEM (Bulian et al., 2022), a question-answer evaluation model. Taking a different approach, Claim-Conditioned Probability (CCP) (Fadeeva et al., 2024) decomposes the output of an LLM into a set of claims and computes the token-level uncertainty of each claim from its constituent tokens. CCP utilizes the OpenAI Chat API (Brown, 2020; Achiam et al., 2023) to identify the main claims in a given response. By examining the component claims, CCP provides finer-grained uncertainty quantification compared to other UQ methods for LLMs.
As described, token-level UQ methods estimate the uncertainty of an LLM based on the conditional distribution associated with each token. Although this approach is effective in general, the conditional distribution of the tokens can be misleading in certain scenarios, especially when an initial token is incorrect but all the succeeding tokens are highly probable given the initial token. Trainable attention-based dependency (TAD) (Vazhentsev et al., 2024) trains a regression model on the conditional dependence between the tokens and applies the predicted factors to improve the estimated uncertainty of the LLM. Lastly, we present token-level UQ methods that use specific prompting strategies to estimate the modelâs confidence. The work in (Kadavath et al., 2022) shows that token-based UQ methods can be particularly effective in estimating the confidence of LLMs when the model is prompted to select an option when given a multiple-choice question. Specifically, the authors show that the modelâs probability distribution over the options in the prompt is well-calibrated, when presented with multiple-choice problems or problems with a True/False answer. Further, the authors fine-tune an LLM with a value head to predict the probability that the model knows the answer to a given question for each token. The probability associated with the LLMâs final token is defined as the confidence of the LLM in its response for the given prompt. The results demonstrate that the LLM predictions of these probability values are well-calibrated, with an improvement in the calibration performance with larger models. Other follow-on work leveraging multiple-choice problems to estimate the uncertainty of LLMs includes (Ren et al., 2023b).
4. Self-Verbalized UQ
Self-verbalized uncertainty quantification methods seek to harness the impressive learning and reasoning capabilities of LLMs to enable an LLM to express its confidence in a given response through natural-language. Self-verbalized uncertainty estimates (e.g., expressed as probabilities) are more easily interpretable to humans, especially when the estimates are provided using widely-used epistemic uncertainty markers (Tang et al., 2024; Yona et al., 2024), e.g., words like I am not sure⊠or This response might be⊠Figure 11 illustrates the use of epistemic markers by an LLM to convey its uncertainty, when asked of the team that won the 2022 NBA Finals. The response of the LLM is actually incorrect; however, by expressing its uncertainty, a user may be more inclined to verify the factuality of the LLMâs response. Prior work has shown that LLMs typically fail to accurately express their confidence in a given response, often using decisive words that suggest confidence, while at the same time being unsure of the accuracy of their response. Empirical studies (Krause et al., 2023) have shown that poor calibration of LLMâs self-verbalized confidence estimates is more pronounced in low-data language settings, e.g., Hindi and Amharic.
<details>
<summary>extracted/6588127/images/self_verbalized_uq/epistemic_uncertainty_markers.png Details</summary>
### Visual Description
# Technical Document Extraction: LLM Epistemic Markers Diagram
## 1. Overview
This image is a flow diagram illustrating the difference in Large Language Model (LLM) outputs when using "Epistemic Markers" versus when they are absent. The diagram follows a top-down linear progression from a user query to a model processing stage, resulting in two comparative output formats.
## 2. Component Isolation
### Region 1: Header (Input Stage)
* **User Icon:** A black silhouette of a person's head and shoulders is positioned to the right of the input box.
* **Input Box:** A blue rounded rectangle containing white text.
* **Text Transcription:** "Which team won the 2022 NBA Finals?"
* **Flow Element:** A downward-pointing black arrow connects the Input Box to the Processing Stage.
### Region 2: Main Processing (LLM Stage)
* **Label:** The text "LLM" is positioned to the left of the central icon.
* **Icon:** A lime-green square with rounded corners containing a black circular emblem with a stylized "X" or interlocking knot design.
* **Flow Element:** A downward-pointing black arrow connects the LLM icon to the Output Stage.
### Region 3: Footer (Output Comparison Stage)
This region is contained within a large, light-blue rounded rectangle and is split into two parallel columns.
#### Column A: With Epistemic Markers (Left)
* **Sub-header Label:** A pale yellow rectangle containing the text: "With Epistemic Markers"
* **Output Box:** A light-blue box with a black border.
* **Transcribed Text:** "I think the Milwaukee Bucks won the 2022 NBA Finals, but I am not sure."
* **Visual Emphasis:** The words "**I think**" and "**but I am not sure**" are highlighted in **red text**, identifying them as the epistemic markers (indicators of uncertainty).
#### Column B: Without Epistemic Markers (Right)
* **Sub-header Label:** A pale yellow rectangle containing the text: "Without Epistemic Markers"
* **Output Box:** A light-blue box with a black border.
* **Transcribed Text:** "The Milwaukee Bucks won the 2022 NBA Finals."
* **Visual Emphasis:** All text is in standard black font, presenting the statement as a definitive fact without qualifiers.
---
## 3. Technical Analysis & Logic Flow
The diagram demonstrates a specific behavior in AI communication:
1. **The Query:** The user asks a factual question about a past event.
2. **The Error:** In both output examples, the LLM provides incorrect information (The Golden State Warriors won the 2022 NBA Finals, not the Milwaukee Bucks).
3. **The Comparison:**
* **Epistemic Version:** By using markers like "I think" and "but I am not sure," the model signals its low confidence in the accuracy of the statement, potentially alerting the user to verify the fact.
* **Non-Epistemic Version:** The model states the incorrect information as an absolute truth, which constitutes a "hallucination" presented with high confidence.
## 4. Text Summary
| Element | Content |
| :--- | :--- |
| **User Query** | Which team won the 2022 NBA Finals? |
| **Processor** | LLM |
| **Output (With Markers)** | **I think** the Milwaukee Bucks won the 2022 NBA Finals, **but I am not sure.** |
| **Output (Without Markers)** | The Milwaukee Bucks won the 2022 NBA Finals. |
</details>
Figure 10. The LLM provides an incorrect response, but communicates its uncertainty using epistemic markers, e.g., âI think.â
\Description
[The LLM provides an incorrect response, but communicates its uncertainty using epistemic markers, e.g., âI think.â]The LLM provides an incorrect response, but communicates its uncertainty using epistemic markers, e.g., âI think.â
<details>
<summary>extracted/6588127/images/self_verbalized_uq/self_verbalized_uncertainty.png Details</summary>
### Visual Description
# Technical Document Extraction: Chat Interface Screenshot
## 1. Image Overview
The image is a digital screenshot of a conversational user interface (chat interface) featuring a dialogue between a human user and an AI assistant. The layout follows a standard messaging format with right-aligned user bubbles and left-aligned assistant bubbles.
## 2. Component Isolation
### Region A: User Input (Top Right)
* **Icon:** A black silhouette of a person's head and shoulders.
* **Bubble Color:** Bright Blue.
* **Text Content:** "What is the capital of Luxembourg?"
* **Spatial Placement:** Top right corner of the interface.
### Region B: Assistant Response (Middle Left)
* **Icon:** A circular logo featuring a black background with a lime-green double-infinity/interlocking knot symbol, enclosed in a lime-green border.
* **Bubble Color:** Medium Blue.
* **Text Content:** "Luxembourg is the capital of Luxembourg."
* **Spatial Placement:** Middle left, positioned below the user input.
### Region C: Metadata/Confidence Score (Bottom Left)
* **Icon:** Identical to Region B (lime-green interlocking knot logo).
* **Bubble Color:** Light Purple/Lavender.
* **Text Content:** "Confidence: I am 98% confident."
* **Spatial Placement:** Bottom left, positioned directly below the assistant's primary response.
## 3. Data Extraction and Transcription
| Role | Element | Content |
| :--- | :--- | :--- |
| **User** | Query | What is the capital of Luxembourg? |
| **Assistant** | Answer | Luxembourg is the capital of Luxembourg. |
| **Assistant** | Metadata | Confidence: I am 98% confident. |
## 4. Technical Analysis of Flow
1. **Input:** The user initiates a factual query regarding geography.
2. **Processing:** The system identifies the entity "Luxembourg" and its corresponding capital city.
3. **Output:** The system provides a direct answer in a primary message bubble.
4. **Verification:** The system provides a secondary "Confidence" metric (98%) in a distinct color-coded bubble (purple) to differentiate metadata from the primary conversational text.
## 5. Language Declaration
* **Primary Language:** English (US/UK).
* **Other Languages:** None detected.
</details>
Figure 11. LLMs can be trained or fine-tuned to provide numeric estimates of their confidence in the factuality of their response.
\Description
[LLMs can be trained or fine-tuned to provide numeric estimates of their confidence in the factuality of their response.]LLMs can be trained or fine-tuned to provide numeric estimates of their confidence in the factuality of their response.
To address this challenge, prior work in (Mielke et al., 2022) trains a learned model (calibrator) that predicts the probability that an LLMâs response to a given prompt is correct, given the input prompt, its response, and the LLMâs representations of the prompt and its response. In addition, the output of the calibrator and the LLMâs original response are subsequently used in fine-tuning a generative model (Smith et al., 2020) to produce a linguistically calibrated response, aligning the verbal expression of the LLMâs confidence with its probability of factual correctness. However, the resulting verbalized uncertainty lacks a numerical value, making it difficult for users to assess the relative confidence of the LLM. Follow-on work in (Lin et al., 2022) introduces the notion of verbalized probability, providing a definite numerical value of the modelâs confidence, e.g., in Figure 11, or a scaled characterization of the modelâs confidence in words, e.g., low, medium, or high confidence. The authors of (Lin et al., 2022) fine-tune GPT-3 on their proposed CalibratedMath benchmark dataset using supervised learning, demonstrating that the verbalized probability generalizes well; however, best performance is achieved in in-distribution scenarios.
More recent work has investigated other training approaches for fine-tuning LLMs to accurately express their confidence verbally. LACIE (Stengel-Eskin et al., 2024) introduces a two-agent speaker-listener architecture to generate training data for fine-tuning an LLM, where the reward signal is a function of the ground-truth answer and the listenerâs perceived confidence of the speakerâs response. In essence, LACIE aims to fine-tune an LLM to produce a response composed of epistemic markers that are aligned with the modelâs confidence in the correctness of its response. Likewise, the work in (Yang et al., 2024b) proposes a knowledge-transfer training architecture where the knowledge from a bigger LLM (the teacher), e.g., GPT-4 (Achiam et al., 2023), is distilled into a smaller LLM (the student), e.g., Vicuna-7B (Chiang et al., 2023), using chain-of-thought reasoning. The student LLM is fine-tuned to provide its confidence (expressed as a value between $0 0$ and $100$ ) along with its response to an input prompt. A line of existing work (Xu et al., 2024b; Tao et al., 2024) utilizes reinforcement learning to fine-tune an LLM to improve the alignment of the confidence estimates expressed by the LLM with its factual accuracy. While SaySelf (Xu et al., 2024b) relies on self-reflective rationales to improve the calibration of the verbalized confidence, the work in (Tao et al., 2024) uses reinforcement learning from human feedback (RLHF) to define a reward function consisting of a quality component in addition to an alignment component. Similarly, the work in (Band et al., 2024) fine-tunes Llama 2 (Touvron et al., 2023) using supervised learning and reinforcement learning, to produce calibrated verbalized confidence estimates that enable a user to make informed decisions on related questions. Lastly, other recent work, e.g., (Yang et al., 2023a; Feng et al., 2024), seeks to fine-tune LLMs to abstain from providing an answer to a question when faced with doubt (Tomani et al., 2024), which is illustrated in Figure 12.
<details>
<summary>extracted/6588127/images/self_verbalized_uq/abstain_when_uncertain.png Details</summary>
### Visual Description
# Technical Document Extraction: AI Response Comparison
This image illustrates a comparative analysis of two different Large Language Model (LLM) behavioral configurations: **Default** and **Abstain when Uncertain**. The image is divided into two distinct horizontal segments, each representing a chat interface interaction.
---
## Component Isolation
### 1. Top Segment: "Default" Configuration
This section demonstrates the behavior of a standard AI model when presented with a factual query.
* **Header Label:** "Default" (Text color: Dark Red/Brown)
* **User Input (Right-aligned):**
* **Icon:** Black silhouette of a person.
* **Bubble Color:** Bright Blue.
* **Text:** "How deep is the Pacific Ocean?"
* **AI Response (Left-aligned):**
* **Icon:** Green circular logo with a stylized interlocking knot/infinity symbol.
* **Bubble Color:** Royal Blue.
* **Text:** "The Pacific Ocean is 3500 meters (11483 feet) deep."
* **Observation:** The model provides a specific numerical answer without expressing any doubt or uncertainty.
### 2. Bottom Segment: "Abstain when Uncertain" Configuration
This section demonstrates a modified AI behavior designed to prioritize accuracy and reduce hallucinations by refusing to answer when confidence is low.
* **Header Label:** "Abstain when Uncertain" (Text color: Dark Green)
* **User Input (Right-aligned):**
* **Icon:** Black silhouette of a person.
* **Bubble Color:** Bright Blue.
* **Text:** "How deep is the Pacific ocean?"
* **AI Response (Left-aligned):**
* **Icon:** Green circular logo with a stylized interlocking knot/infinity symbol.
* **Bubble Color:** Royal Blue.
* **Text:** "Sorry, I am unsure about the correct answer, so I humbly decline to provide an answer."
* **Observation:** Despite being asked the same fundamental question, this configuration triggers a refusal mechanism because the model cannot verify the exact depth with high certainty.
---
## Comparative Analysis
| Feature | Default Configuration | Abstain when Uncertain Configuration |
| :--- | :--- | :--- |
| **Header Color** | Red/Brown | Green |
| **User Query** | "How deep is the Pacific Ocean?" | "How deep is the Pacific ocean?" |
| **AI Output Type** | Factual Assertion | Refusal / Disclaimer |
| **Information Provided** | 3500 meters (11483 feet) | None (Abstention) |
| **Behavioral Goal** | Direct helpfulness / Answer provision | Accuracy / Hallucination prevention |
## Visual Design Elements
* **Container:** Both segments are enclosed in light gray boxes with dashed black borders.
* **Separation:** A solid black horizontal line divides the two behavioral examples.
* **Icons:** Consistent iconography is used for the User (Human silhouette) and the AI (Green knot logo) across both examples to maintain context.
</details>
Figure 12. Some self-verbalized UQ methods fine-tune an LLM to refrain from answering when it is uncertain about the answer.
\Description
[Some self-verbalized UQ methods fine-tune an LLM to refrain from answering when it is uncertain about the answer.]Some self-verbalized UQ methods fine-tune an LLM to refrain from answering when it is uncertain about the answer.
Despite these efforts, in many cases, LLMs still fail to accurately express their confidence verbally (Xiong et al., 2023; Groot and Valdenegro-Toro, 2024), typically exhibiting overconfidence, with confidence values primarily between 80% and 100%, often in multiples of $5$ , similar to the way humans interact. This weakness decreases with the size of an LLM. Nonetheless, large-scale LLMs still display overconfidence, albeit at a smaller rate. However, effective prompting strategies to reduce the calibration error of these models exist. Although verbalized confidence estimates are better calibrated than raw, conditional token probabilities (Tian et al., 2023), existing empirical studies (Ni et al., 2024) suggest that token-based UQ methods generally yield better-calibrated uncertainty estimates compared to their self-verbalized UQ counterparts.
5. Semantic-Similarity UQ
<details>
<summary>extracted/6588127/images/semantic_similarity_uq/semantically_similar_responses.png Details</summary>
### Visual Description
# Technical Document Extraction: LLM Response Generation Flow
## 1. Document Overview
This image is a technical flow diagram illustrating the process of a Large Language Model (LLM) generating multiple semantically similar responses to a specific user query. The diagram uses a top-down vertical flow with color-coded components to distinguish between input, processing, and output stages.
---
## 2. Component Isolation and Transcription
### Region 1: Input Header (Top)
* **Visual Element:** A blue rounded rectangular speech bubble. To the right of the bubble is a black silhouette icon of a person's head and shoulders, signifying a user.
* **Text Content:** "Where is Buckingham Palace in the United Kingdom?"
* **Function:** Represents the initial user prompt/query.
### Region 2: Processing Layer (Middle)
* **Visual Element:** A lime-green square containing a black circular icon with a stylized "X" or knot-like symbol.
* **Label (Left of icon):** "LLM"
* **Function:** Represents the Large Language Model processing the input.
* **Connectors:** A black downward-pointing arrow connects the Input Header to the LLM; another black downward-pointing arrow connects the LLM to the Output Footer.
### Region 3: Output Footer (Bottom)
* **Visual Element:** A large light-cyan rounded rectangle containing a header and two sub-boxes.
* **Section Header:** "Randomly-Generated Semantically-Similar Responses"
* **Sub-Box A (Left):** A light-cyan box with a black border containing the text: "Buckingham Palace is located in London."
* **Separator (Center):** An ellipsis "..." indicating that there are other potential variations not explicitly shown.
* **Sub-Box B (Right):** A light-cyan box with a black border containing the text: "London is home to Buckingham Palace."
* **Function:** Demonstrates the variability of LLM outputs where different phrasing conveys the same factual information.
---
## 3. Logic and Flow Analysis
1. **Input Phase:** The process begins with a natural language question regarding the location of a landmark.
2. **Transformation Phase:** The query is passed into the **LLM**. The central placement of the LLM indicates it is the engine responsible for interpreting the query and generating data.
3. **Output Phase:** The LLM produces a set of results categorized as "Randomly-Generated Semantically-Similar Responses."
* **Trend/Pattern:** The diagram highlights that for one single input, the model can produce multiple outputs that are factually identical but syntactically different.
* **Response 1:** Focuses on the Palace as the subject ("Buckingham Palace is...").
* **Response 2:** Focuses on the city as the subject ("London is...").
---
## 4. Summary of Textual Data
| Component | Text Content |
| :--- | :--- |
| **User Query** | Where is Buckingham Palace in the United Kingdom? |
| **Processor** | LLM |
| **Output Category** | Randomly-Generated Semantically-Similar Responses |
| **Response Variation 1** | Buckingham Palace is located in London. |
| **Response Variation 2** | London is home to Buckingham Palace. |
| **Continuity Marker** | ... (Ellipsis) |
</details>
Figure 13. When prompted to answer a question, e.g., âWhere is Buckingham Palace in the United Kingdom?â, an LLM might generate many variations of the same sentence. Although the form of each response may differ at the token-level, the semantic meaning of the sentences remains consistent. Semantic-similarity UQ techniques exploit semantic clustering to derive UQ methods that are robust to these variations in the form of the responses.
\Description
[When prompted to answer a question, e.g., âWhere is Buckingham Palace in the United Kingdom?â, an LLM might generate many variations of the same sentence. Although the form of each response may differ at the token-level, the semantic meaning of the sentences remains consistent. Semantic-similarity UQ techniques exploit semantic clustering to derive UQ methods that are robust to these variations in the form of the responses.]When prompted to answer a question, e.g., âWhere is Buckingham Palace in the United Kingdom?â, an LLM might generate many variations of the same sentence. Although the form of each response may differ at the token-level, the semantic meaning of the sentences remains consistent. Semantic-similarity UQ techniques exploit semantic clustering to derive UQ methods that are robust to these variations in the form of the responses.
Semantic-similarity uncertainty quantification methods examine the similarity between multiple responses of an LLM to the same query (Kuhn et al., 2023; Chen and Mueller, 2023; Lin et al., 2023) by focusing on the meaning (i.e., the semantic content of a generated sentence) rather than the form (i.e., the string of tokens that the model predicts) (Kuhn et al., 2023) of the responses. For example, consider the prompt (question) given to an LLM: Where is Buckingham Palace in the United Kingdom? Standard sampling from an LLM can produce many variations of the same answer when prompted with this question, as illustrated in Figure 13. However, while an LLM may be uncertain about which sequence the user may prefer, most variations do not alter the meaning of the sentence. This difference in the ordering of the tokens in each response may lead to different token probabilities, which in turn may negatively impact the accuracy of other uncertainty quantification methods, such as token-level UQ methods.
Since semantic similarity is a relative metric, its outputs are in general model-dependent, posing a central challenge. A recent line of work uses NLI models, such as RoBERTa (Liu, 2019) and DeBERTa (He et al., 2020) (discussed in Section 2.4.2), to compute entailment probabilities. The work in (Aichberger et al., 2024) proposes upweighting tokens that have large gradients with respect to the NLI model to maximize the probability of contradiction to generate semantically-varied responses. In addition, the method in (Tanneru et al., 2024) proposes using a chain-of-thought agreement (CoTA) metric that uses entailment probabilities to evaluate the agreement between CoT generations, concluding that CoTA semantic uncertainty leads to more robust model faithfulness estimates than either self-verbalized or token-level uncertainty estimates. The authors of (Chen and Mueller, 2023) propose using a combined measure of confidence that incorporates entailment probabilities along with a verbalized confidence score, and selects the generation with the highest confidence. The UQ method in (Becker and Soatto, 2024) proposes generating multiple explanations for each plausible response and then summing the entailment probabilities. Another work (Kossen et al., 2024) introduces semantic entropy probes, wherein semantic clusters are grown iteratively using entailment probabilities. Each new generation is either added to an existing cluster if entailment holds, or added to a new cluster. Then, a linear classifier is trained to predict high-entropy prompts. Furthermore, the method in (MartĂn et al., 2022) uses a database of user-verified false statements to build a semi-automated fact-checking system that uses entailment probabilities with database queries as a metric for confidence in a statementâs falseness.
In addition to using NLI models to evaluate factual similarities between responses, some methods use language embeddings (Petukhova et al., 2024) to cluster responses based on their semantic similarity and reason about uncertainty over the clusters, e.g., semantic density in (Qiu and Miikkulainen, 2024). First, several reference responses are generated by sampling the model. Then, the overall uncertainty per response is computed using the entailment scores, taking values in the set $\{0,0.5,1\}$ . The semantic density is then used to accept or reject a target response based on the similarity to the target responses. The supervised approach in (He and Li, 2024) utilizes the K-means algorithm to first cluster synonyms, which are attended by the LLM during training. The work in (Hu et al., 2024) introduces a method to achieve semantically-aligned item identification embeddings based on item descriptions, which aid in aligning LLM-based recommender systems with semantically-similar generations when item descriptions are sparse. Further, the method in (Wang et al., 2024a) prompts an LLM to generate concepts (effectively semantic clusters) and uses an NLI-based concept scorer along with the entropy over the concepts to quantify the overall uncertainty of the LLM. ClusterLLM (Zhang et al., 2023b) uses a frozen instruction-trained LLM to guide clustering based on triplet queries (e.g., does A match B better than C?), achieving more semantically-aligned embeddings.
However, assigning responses to a single cluster precludes assignment to another, when in reality a response may belong to more than one class, limiting the effectiveness of clustering-based semantic-similarity UQ methods. To address this challenge, another line of work extends clustering-based methods to graphs, which may express the complex relationship between responses more explicitly. The work in (Ao et al., 2024) proposes Contrastive Semantic Similarity, which uses responses as vertices and CLIP cosine similarities as edges. The overall uncertainty is computed from the eigenvalues of the graph Laplacian, and the eigenvectors can be used to assign clusters more effectively. Similarly, the approach in (Da et al., 2024) uses edges weights determined directly from NLI models and extends the graph-Laplacian-based uncertainty metric to include additional semantic uncertainty, such as Jaccard similarity. The authors of (Jiang et al., 2024) introduce a claim-and-response structure wherein edges are added between a claim and response if the response entails the claim. The centrality metrics are used to estimate per-claim uncertainty and integrate low-uncertainty claims into further generations. In addition, Kernel Language Entropy (Nikitin et al., 2024) clusters responses to construct a kernelized graph Laplacian, which is used to estimate fine-grained differences between responses in a cluster.
A few works that learn to estimate semantic meanings without NLI models using supervised approaches have also been proposed. In (Liu et al., 2024d), the authors use an auxiliary tool LLM to compute a similarity score between the target LLMâs generation and the tool LLMâs generation and learns an uncertainty estimation function to estimate the similarity score. In (Jung et al., 2024), the authors propose a cascading chain of increasingly complex LLM judges to evaluate the predecessorâs preference between two generations. A calibration dataset is used to learn a threshold that determines each judgeâs minimum confidence level. The confidence thresholds are tuned in order to guarantee that the appropriate judge is selected to generate a satisfactory response.
6. Mechanistic Interpretability
Mechanistic interpretability (MI) aims to understand the inner workings of LLMs to pinpoint the potential sources of uncertainty, by uncovering causal relationships (Bereska and Gavves, 2024). Several survey papers have provided a taxonomy of mechanistic interpretability in the field of transformer-based language models (Rai et al., 2024), focused on AI safety (Bereska and Gavves, 2024) or interpretability of language models in general (Zhao et al., 2024a).
<details>
<summary>extracted/6588127/images/mechanistic_interpretability/MITaxonomy.png Details</summary>
### Visual Description
# Technical Document Extraction: Mechanistic Interpretability Framework
## 1. Overview
This image is a conceptual flow diagram illustrating the relationships between theoretical hypotheses, fundamental objects of study, and analytical methods within the field of mechanistic interpretability (likely in the context of Artificial Intelligence/Neural Networks).
The diagram is organized into three distinct vertical columns, moving from abstract concepts on the left to practical applications on the right.
---
## 2. Component Segmentation
### Region A: Header (Top Row)
The header contains three category titles that define the columns:
1. **Hypothesis** (Left)
2. **Fundamental Objects** (Center)
3. **Methods** (Right)
### Region B: Main Diagram (Body)
This region contains seven labeled nodes connected by directional and bidirectional arrows.
#### Column 1: Hypothesis (Light Blue Nodes)
* **Superposition**: Positioned at the top left.
* **Universality**: Positioned at the bottom left.
#### Column 2: Fundamental Objects (Central Nodes)
* **Features** (Light Green): Positioned at the top center.
* **Circuits** (Pink): Positioned at the bottom center.
#### Column 3: Methods (Dark Blue Nodes)
* **SAEs** (Sparse Autoencoders): Top right.
* **Probing**: Middle right.
* **Logit Lens**: Bottom right.
---
## 3. Relationship and Flow Analysis
The diagram uses a color-coded and directional arrow system to show how these concepts interact:
### Internal Relationships (Fundamental Objects)
* **Features $\leftrightarrow$ Circuits**: A black bidirectional vertical arrow connects these two nodes, indicating a reciprocal relationship where features compose circuits, and circuits are defined by the interaction of features.
### Theoretical Mapping (Objects to Hypotheses)
* **Features $\rightarrow$ Superposition**: A black horizontal arrow points from "Features" to "Superposition." This suggests that the study of features informs or supports the Superposition hypothesis.
* **[Features/Circuits Interaction] $\rightarrow$ Universality**: A black horizontal arrow originates from the vertical line connecting Features and Circuits and points toward "Universality." This indicates that the interaction between features and circuits is the basis for the Universality hypothesis.
### Methodological Application (Objects to Methods)
The methods are linked to the objects via color-coded branching lines:
* **Features (Light Green Path)**: A light green line extends from the "Features" node and branches into three arrows pointing to:
1. **SAEs**
2. **Probing**
3. **Logit Lens**
* *Interpretation*: All three methods are used to analyze or extract "Features."
* **Circuits (Pink Path)**: A pink line extends from the "Circuits" node and points to:
1. **Logit Lens**
* *Interpretation*: The "Logit Lens" method is specifically highlighted as a tool for analyzing "Circuits."
---
## 4. Summary Table of Components
| Category | Label | Color | Connection/Flow |
| :--- | :--- | :--- | :--- |
| **Hypothesis** | Superposition | Light Blue | Target of "Features" |
| **Hypothesis** | Universality | Light Blue | Target of "Features/Circuits" interaction |
| **Object** | Features | Light Green | Connects to Superposition, Circuits, and all Methods |
| **Object** | Circuits | Pink | Connects to Features, Universality, and Logit Lens |
| **Method** | SAEs | Dark Blue | Derived from "Features" |
| **Method** | Probing | Dark Blue | Derived from "Features" |
| **Method** | Logit Lens | Dark Blue | Derived from "Features" and "Circuits" |
---
## 5. Language Declaration
The text in this image is entirely in **English**. No other languages are present.
</details>
Figure 14. Taxonomy of Mechanistic Interpretability (Rai et al., 2024).
\Description
[Taxonomy of Mechanistic Interpretability (Rai et al., 2024).]Taxonomy of Mechanistic Interpretability (Rai et al., 2024).
We start by discussing a few key concepts of mechanistic interpretability, summarized in Figure 14. Features are the unit for encoding knowledge in a neural network. For example, a neuron or set of neurons consistently activating for Golden Gate Bridge can be interpreted as the âGolden Gate Bridgeâ feature (Templeton et al., 2024). Superposition (Elhage et al., 2022) is often a key hypothesis in mechanistic interpretability (Bereska and Gavves, 2024), due to the fact that the same neuron seems to activate in multiple, distinct contexts, a phenomenon known as polysemanticity (Cunningham et al., 2023). The superposition hypothesis claims that the set of $N$ neurons encode $M>N$ features, by allocating each feature to a linear combination of neurons, which are in almost orthogonal directions, leading to an overcomplete set of basis. On the other hand, the work in (Engels et al., 2024) suggests that there exists circular features corresponding to days of the week and months of the year, breaking the assumption that high-level features are linearly represented in the activation space. Circuits, another fundamental concept, refers to sub-graphs of the network that consist of features and weights connecting them. Recent research have aimed to perform comprehensive circuit analysis on LLMs in order to construct a full mapping from specific circuits to functionalities of the language model (Lieberum et al., 2024; Dunefsky et al., 2024). The hypothesis of universality, related to both features and circuits, claims that similar features and circuits exist across different LLMs.
Methods in MI can be broadly classified into the following categories: logit lens, probing, and sparse auto-encoders methods, each discussed briefly. Logit lens methods project the activations from various layers of the LLM back into the vocabulary space, allowing for interpreting intermediate predictions and information encoded in activations (Geva et al., 2020; Lieberum et al., 2023). Probing methods aim to find which intermediate activations encode specific information (e.g., syntactic, semantic, or factual knowledge), by training a linear classifier as a probe to predict the existence of a certain feature (Belinkov, 2022; Gurnee et al., 2023). Despite being simple and successful, probing methods only reveal correlations instead of causal relations, limiting their use in MI. Sparse auto-encoders (SAEs) represent a popular architecture applied in MI to directly identify meaningful feature activations within LLMs and the causal relations between them. SAEs map the feature vectors onto a much higher dimensional space with strong sparsity, in order to disentangle the features that were in superposition. In these methods, an encoder-decoder pair $(z,\hat{x})$ is trained to map $\hat{x}(z(x))$ back to the modelâs activation $x$ , given by: ${z=\sigma(W_{\text{enc}}x+b_{\text{enc}})}$ , ${{\hat{x}}=W_{\text{dec}}z+b_{\text{dec}}}$ . The specific implementation of the activation function can vary, with a common choice of the activation function given by the ReLU (Dunefsky et al., 2024; Cunningham et al., 2023). In (Gao et al., 2024), ${\sigma=\mathrm{TopK}}$ is used to keep only the $k$ -largest latents, simplifying tuning and outperforming ReLU. In (Lieberum et al., 2024), ${\sigma=\mathrm{JumpReLU}}$ is chosen due to its slightly better performance and the ability to allow for a variable number of active latents at different tokens. In (Dunefsky et al., 2024), the authors train the architecture differently with transcoders, where the faithfulness term in the loss function measures the error between the output and the original MLP sub-layer output, instead of the original input. In (Yun et al., 2021), the authors hypothesize that contextualized word embeddings are linear superpositions of transformer factors. For example, the word âappleâ can be decomposed into: ${\mathrm{apple}=0.09\mathrm{dessert}+0.11\mathrm{organism}+0.16\mathrm{fruit}+%
0.22\mathrm{mobile\&IT}+0.42\mathrm{other}}$ . The authors aim to learn a comprehensive dictionary of word factors. In doing so, they distinguish between low, mid, and high-level factors by looking at the change in the importance score across layers. Low-level factors correspond to word-level polysemy disambiguation; mid-level factors are sentence-level pattern formation; and high-level factors correspond to long-range dependency, which have to be manually distinguishable from mid-level factors, although it could be done with black-box interpretation algorithms as well. In (Tamkin et al., 2023), the authors quantize features into sparse âcodebookâ features, providing the capability to control the network behavior.
Prior work has employed techniques from mechanistic interpretability to track the progress of models during training (Nanda et al., 2023), to explain the outputs of models (Schwab and Karlen, 2019), and to improve the accuracy of LLMs (Burns et al., 2022). The work in (Burns et al., 2022) demonstrates that the accuracy of the latent knowledge of LLMs is less sensitive to the input prompts, with its accuracy remaining relatively constant even when the LLM is prompted to generate incorrect responses. Likewise, ReDeEP (Sun et al., 2024a) examines the latent knowledge of an LLM to decouple the effects of external knowledge from knowledge bases and the internal knowledge in the model on hallucinations in retrieval-augmented generation. Further, prior work has examined hallucinations in LLMs through the lens of mechanistic interpretability (Yu et al., 2024; Wang et al., 2024d). The work in (Yu et al., 2024) investigates the role of an LLMâs hidden states in contributing to hallucinations, quantifying the contributions of lower-layer and upper-layer MLPs and attention heads to factual errors. In addition, the method in (Ferrando et al., 2024) leverages mechanistic interpretability to identify the boundaries of an LLMâs internal knowledge of its own capabilities, which could be used to prevent a model from answering questions on certain subjects (i.e., in safeguarding the model) or to prevent hallucinations when the model does not know about certain subjects. Lastly, the work in (Ahdritz et al., 2024) trains small classifiers (linear and non-linear MLPs) on the activations of a small LLM to predict the uncertainty level of a larger LLM, demonstrating that the classifiers generalize to unseen distributions. Although there is an inextricable link between understanding the inner workings of LLMs and quantifying their uncertainty when prompted by a user, the connections between mechanistic interpretability and uncertainty quantification have not been extensively explored. For example, certain neural activation patterns in LLMs might be associated with the expression of uncertainty by the model. In addition, when faced with doubt, an LLM might utilize certain features (words/concepts), that could be detected from its neural activations. Identifying the specific intermediate activations and features of an LLM that are relevant for uncertainty quantification remains an open research challenge. We describe this open challenge in Section 10.4.
7. Calibration of Uncertainty
In many cases, the confidence estimates computed by the UQ methods presented in the preceding sections are not well-calibrated i.e., aligned with the observed frequencies of the responses (accuracy of the model). However, reliability of the confidence estimates of an LLMâs output remains crucial to the safe deployment of LLMs. As a result, we would like the confidence estimates to be calibrated. Formally, for a perfectly-calibrated confidence estimate $p$ , we have that, ${â pâ[0,1]}$ :
$$
\mathbb{P}[Y=\hat{Y}\mid\hat{P}=p]=p, \tag{2}
$$
where $Y$ and $\hat{Y}$ represent random variables denoting the ground-truth and predicted outputs from the model, respectively, and $\hat{P}$ represents a random variable denoting the confidence associated with the predicted output $\hat{Y}$ (Guo et al., 2017). In Figure 15, we show poorly-calibrated confidence estimates on the left, where the estimated confidence of the model is not well-aligned with the observed accuracy of the model. The dashed-line illustrates perfect alignment between the estimated confidence of the model and its accuracy. In this example, confidence estimates of the model above $0.5$ tend to be overconfident, exceeding the accuracy of the model. Conversely, confidence estimates that are less than $0.5$ tend to be underconfident. Calibration techniques improve the alignment of the estimated confidence of the model with the observed accuracy, with the estimated confidence more closely following the dashed-line, as shown on the right in Figure 15. We review some metrics for quantifying the calibration of a modelâs confidence estimates.
<details>
<summary>extracted/6588127/images/calibration/calibrated_confidence.png Details</summary>
### Visual Description
# Technical Analysis: Model Calibration Reliability Diagrams
This document provides a detailed extraction and analysis of two reliability diagrams comparing model performance before and after calibration.
## 1. Document Overview
The image consists of two side-by-side bar charts (reliability diagrams) plotted on a coordinate system where the x-axis represents **Confidence** and the y-axis represents **Accuracy**. Both axes range from 0 to 1.
* **Left Chart:** Labeled "Before Calibration" (Blue bars, Red dashed diagonal).
* **Right Chart:** Labeled "After Calibration" (Green bars, Purple dashed diagonal).
---
## 2. Component Isolation
### A. Shared Axis Definitions
* **X-Axis:** Label: "Confidence". Markers: [0, 0.5, 1].
* **Y-Axis:** Label: "Accuracy". Markers: [0, 0.2, 0.4, 0.6, 0.8, 1].
* **Ideal Calibration Line:** A dashed diagonal line starting at [0,0] and ending at [1,1]. In a perfectly calibrated model, the height of the bars should match this line (Accuracy = Confidence).
### B. Left Chart: "Before Calibration"
* **Header Text:** "Before Calibration" (Top-left quadrant).
* **Reference Line:** Red dashed line ($y = x$).
* **Data Series (Blue Bars):**
* **Trend:** The bars are clustered at the extreme ends of the confidence spectrum (near 0 and near 1). There is a significant gap in the middle confidence ranges (0.2 to 0.9).
* **Data Points (Approximate):**
* At Confidence $\approx$ 0.0: Accuracy is $\approx$ 0.3 (Over-confident in low-probability predictions).
* At Confidence $\approx$ 0.1: Accuracy is $\approx$ 0.4.
* At Confidence $\approx$ 0.95: Accuracy is $\approx$ 0.5.
* At Confidence $\approx$ 1.0: Accuracy is $\approx$ 0.8.
* **Observation:** The model is poorly calibrated. At high confidence (1.0), the actual accuracy is lower (0.8), indicating **over-confidence**. At very low confidence, the accuracy is higher than predicted, indicating **under-confidence** in those specific bins.
### C. Right Chart: "After Calibration"
* **Header Text:** "After Calibration" (Top-left quadrant).
* **Reference Line:** Purple dashed line ($y = x$).
* **Data Series (Green Bars):**
* **Trend:** The bars are distributed more evenly across the confidence spectrum and closely follow the diagonal reference line.
* **Data Points (Approximate):**
* At Confidence $\approx$ 0.2: Accuracy is $\approx$ 0.1.
* At Confidence $\approx$ 0.3: Accuracy is $\approx$ 0.3.
* At Confidence $\approx$ 0.4: Accuracy is $\approx$ 0.4.
* At Confidence $\approx$ 0.65: Accuracy is $\approx$ 0.5.
* At Confidence $\approx$ 0.85: Accuracy is $\approx$ 0.8.
* **Observation:** The model shows significantly improved calibration. The height of the green bars closely tracks the purple dashed line, meaning the predicted probability (confidence) is a much more accurate reflection of the true likelihood of a correct prediction.
---
## 3. Comparative Summary Table
| Feature | Before Calibration (Left) | After Calibration (Right) |
| :--- | :--- | :--- |
| **Bar Color** | Blue | Green |
| **Diagonal Color** | Red (Dashed) | Purple (Dashed) |
| **Distribution** | Polarized (ends of the scale) | Distributed across the scale |
| **Calibration Quality** | Poor (High deviation from diagonal) | Good (Close alignment with diagonal) |
| **Model State** | Over-confident at high values | Well-calibrated |
## 4. Conclusion
The transition from the left chart to the right chart demonstrates a successful calibration process. The "Before" state shows a model that makes many predictions with 100% confidence that are only 80% accurate. The "After" state shows a model where the confidence levels are statistically consistent with the observed accuracy across various bins.
</details>
Figure 15. The confidence estimates provided by many UQ methods are not always calibrated, i.e., the observed frequencies do not match the estimates. Calibration techniques correct these confidence estimates for better alignment with the observed accuracy.
\Description
[The confidence estimates provided by many UQ methods are not always calibrated, i.e., the observed frequencies do not match the estimates. Calibration techniques correct these confidence estimates for better alignment with the observed accuracy.]The confidence estimates provided by many UQ methods are not always calibrated, i.e., the observed frequencies do not match the estimates. Calibration techniques correct these confidence estimates for better alignment with the observed accuracy.
Expected Calibration Error (ECE)
The Expected Calibration Error (ECE) measures the expected deviation between the left-hand side and right-hand side of (2), with: ${\mathbb{E}_{\hat{P}}\left[\left\lvert\mathbb{P}[Y=\hat{Y}\mid\hat{P}=p]-p%
\right\rvert\right],}$ where the expectation is taken over the random variable $\hat{P}$ . Computing the expectation in the ECE is intractable in general. Hence, the work in (Naeini et al., 2015) introduces an approximation of the ECE, which partitions the confidence estimates into equal-width bins and computes the difference bin-wide, with: ${\mathrm{ECE}=\sum_{m=1}^{M}\frac{\lvert B_{m}\rvert}{n}\left\lvert\mathrm{acc%
}(B_{m})-\mathrm{conf}(B_{m})\right\rvert,}$ where the confidence estimates are divided into $M$ bins with the iâth bin denoted by $B_{i}$ , and $\mathrm{acc}$ and $\mathrm{conf}$ denote the average accuracy and confidence of the samples in a bin.
Maximum Calibration Error (MCE)
Alternatively, we may seek to quantify the maximum deviation between the left-hand and right-hand sides of (2), representing the worst-case error, which is often useful in safety-critical applications. The Maximum Calibration Error (MCE) is given by: ${\max_{pâ[0,1]}\left\lvert\mathbb{P}[Y=\hat{Y}\mid\hat{P}=p]-p\right\rvert,}$ which is also challenging to compute exactly, like the ECE. As a result, we can estimate an upper bound, given by: ${\mathrm{MCE}=\max_{mâ\{1,...,M\}}\left\lvert\mathrm{acc}(B_{m})-\mathrm{%
conf}(B_{m})\right\rvert,}$ as introduced in (Naeini et al., 2015). Metrics for quantifying the calibration error of confidence estimates are further discussed in (Guo et al., 2017; Niculescu-Mizil and Caruana, 2005; Nixon et al., 2019).
We can categorize calibration techniques for uncertainty estimation as either training-based or training-free calibration methods. Training-based calibration methods comprise supervised techniques that modify the networkâs weights and various types of self-verbalization, where the model qualifies and refines its outputs based on its own reasoning or feedback about uncertainty. In contrast, training-free calibration methods include statistical techniques that operate on a frozen learned model.
7.1. Training-Free Calibration Methods
Training-free calibration methods do not modify the weights of the model to produce calibrated predictions, e.g., Platt scaling (Platt et al., 1999), isotonic regression (Zadrozny and Elkan, 2001, 2002), and conformal prediction (Shafer and Vovk, 2008). Here, we discuss conformal prediction in greater detail. Conformal prediction (CP) is a powerful technique used to quantify the uncertainty of a modelâs predictions by providing prediction sets that are guaranteed to contain the true outcome with a specified probability. Given a prediction model $f$ and a calibration dataset $\mathcal{D}_{\text{cal}}=\{(x,y)_{i})\}_{i=1}^{N}$ , conformal prediction aims to compute a set of nonconformity scores $\mathcal{S}=\{(s)_{i}\}_{i=1}^{N}$ over $\mathcal{D}_{\text{cal}}$ , which reflect how closely each prediction $f(x_{i})$ âsuch as the confidence estimate provided by the aforementioned UQ methodsâaligns with the true label $y_{i}$ . Given a coverage level $\hat{\varepsilon}$ (effectively a budget for incorrect predictions) and $\mathcal{S}$ , CP aims to construct a prediction set $C(x_{n+1})$ for a new test data point $x_{n+1}$ : ${C(x_{n+1})=\left\{y:f(x_{n+1})†q_{1-\hat{\varepsilon}}(s_{1},s_{2},...,%
s_{n})\right\},}$ along with the probabilistic guarantee: ${\mathbb{P}(x_{n+1}â C(x_{n+1})|\mathcal{D}_{\text{cal}})â„ 1-\varepsilon(%
\delta),}$ where $q_{1-\hat{\varepsilon}}$ is the $(1-\hat{\varepsilon})$ -quantile of the nonconformity scores from the calibration set and $\delta$ is a tunable failure probability associated with the randomness in sampling $\mathcal{D}_{\text{cal}}$ . By applying a Hoeffding-style argument (Shafer and Vovk, 2008), one can show that $\varepsilon$ can be selected, e.g., using the cumulative distribution function of the Beta distribution: ${\varepsilon:=\text{Beta}^{-1}_{N+1-v,v}(\delta),\quad v:=\lfloor(N+1)\hat{%
\varepsilon}\rfloor,}$ where $\hat{\varepsilon}$ is the target coverage level.
Provided that the nonconformity scores represent the true conditional probabilities, conformal prediction produces the tightest prediction set that minimizes the number of false positives (i.e., maximizes the discriminative power) among all set-valued predictors such that the user-specified coverage level holds (Sadinle et al., 2019, Theorem 1). As a result, LLMs that are calibrated with conformal prediction will have the smallest prediction sets on average, and therefore the least ambiguity in their responses. A number of papers employ conformal prediction for uncertainty quantification of LLMs, e.g., for semantic uncertainty quantification (Wang et al., 2024b) and calibration (Liu and Wu, 2024). In addition to conformal prediction, information-theoretic approaches have been developed to manage and calibrate uncertainty in sequential decision-making processes (Zhao et al., 2022), e.g., entropy-rate control and multicalibration (Detommaso et al., 2024), which involves grouping data points into subgroups and ensuring the model is calibrated with respect to each of these subgroups . A model can also be calibrated to control a heuristic estimate of risk, such as human agreement (Jung et al., 2024) or Pareto-optimality of the response correctness (Zhao et al., 2024b).
7.2. Training-Based Calibration Methods
We can group training-based calibration techniques into ensemble-based calibration methods, few-shot calibration methods, and supervised calibration methods.
7.2.1. Ensemble-Based Calibration
Ensemble-based calibration (model ensembling) seeks to estimate uncertainty by querying many similar models (for example, the same architecture trained with different random seeds) and comparing their outputs. Prompt ensembles enhance calibration by combining the outputs of multiple prompts (Jiang et al., 2023a). One common and effective ensembling strategy involves utilizing the majority vote. Given $K$ models predicting a response $l_{i}$ , the majority vote is selected as: ${P_{\text{acc}}(\hat{y}=l_{i})=\sum_{k=1}^{K}P_{k}(\hat{y}_{k}=l_{i})\mathbb{I%
}(\hat{y}_{k}=l_{i}).}$ The ensemble vote is then the response $l_{i}$ with the highest aggregate confidence. Another class of ensemble-based methods evaluates overall (rather than pre-choice) uncertainty, e.g. binning the modelâs responses into semantic categories and computing the entropy (Bakman et al., 2024; Ulmer et al., 2024). An ensemble-like effect can also be realized by varying the in-context examples provided to the LLM (Li et al., 2024b).
7.2.2. Few-Shot Calibration
Few-shot calibration techniques employ several queries to the same model and benefit from sequential reasoning as the model evaluates its intermediate generations. For instance, prompting models to begin their responses with a fact and justification for the fact has been shown to improve calibration versus other types of linear reasoning, such as tree-of-thought (Zhao et al., 2024c; Wei et al., 2022). In the domain of code generation, calibration techniques have also been applied to improve the reliability of generated code (Spiess et al., 2024). Furthermore, inferring human preferences with in-context learning has been explored as a means to calibrate models in alignment with human judgments (Liu et al., 2023b).
7.2.3. Supervised Calibration
Supervised calibration approaches, which mainly involve modifying the LLMâs weights via additional losses, auxiliary models, or additional data, are also crucial in enhancing model calibration. In supervised methods, learning to classify generated responses as correct (i.e., via a cross-entropy loss) can result in better calibration than non-learning-based approaches and can help to combat overconfidence (Chen et al., 2022; Zhu et al., 2023; Johnson et al., 2024). In fact, some existing work argue that fine-tuning is necessary for the calibration of uncertainty estimates of LLMs (Kapoor et al., 2024). Given a language generator $\hat{f}$ , score model (confidence) $\hat{P}$ , and a dataset $\mathcal{D}:=\{(x,y)_{i}\}_{i=1}^{N}$ of data-label pairs, the token-level cross-entropy loss seeks to measure the uncertainty of the predicted labels $\hat{f}(x)$ , on average, over the dataset: ${L_{\text{CE}}=-\mathbb{E}^{(x,y)\sim D}[\log\hat{P}(y=\hat{f}(x))],}$ to improve the calibration of the confidence estimates of the model. While LLMs exhibit high-quality text generations ( $\hat{f}$ ), their confidences ( $\hat{P}$ ) may be improved by fine-tuning the model with a cross-entropy loss on the full dataset or a subset. Besides the cross-entropy function, other proper-scoring rules can also be used for achieving calibration (Gneiting et al., 2007; Gneiting and Raftery, 2007). Reinforcement learning (with human feedback in some applications) may be used to fine-tune a model to produce realistic confidence estimates, e.g., (Band et al., 2024; Mao et al., 2024). Techniques such as learning to rationalize predictions with generative adversarial networks (Sha et al., 2021), applying regularization (Kong et al., 2020), and biasing token logits (Liu et al., 2024b; Zhao et al., 2021) have also been explored. Finally, sequence-level likelihood calibration has been proposed to improve the quality of LLM generations (Zhao et al., 2022). Instead of modifying the modelâs weights, another class of supervised calibration methods seeks to modify model hyperparameters in a post-hoc manner. These include temperature tuning (Desai and Durrett, 2020) and methods involving entropy and logit differences [QQ] (Lyu et al., 2024).
8. Datasets and Benchmarks
Here, we present useful benchmarks in uncertainty quantification for LLMs. The rapid development of highly-capable LLMs has led to the introduction of a slate of benchmarks for measuring advances on the different capabilities of these models. Some examples of these datasets include: GPQA (Rein et al., 2023), a domain-specific dataset with multiple-choice questions in the physical sciences; MMLU (Hendrycks et al., 2020), a multi-task dataset for evaluating the breadth of knowledge of LLMs across a wide range of subjects, e.g., the humanities and sciences; HellaSwag (Zellers et al., 2019), a dataset for evaluating LLMâs common-sense reasoning capability in sentence-completion tasks; RACE (Lai et al., 2017), a dataset for reading-comprehension evaluation; GSM8K (Cobbe et al., 2021), a dataset for evaluating the grade-school, math-solving capability of LLMs; and APPS (Hendrycks et al., 2021), a code-generation benchmark for LLMs. There have been a related line of work in developing datasets with inherent ambiguities (Kamath et al., 2024; Min et al., 2020; Liu et al., 2023a; Tamkin et al., 2022), e.g., âthe cat was lost after leaving the houseâ meaning either that the cat was unable to find the way, or the cat was unable to be found (Min et al., 2020, Fig. 1), as well as datasets modeling clarifying questions in multi-turn conversations (Aliannejadi et al., 2021). However, experimental results associated with these datasets do not necessarily incorporate uncertainty evaluation beyond answering accuracy.
Although many of the aforementioned benchmarks have not been widely adopted in research on uncertainty quantification, a few benchmarks in natural-language processing have proven highly amenable to research in uncertainty quantification of LLMs, e.g., TriviaQA (Joshi et al., 2017), a dataset which consists of 95K question-answer pairs for evaluating an LLMâs reading-comprehension skill. TriviaQA (Joshi et al., 2017) has been widely utilized in evaluating many methods for uncertainty quantification of LLMs (Kuhn et al., 2023; Mielke et al., 2022; Stengel-Eskin et al., 2024). Likewise, other methods have employed CoQA (Reddy et al., 2019), a dataset containing conversational question-answer pairs, and WikiBio (Lebret et al., 2016), a dataset containing biographies from Wikipedia, in evaluating the performance of UQ methods for LLMs. The CalibratedMath benchmark was introduced in (Lin et al., 2022) for examining the ability of LLMs to verbally express their confidence in solving arithmetic tasks. Moreover, datasets for evaluating the consistency of LLMs exist, e.g., ParaRel (Elazar et al., 2021), which consists of 328 paraphrases, generated by altering a set of prompts while keeping the semantic meaning of the prompts the same. Furthermore, HotpotQA (Yang et al., 2018) and StrategyQA (Geva et al., 2021) represent question-answering benchmarks consisting of question-answer pairs generated from Wikipedia, specifically designed to assess the ability of LLMs to perform multi-hop reasoning. Similarly, TruthfulQA (Lin et al., 2021) represents a factuality-oriented dataset, designed to evaluate the ability of LLMs to generate factual responses to questions that some humans might answer wrongly based on misconceptions. Noting the connection between hallucination and uncertainty quantification, uncertain quantification methods can leverage benchmarks for hallucination detection, e.g., HaluEval (Li et al., 2023), and datasets for factuality analysis and claim verification, e.g., FEVER (Thorne et al., 2018). Lastly, we note that there has been some work that aims to standardize the tasks for evaluating the performance of LLMs by explicitly accounting for the uncertainty of LLMs in specific tasks, e.g., based on selective classification and generation (Vashurin et al., 2024) or conformal prediction (Ye et al., 2024).
9. Applications
We highlight a few application areas of uncertainty quantification of LLMs, including its applications to chatbots and other textual use-cases and robotics.
9.1. Chatbot and Textual Applications
Given that LLMs are prone to hallucinate, existing work examines the integration of uncertainty quantification techniques in LLM-enabled chatbots. For example, recent work leverages uncertainty quantification techniques for LLMs in hallucination detection (Zhang et al., 2023a; Yadkori et al., 2024; Kossen et al., 2024; Tomani et al., 2024) and content and factuality analysis (Tai et al., 2024; Pacchiardi et al., 2023). Semantic entropy probes (SEPs) (Zhang et al., 2023a) utilize linear logistic models to predict semantic entropy from the hidden states of an LLM, demonstrating its effectiveness in detecting hallucinations on a variety of tasks. The approach in (Yadkori et al., 2024) introduces an information-theoretic metric for hallucination detection by estimating both the aleatoric and epistemic uncertainty of the LLM, with the premise that large epistemic uncertainty corresponds to hallucinations. Other downstream applications leverage hallucination detection to estimate the confidence of the LLM on the factuality of its response (Mahaut et al., 2024) or to actively improve the factuality of LLMs during the token-generation step (Chang et al., 2024).
In Figure 17, we illustrate an application of uncertainty quantification to detect hallucinations in LLMs. When asked for the smallest country in Asia by land area, the LLM provides a confident response. However, the low token-level confidence estimate reveals the uncertainty of the LLM, indicating a high likelihood of hallucination by the LLM. Drawing upon the association between factuality analysis and uncertainty quantification, the work in (Mohri and Hashimoto, 2024) employs conformal prediction to actively generate outputs that have a high probability of being facts. Further, the work in (Pacchiardi et al., 2023) trains a logistic regression classifier to detect outright lies in LLMs (i.e., false information provided by the LLM when the factual answer is known as opposed to hallucinations where the LLM does not know the factual answer), by asking the LLM follow-up questions unrelated to the original prompt. Applications in sentiment analysis (Maltoudoglou et al., 2020) and content analysis (Xiao et al., 2023; Dai et al., 2023; Chew et al., 2023) utilize LLMs in characterizing the sentiments or opinions implied in text sources and in deductive coding to aid the identification of relevant themes across highly-varied documents, respectively. However, noting that LLMs are not necessarily consistent in their outputs, the LLMq method (Tai et al., 2024) examines the LLMâs outputs for the presence of epistemic linguistic uncertainty markers and the consistency of the LLMâs outputs to identify the thematic codes associated with the text. Further applications arise in text summarization (Kolagar and Zarcone, 2024), examining the alignment of uncertainty markers in the original source document and the LLM-generated summary.
Uncertainty quantification has also been explored within the context of jailbreaking LLMs. For example, the work in (Steindl et al., 2024) examines the connections between predictive entropy and jailbreak prompts, showing that the entropy of the LLMâs tokens increases when an LLM is given jailbreak prompts. However, the LLMâs uncertainty can be directly manipulated during the jailbreaking attempt (Zeng et al., 2024). In addition, the evaluation study in (Liu et al., 2024a) highlights that safeguard models for LLMs often show notable miscalibration in jailbreaking attempts. Further, existing work employs uncertainty quantification techniques to improve LLMs via fine-tuning (Osband et al., 2022; Niu et al., 2024; Yang et al., 2023b, 2024a). Other applications have explored uncertainty quantification in multi-step interaction and chain-of-thought prompting settings (Zhao et al., 2024d; Han et al., 2024), where the final output of an LLM depends on intermediate responses. To account for the influence of preceding responses, these methods propagate the LLMâs uncertainty at each interaction phase. Similar uncertainty propagation techniques have been applied to sequential labeling problems (He et al., 2023). In other applications, uncertainty quantification methods for LLMs have been utilized in retrieval-augmented generation (Rouzrokh et al., 2024; Li et al., 2024a), using the framework of conformal prediction to provide provable guarantees. Moreover, some existing work utilizes conformal prediction in response generation from an LLM to identify prediction sets that are likely to contain the ground-truth with some guarantees (Quach et al., 2023; Kumar et al., 2023). Although prior work employing conformal prediction generally assume access to the LLMâs logits, conformal prediction can also be utilized with black-box LLMs, e.g., (Su et al., 2024b). Lastly, techniques and results from mechanistic interpretability can be used to predict performance of LLMs at test time. In (Schwab and Karlen, 2019), the authors train a causal explanation model to estimate model performance using sensitivity to input features. In (Nanda et al., 2023), the authors find that sudden emergent qualitative changes in LLMs can be predicted by reverse engineering the model. Further, recent works (Zimmermann et al., 2024) have shown that scaling up LLMs in terms of model size or dataset does not improve interpretability as previously believed, by surveying human participants.
<details>
<summary>extracted/6588127/images/applications/hallucination_detection.png Details</summary>
### Visual Description
# Technical Document Extraction: AI Hallucination Analysis Diagram
## 1. Overview
This image is a flow-based diagram illustrating a Large Language Model (LLM) interaction and the subsequent metadata analysis regarding the accuracy of the generated response. It depicts a user query, an incorrect AI response, and two diagnostic metrics indicating a high probability of error.
## 2. Component Isolation and Transcription
### Region 1: User Input (Top Right)
* **Visual Element:** A blue speech bubble associated with a black silhouette icon of a person.
* **Transcribed Text:** "What is the most smallest country in Asia, by land area?"
* **Note:** The query contains a grammatical redundancy ("most smallest").
### Region 2: AI Response (Middle Left)
* **Visual Element:** A royal blue speech bubble associated with a green circular logo containing a stylized knot/interlocking pattern.
* **Transcribed Text:** "Nepal is the smallest country in Asia, by land area."
* **Fact Check:** This statement is factually incorrect (Maldives is the smallest country in Asia), serving as the example for the following metrics.
### Region 3: Confidence Metric (Center)
* **Visual Element:** A pink rectangular block connected to the AI Response by a downward-pointing black arrow.
* **Transcribed Text:** "Token-Level Confidence Estimate: 13%"
* **Trend/Data Point:** This represents a very low confidence score from the model for the generated tokens.
### Region 4: Hallucination Metric (Bottom)
* **Visual Element:** An orange/tan rectangular block connected to the Confidence Metric by a downward-pointing black arrow.
* **Transcribed Text:** "Hallucination Score: 80%"
* **Trend/Data Point:** This represents a high probability that the information provided in the AI Response is fabricated or incorrect.
## 3. Diagram Flow and Logic
The diagram follows a vertical and diagonal flow to demonstrate a "Detection Pipeline":
1. **Input:** User asks a factual question.
2. **Output:** The AI provides a factually incorrect answer (Nepal).
3. **Analysis Step 1:** The system evaluates the internal confidence of the tokens generated, resulting in a low **13%**.
4. **Analysis Step 2:** Based on the low confidence and potentially other cross-referencing, the system assigns a high **80% Hallucination Score**.
## 4. Summary of Data Points
| Metric | Value | Interpretation |
| :--- | :--- | :--- |
| **Token-Level Confidence** | 13% | Extremely Low; indicates the model is "unsure" of its word choice. |
| **Hallucination Score** | 80% | High; indicates a high likelihood of factual error. |
| **Subject Matter** | Geography | Specifically Asian land area. |
**Language Declaration:** All text in this image is in **English**.
</details>
Figure 16. Uncertainty quantification methods for LLMs have been employed in hallucination detection. LLMs tend to be less confident when hallucinating (measured via token-based metrics), although their responses may sound overly confident. In this example, although the LLM provides a confident response to the prompt, a token-level UQ method indicates that the LLM is uncertain, enabling hallucination detection.
\Description
[Uncertainty quantification methods for LLMs have been employed in hallucination detection. LLMs tend to be less confident when hallucinating (measured via token-based metrics), although their responses may sound overly confident. In this example, although the LLM provides a confident response to the prompt, a token-level UQ method indicates that the LLM is uncertain, enabling hallucination detection.]Uncertainty quantification methods for LLMs have been employed in hallucination detection. LLMs tend to be less confident when hallucinating (measured via token-based metrics), although their responses may sound overly confident. In this example, although the LLM provides a confident response to the prompt, a token-level UQ method indicates that the LLM is uncertain, enabling hallucination detection.
<details>
<summary>extracted/6588127/images/applications/uq_llm_robotics.png Details</summary>
### Visual Description
# Technical Document Extraction: Robotic Task Inference Interface
## 1. Image Overview
This image depicts a user interface (UI) flow for a multimodal AI system, likely a Large Language Model (LLM) integrated with robotics. The flow illustrates an image input, a natural language query, and a ranked list of predicted robotic actions with associated confidence scores.
---
## 2. Component Segmentation
### Region A: Visual Input (Top Right)
* **Type:** Photographic Image.
* **Content:** A white collaborative robot arm (cobot) positioned in a kitchen or office breakroom environment.
* **Scene Details:**
* The robot is positioned next to a grey cabinet with three drawers.
* On top of the cabinet are two bowls: one white ceramic bowl and one smaller metal bowl.
* The background includes wooden slat walls, shelving with containers (nuts/grains), and a potted plant.
* **Flow Indicator:** A black downward-pointing arrow connects this image to the user query.
### Region B: User Query (Middle Right)
* **Type:** Chat Bubble (Blue).
* **Icon:** A black silhouette of a person's head and shoulders is positioned to the right of the bubble.
* **Transcribed Text:** "How do I clear the table?"
### Region C: System Response (Bottom)
* **Type:** Chat Bubble (Blue).
* **Icon:** A green circular logo with a stylized "X" or knot pattern is positioned to the left of the bubble.
* **Content Type:** A list of four potential actions, each preceded by a numerical value (confidence score) and a dash.
---
## 3. Data Extraction: Action Predictions
The system provides a ranked list of actions. The numerical values represent probability or confidence scores (summing to 1.00).
| Confidence Score | Color Code | Action Description |
| :--- | :--- | :--- |
| **0.51** | Green | Move the ceramic bowl into the top drawer. |
| **0.33** | Orange | Move the metal bowl into the middle drawer. |
| **0.10** | Pink/Red | Move the table to the living room. |
| **0.06** | Red/Orange | Remove the top drawer. |
---
## 4. Trend and Logic Analysis
* **Primary Intent:** The system identifies "Move the ceramic bowl into the top drawer" as the most likely intended action (51% confidence).
* **Secondary Intent:** "Move the metal bowl into the middle drawer" is the second most likely (33% confidence).
* **Outlier/Low Confidence:** The system assigns very low probability to moving the entire table (10%) or removing the drawer itself (6%), suggesting these are interpreted as less logical responses to the command "clear the table" in this context.
* **Spatial Grounding:** The system correctly identifies objects in the image (ceramic bowl, metal bowl, top drawer, middle drawer) and maps them to the linguistic command.
---
## 5. Technical Summary
This document represents a **Multimodal Task Planning** interface. It demonstrates the translation of a high-level human instruction ("clear the table") into discrete, executable robotic sub-tasks based on visual context. The output format suggests a probabilistic model where multiple hypotheses are generated and ranked.
</details>
Figure 17. Robotics applications utilize UQ methods to estimate the LLMâs confidence in the sub-tasks proposed by the LLM, to determine when human assistance is required.
\Description
[Robotics applications utilize UQ methods to estimate the LLMâs confidence in the sub-tasks proposed by the LLM, to determine when human assistance is required.]Robotics applications utilize UQ methods to estimate the LLMâs confidence in the sub-tasks proposed by the LLM, to determine when human assistance is required.
9.2. Robotics
Endowing LLMs with an embodiment (physical form) presents unique challenges, as is the case in robotics. Such embodiment essentially empowers LLMs to be agents of physical change, which can lead to potentially disastrous outcomes if the outputs of the LLMs are not reliable or trustworthy. Although LLMs (and vision-language models) have found widespread applications in robotics, e.g., robotic manipulation (Ahn et al., 2022; Brohan et al., 2022, 2023; Kim et al., 2024b), robotic navigation and exploration (Shah et al., 2023; Dorbala et al., 2023; Ren et al., 2024), and multi-robot collaboration (Kannan et al., 2023; Chen et al., 2024; Mandi et al., 2024), only a few of these applications explicitly consider the uncertainty of the LLMs to ensure safety, although other existing work (Wang et al., 2024c) utilize LLMs to assess the success of a task without explicitly reasoning about the confidence of the LLM.
The work in (Tsai et al., 2024) fine-tunes the Mistral-7B LLM (Jiang et al., 2023b) to generate possible next actions for a decision-making agent and trains a neural point-wise dependency estimator to predict the compatibility score between a user-provided prompt and all generated actions. Subsequently, the authors employ conformal prediction to identify more likely actions for a given prompt, which is presented to the user to select the next action. A collection of LLM-based task planning work for robots examine the confidence an LLM assigns to its generated next-step plans to determine when human assistance or verification is required, illustrated in Figure 17. To determine when an LLM requires clarification from a human, KnowNo (Ren et al., 2023a) utilizes a token-based UQ approach to estimate the uncertainty of the LLM in generating possible next steps for a robot given a task, by examining the token probability assigned to each option in the list of possible next steps. Further, KnowNo employs conformal prediction to generate prediction sets over the possible next steps, with provable theoretical guarantees, prompting the human for help, if the prediction set consists of more than one possible action. HERACLEs (Wang et al., 2023b) presents a similar pipeline within a Linear Temporal Logic framework, with multiple high-level sub-goals.
IntroPlan (Liang et al., 2024) extends KnowNo (Ren et al., 2023a) through introspective planning, where, given a task, the LLM retrieves the most relevant instance from a knowledge base constructed from few-shot, human-provided examples and reasons about the feasibility of the possible next actions. Introspective planning enables IntroPlan to generate prediction sets with tighter confidence bounds, minimizing human intervention. LAP (Mullen Jr and Manocha, 2024) further introduces an action-feasibility metric to improve the alignment of the LLMâs confidence estimate with the probability of success, resulting in fewer clarification queries. S-ATLAS (Wang et al., 2024c) extends KnowNo to LLM-based multi-robot task planning, where a team of robot collaborate to complete a task. In addition, KnowLoop (Zheng et al., 2024) utilizes a multi-modal large language model (MLLM), e.g., LLaVa (Liu et al., 2024c) or ChatGPT-4V, for failure detection in LLM-based task planning. The MLLM evaluates the success of the task, given images of the environment at each stage, providing its feedback along with its estimated confidence, using either a self-verbalized approach or a token-level UQ method. KnowLoop (Zheng et al., 2024) demonstrates that token-level UQ approaches yield better-aligned uncertainty estimates compared to a self-verbalized UQ approach. Lastly, TrustNavGPT (Sun et al., 2024b) employs a similar architecture to evaluate the trustworthiness of human commands to an LLM in LLM-based, audio-guided robot navigation.
10. Open Research Challenges
We enumerate a number of open research challenges, hoping to drive future research to address these challenges.
10.1. Consistency is not Factuality
Many uncertainty quantification methods for LLMs rely on evaluating the consistency between multiple realizations of the response generated by LLMs. This approach faces fundamental limitations, since consistency is not necessarily aligned with factuality. For example, in Figure 19, when prompted to provide a response to the question: âWhat happened to Google in June 2007, in a single sentence?â GPT-4 claims that Google announced its mobile operating system Android in June 2007, which is incorrect, given that Android was launched in November 2007. In fact, when creating the set of responses for uncertainty quantification, multiple queries to GPT-4 generate the same incorrect response, which can lead to a miscalibrated confidence estimate. Notably, black-box methods that rely entirely on consistency are most susceptible to this challenge.
Nonetheless, consistency is often a good predictor of factuality, especially when given a sufficiently large number of samples. However, many existing methods do not rigorously examine the number of samples required to define a reliable set of responses when evaluating the consistency of an LLM on a given prompt, which constitutes a critical component for any guarantee on the estimated confidence of the model or factuality of the modelâs response. Moreover, this challenge might be mitigated by a principled selection of the temperature parameter in an LLM to increase the randomness of the mode; however, the effectiveness of this strategy is quite limited, as excessive randomness in the LLMâs outputs defeats the purpose of examining the confidence of the model on a given prompt.
<details>
<summary>extracted/6588127/images/open_research_challenges/consistency_and_factuality.png Details</summary>
### Visual Description
# Technical Document Extraction: LLM Hallucination and Consistency Diagram
## 1. Document Overview
This image is a flow diagram illustrating a technical concept in Artificial Intelligence, specifically regarding Large Language Models (LLMs). It demonstrates a scenario where a model provides highly consistent but factually incorrect information (a "hallucination").
## 2. Component Isolation and Flow Analysis
The diagram follows a vertical top-to-bottom linear flow, segmented into four primary stages:
### Stage 1: Input (Header)
* **Visual Element:** A blue rounded rectangular box with a black silhouette icon of a person's head and shoulders in the top right corner.
* **Transcribed Text:** "What happened to Google in June 2007?"
* **Function:** Represents the user query or prompt being fed into the system.
### Stage 2: Processing (The Model)
* **Visual Element:** A lime green square box containing a black circular icon with an interlocking "X" or knot-like symbol.
* **Label:** To the left of the box, the text "LLM" is present.
* **Function:** Represents the Large Language Model processing the input.
### Stage 3: Output Generation (Main Content)
* **Visual Element:** A large light-cyan rounded rectangular container labeled "Randomly-Generated Responses". Inside this container are two smaller white boxes with black borders, separated by an ellipsis (...).
* **Left Response Box Text:** "In June 2007, Google introduced Android, its mobile operating system."
* **Right Response Box Text:** "Google launched its open-source mobile operating system Android in June 2007."
* **Function:** Shows that the model generated multiple variations of the same claim.
### Stage 4: Evaluation (Footer)
* **Visual Element 1:** A pink rounded rectangular box.
* **Transcribed Text:** "Consistency Estimate: 99%"
* **Visual Element 2:** A large brown "X" mark followed by text.
* **Transcribed Text:** "Fact-Check: False"
* **Function:** This stage highlights the discrepancy between internal model confidence (consistency) and external truth (factuality).
## 3. Logic and Trend Verification
* **Flow Direction:** Indicated by four downward-pointing black arrows connecting each stage.
* **Trend Analysis:** The diagram illustrates a "High Consistency, Low Accuracy" failure mode.
* The LLM generates multiple responses that are semantically identical (both claim Android launched in June 2007).
* Because the responses match, the "Consistency Estimate" is nearly perfect (99%).
* However, the final "Fact-Check" reveals the information is "False" (Android was actually announced in November 2007).
## 4. Summary of Textual Data
| Component | Text Content |
| :--- | :--- |
| **User Prompt** | What happened to Google in June 2007? |
| **Processor** | LLM |
| **Process Type** | Randomly-Generated Responses |
| **Response A** | In June 2007, Google introduced Android, its mobile operating system. |
| **Response B** | Google launched its open-source mobile operating system Android in June 2007. |
| **Metric** | Consistency Estimate: 99% |
| **Verification** | Fact-Check: False |
## 5. Language Declaration
The primary and only language present in this image is **English**. No other languages were detected.
</details>
Figure 18. Consistency is not factuality. Semantic-similarity UQ methods for LLMs might provide misleading confidence estimates, e.g., when multiple random responses from the LLM are consistent but false. In this example, the LLM consistently claims that Google introduced Android in June 2007, which is incorrect, given that Android was introduced in November 2007.
\Description
[Consistency is not factuality. Semantic-similarity UQ methods for LLMs might provide misleading confidence estimates, e.g., when multiple random responses from the LLM are consistent but false. In this example, the LLM consistently claims that Google introduced Android in June 2007, which is incorrect, given that Android was introduced in November 2007.]Consistency is not factuality. Semantic-similarity UQ methods for LLMs might provide misleading confidence estimates, e.g., when multiple random responses from the LLM are consistent but false. In this example, the LLM consistently claims that Google introduced Android in June 2007, which is incorrect, given that Android was introduced in November 2007.
<details>
<summary>extracted/6588127/images/open_research_challenges/entropy_and_factuality.png Details</summary>
### Visual Description
# Technical Document Extraction: AI Hallucination and Fact-Checking Diagram
## 1. Overview
This image is a conceptual diagram illustrating the discrepancy between an AI model's internal confidence (based on token probability) and external factual accuracy. It depicts a conversational interface where an AI provides a factually incorrect answer despite having high statistical confidence.
---
## 2. Component Isolation and Transcription
### Region 1: User Input (Header Right)
* **Visual Element:** A blue speech bubble associated with a black silhouette icon of a person.
* **Text Transcription:** "What is the most populous country in the world in 2024?"
### Region 2: AI Response (Main Body)
* **Visual Element:** A blue speech bubble associated with a green circular logo containing a stylized "X" or knot symbol.
* **Text Transcription:** "The United States of America with a population of 345,426,571."
* **Embedded Data Visualization:** Beneath the text, within the same blue bubble, is a series of colored rectangles representing token-level confidence:
* **Green Rectangles:** 10 units.
* **Yellow Rectangle:** 1 unit (positioned under the word "States").
* **Sequence:** [Green, Yellow, Green, Green, Green, Green, Green, Green, Green, Green].
* **Trend:** The visualization shows that almost all tokens in the generated sentence have high probability (Green), with only one token showing moderate/lower probability (Yellow).
### Region 3: Confidence Metric (Center)
* **Visual Element:** A pink rectangular box connected to the AI response by a downward-pointing black arrow.
* **Text Transcription:** "Confidence Estimate from Token Probability: 91%"
* **Analysis:** This represents the aggregate statistical confidence the model has in its generated string.
### Region 4: Fact-Check Result (Footer)
* **Visual Element:** A large red "X" mark.
* **Text Transcription:** "Fact-Check: False"
* **Color Coding:** The word "False" is highlighted in a dark red/brown color to match the "X" mark.
---
## 3. Technical Flow and Logic Analysis
1. **Query:** The user asks a factual question regarding global population.
2. **Generation:** The AI generates a specific answer.
3. **Internal Metric:** The system calculates a **91% confidence score** based on the mathematical probability of the tokens selected during generation. The visual markers show that the model "believes" its output is highly likely to be correct.
4. **External Verification:** Despite the high internal confidence (91%), the statement is objectively incorrect (as India and China have significantly larger populations than the USA).
5. **Conclusion:** The diagram serves as a technical warning that high token probability/confidence estimates do not guarantee factual truth, illustrating the phenomenon of "confident hallucination."
---
## 4. Data Summary Table
| Category | Value / Content |
| :--- | :--- |
| **User Query** | What is the most populous country in the world in 2024? |
| **AI Output** | The United States of America with a population of 345,426,571. |
| **Token Probability Visual** | 9 Green blocks, 1 Yellow block |
| **Aggregate Confidence** | 91% |
| **Factual Status** | False |
</details>
Figure 19. Using the conditional distribution of tokens for uncertainty quantification (e.g., in token-level UQ methods) can lead to misleading uncertainty estimates. In this example, the uncertainty of the LLM is notably low, since the succeeding tokens are highly likely given the preceding tokens. However, the claim is incorrect. The most populous country in the world in 2024 is India, not the United States of America. The bars denote the probability of each token.
\Description
[Using the conditional distribution of tokens for uncertainty quantification (e.g., in token-level UQ methods) can lead to misleading uncertainty estimates. In this example, the uncertainty of the LLM is notably low, since the succeeding tokens are highly likely given the preceding tokens. However, the claim is incorrect. The most populous country in the world in 2024 is India, not the United States of America. The bars denote the probability of each token.]Using the conditional distribution of tokens for uncertainty quantification (e.g., in token-level UQ methods) can lead to misleading uncertainty estimates. In this example, the uncertainty of the LLM is notably low, since the succeeding tokens are highly likely given the preceding tokens. However, the claim is incorrect. The most populous country in the world in 2024 is India, not the United States of America. The bars denote the probability of each token.
10.2. Entropy is not Factuality
Entropy and other token-based UQ metrics of the token probability distribution in an LLMâs output are not necessarily aligned with the factuality of the modelâs output, although entropy and factuality are often aligned. In particular, the distribution over the tokens is a function of the size of the LLM (including its dictionary of tokens) and the diversity and size of the training data, which can influence the alignment of entropy and factuality. Hence, token-based UQ methods might produce highly miscalibrated confidence estimates for a given prompt, when these estimates are computed entirely from the distribution over the tokens. For example, in a worst-case scenario where the training data is corrupted or insufficient, an LLM might assign most of its probability to an incorrect answer (token) which is most closely related to the training data, leading to a miscalibrated estimate of its confidence. Moreover, reinforcement learning with human feedback (RLHF), which is utilized in fine-tuning LLMs, generally reduces the calibration of the LLMâs confidence estimates (Achiam et al., 2023). Further, the conditional distribution of each token might not be indicative of the factuality of an LLMâs response at the claim-level (sentence-level), i.e., although each generated token might be highly likely given the preceding token, the overall claim expressed by the LLM might not be correct (Vazhentsev et al., 2024), as illustrated in Figure 19.
Future research should explore aligning the entropy of tokens with the factuality of the claims expressed by LLMs and examine augmentation strategies that consider the training distribution of LLMs to better account for the influence of the training data on the probability distribution associated with the generated tokens to ultimately improve the alignment of entropy and other token-based measures of uncertainty with factuality. Moreover, the probability distributions over the tokens of an LLM can be manipulated in jailbreaking attacks, leading to misleading confidence estimates and, in some cases, non-factual responses (Zeng et al., 2024). Future research should seek to improve the robustness of token-level uncertainty quantification methods to adversarial attacks. Further, few existing methods explore uncertainty quantification of LLMs in text summarization, which is critical to the preservation of factual records, constituting an important direction for future research.
10.3. Applications in Interactive LLM-Enabled Agents
Although some existing applications explore uncertainty quantification in LLM-enabled agents, e.g., see Section 9.2, many of these applications only estimate the LLMâs uncertainty at each episode without considering the history of the agentâs interaction with the LLM. However, many practical applications require multi-episode interactions, where the LLM generates successive responses based on the information from preceding episodes with the agent. For example, in the scenario depicted in Figure 17, the robot may be asked to prepare a meal for a user, which would require multi-episode interactions, where each episode corresponds to a given sub-task, such as dicing some vegetables before sautĂ©ing it. Note that utilizing many existing techniques for uncertainty quantification would require the assumption that the LLMâs uncertainty at each episode is independent of its prior interaction history, an assumption that is generally not satisfied in real-world applications. Rigorous uncertainty quantification of the LLMâs outputs requires the consideration of the history of the agentâs interaction with the LLM and its observations (e.g., camera images), in the case of VLMs. This yet-unexplored research area constitutes an exciting direction for future research.
10.4. Applications of Mechanistic Interpretability to Uncertainty Quantification
The connections between interpretability of LLMs and uncertainty quantification have been relatively unexplored, despite the intuitive relationship between both concepts. Mechanistic interpretability holds notable potential in exploiting the synergy between both areas to derive solutions to some of the aforementioned research challenges. For example, the work in (Ahdritz et al., 2024) predicts the token-level confidence of large LLMs using small linear probes (models) trained on the embeddings of frozen pretrained models. This work suggests the existence of a relationship between the internal states of LLMs and their confidence. The authors indicate that their findings suggest that information on the internal state of an LLM could be utilized in distinguishing epistemic uncertainty of the model from aleatoric uncertainty. However, this research area is relatively unexplored, presenting a potentially fruitful direction for future research.
10.5. Datasets and Benchmarks
Although a number of datasets and benchmark for uncertainty quantification exists (Joshi et al., 2017; Reddy et al., 2019; Yang et al., 2018; Lin et al., 2021), to the best of our knowledge, no dataset exists for uncertainty quantification of LLMs in multi-episode interaction scenarios. Future research should examine the creation of versatile, standardized datasets that aid research on uncertainty quantification of LLMs, taking into consideration the history of interaction between a user and an LLM. Moreover, benchmarks on uncertainty quantification of LLMs can help inform researchers on the relative performance of their proposed methods. Unfortunately, widely-accepted benchmarks for uncertainty quantification of LLMs do not exist, although some work has been devoted to developing such benchmarks. Future work should seek to create suitable benchmarks for this purpose, especially benchmarks that evaluate the calibration, tightness (conservativeness), and interpretability of uncertainty quantification methods. However, benchmarks can also introduce other challenges, by disconnecting research from practical concerns, overly simplifying the assessment of research advances to outperforming existing work on some metric defined in a benchmark. Hence, care must be taken to ensure that benchmarks remain relevant to practical effectiveness.
11. Conclusion
In this survey, we provide a comprehensive review of existing uncertainty quantification methods for LLMs, including relevant background information necessary for readers. We categorize UQ methods for LLMs into four broad classes based on the underlying technique employed by these methods, namely: token-based UQ methods, self-verbalized UQ methods, semantic-similarity-based methods, and mechanistic interpretability. Token-based UQ methods rely on access to an LLMâs intermediate outputs or architecture to estimate the confidence an LLM, whereas in self-verbalized UQ methods, the LLM provides its estimated confidence in natural-language. Many semantic-similarity-based methods are black-box methods which only require access to the modelâs natural-language output, relying on consistency metrics to estimate the LLMâs confidence. In contrast, mechanistic interpretability requires access to the LLMâs internal activations to identify latent features that explain its activation patterns. Furthermore, we identify relevant datasets and applications for uncertainty quantification of LLMs and highlight open research challenges to inspire future research.
Acknowledgements. We would like to acknowledge Apurva S. Badithela and David Snyder for their contributions. This work was partially supported by the NSF CAREER Award [#2044149], the Office of Naval Research [N00014-23-1-2148], and the Sloan Fellowship. Justin Lidard was supported by a National Science Foundation Graduate Research Fellowship.
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