2412.05563v2

# 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 ## Chat Dialogue: Antarctica Temperature Fact-Check ### Overview The image depicts a chat dialogue between two entities, one represented by a human silhouette and the other by a stylized logo. The dialogue concerns the lowest-ever recorded temperature in Antarctica. The response provided by the logo-represented entity is fact-checked and found to be false. ### Components/Axes * **Chat Bubbles:** Blue chat bubbles contain the dialogue. * **Avatars:** A human silhouette represents one participant, and a stylized logo represents the other. * **Fact-Check Label:** Located at the bottom, indicating the veracity of the information. * **Fact-Check Symbol:** A red "X" symbol indicates the information is false. ### Detailed Analysis or ### Content Details 1. **First Chat Bubble (Top-Right):** * Text: "What is the lowest-ever temperature recorded in Antarctica?" * Speaker: Human silhouette avatar. 2. **Second Chat Bubble (Top-Left):** * Text: "The lowest-ever temperature recorded in Antarctica was a staggering -128.6°F (-89.2°C)." * Speaker: Stylized logo avatar. 3. **Third Chat Bubble (Center-Right):** * Text: "How confident are you?" * Speaker: Human silhouette avatar. 4. **Fourth Chat Bubble (Bottom-Left):** * Text: "I am 100% confident." * Speaker: Stylized logo avatar. 5. **Fact-Check (Bottom):** * Symbol: Red "X". * Text: "Fact-Check: False" ### Key Observations * The stylized logo avatar confidently provides a specific temperature as the lowest ever recorded in Antarctica. * The fact-check indicates that the provided temperature is incorrect. ### Interpretation The image illustrates a scenario where a system (represented by the stylized logo) confidently provides information that is later determined to be false. This highlights the importance of fact-checking and verifying information, even when it is presented with high confidence. The dialogue suggests a potential flaw in the system's knowledge base or information retrieval process. The fact that the system claims 100% confidence despite providing incorrect information raises concerns about its reliability and the potential for misinformation. </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 “100% confident,” the LLM’s 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 “100% confident,” the LLM’s 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 “100% confident,” the LLM’s 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: “What is the lowest-ever temperature recorded in Antarctica?”, to which the LLM responds definitively. Even when prompted for its confidence in its answer, the LLM claims that it is “100% confident.” However, the LLM’s 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 LLM’s 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 user’s 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 model’s 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: “The Ultimate Guide to Cooking for One.” 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: “The Jewish Cookbook.” This book is authored by Leah Koenig, not Jamie Feldman. Uncertainty quantification (UQ) methods aim to provide a more rigorous estimate of the model’s 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 ## Chat Log: Cooking Book Recommendation ### Overview The image presents a chat log between two users. One user asks for a recommendation for the best cooking book written by Jaime Feldman. The other user responds by recommending "The Ultimate Guide to Cooking for One" by Jamie Feldman, highlighting its focus on recipes for solo cooks. ### Components/Axes * **User 1 (Top-Right):** Asks the question. Represented by a silhouette icon. * **User 2 (Bottom-Left):** Provides the answer. Represented by a circular icon with a stylized "88" inside. * **Chat Bubbles:** Blue rounded rectangles containing the text of the conversation. ### Detailed Analysis or ### Content Details * **User 1's Question:** "What is the best cooking book written by Jaime Feldman?" * **User 2's Response:** "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!" ### Key Observations * The response directly answers the question by recommending a specific book by Jamie Feldman. * The response provides additional information about the book's content and target audience. ### Interpretation The chat log demonstrates a successful exchange where one user seeks a recommendation and another user provides a relevant and informative response. The recommended book, "The Ultimate Guide to Cooking for One," is positioned as a practical resource for individuals cooking for themselves, emphasizing simplicity, flavor, and waste reduction. </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 ## Chat Conversation: Jamie Feldman ### Overview The image depicts a chat conversation between two individuals. The conversation revolves around Jamie Feldman, specifically regarding a response about her. The first message asks if the other party is confident about their response on Jamie Feldman. The second message apologizes for confusion and clarifies that Jamie Feldman is best known for "The Jewish Cookbook," which features a wide range of traditional and modern Jewish recipes. It suggests the cookbook as a comprehensive guide to Jewish cooking and thanks the other party for their patience. ### Components/Axes * **Message 1 (Top-Right):** "Are you confident about your response on Jamie Feldman?" - This message is from a user represented by a silhouette icon. * **Message 2 (Bottom-Left):** "I apologize for the confusion. Jamie Feldman is best known for "The Jewish Cookbook," which features a wide range of traditional and modern Jewish recipes. If you're looking for a comprehensive guide to Jewish cooking, that would be the book to check out! Thank you for your patience." - This message is from a user represented by an icon with a green circle and a white design inside. ### Detailed Analysis or ### Content Details * **Message 1:** The question is a direct inquiry about the confidence level regarding a response related to Jamie Feldman. * **Message 2:** The response provides context about Jamie Feldman, identifying her as the author of "The Jewish Cookbook." It also positions the cookbook as a valuable resource for Jewish recipes. ### Key Observations * The conversation suggests a prior exchange or question about Jamie Feldman that may have caused confusion. * The response aims to clarify Jamie Feldman's identity and her association with Jewish cuisine. ### Interpretation The chat likely stems from a question or task where Jamie Feldman was mentioned. The initial response may have been unclear or inaccurate, prompting the follow-up question about confidence. The second message serves to correct any misunderstanding by providing specific information about Jamie Feldman's work and expertise. The conversation highlights the importance of accurate information and the value of providing context when discussing individuals or their work. </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 “What will the temperature be tomorrow?”, the uncertainty associated with the LLM’s 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 “ignorance” 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 LLM’s 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 model’s 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; Buciluǎ 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 model’s 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 (Buciluǎ 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 model’s 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 model’s 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 model’s 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 $j$ ’th token in the $i$ ’th sentence of an LLM’s 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 ## Diagram: Transformer Architecture - Encoder Block ### Overview The image is a diagram illustrating the architecture of a single encoder block within a Transformer model. It shows the flow of data through various layers, including embedding, positional encoding, normalization, multi-head attention, feed-forward networks, and residual connections. ### Components/Axes * **Blocks:** The diagram consists of several blocks representing different layers or operations. These blocks are arranged horizontally, indicating the flow of data. * **Arrows:** Arrows indicate the direction of data flow between the blocks. * **Addition Symbols (+):** These symbols represent residual connections, where the input of a layer is added to its output. * **Text Labels:** Each block is labeled with the name of the corresponding layer or operation. * **Colors:** Different colors are used to distinguish the different types of layers. * **Green:** Positional Encoding * **Light Blue:** Embedding * **Dark Blue:** Norm * **Purple:** Masked Multi-Head Attention * **Orange:** Feed-Forward * **Pink:** Linear * **Red:** Softmax ### Detailed Analysis 1. **Positional Encoding:** A green block labeled "Positional Encoding" is located at the top-left of the diagram. A sine wave symbol is to the left of the block. 2. **Embedding:** A light blue block labeled "Embedding" is located below the "Positional Encoding" block. 3. **Addition:** The outputs of "Positional Encoding" and "Embedding" are combined using an addition operation. 4. **N multi-head attention sub-blocks:** A dashed rectangle surrounds the core multi-head attention sub-blocks. The text "N multi-head attention sub-blocks" is at the top of the rectangle. 5. **Norm (1st):** A dark blue block labeled "Norm" follows the addition operation. 6. **Masked Multi-Head Attention:** A purple block labeled "Masked Multi-Head Attention" follows the first "Norm" block. 7. **Addition (2nd):** The output of the "Masked Multi-Head Attention" block is added to the input of the first "Norm" block. 8. **Norm (2nd):** A dark blue block labeled "Norm" follows the second addition operation. 9. **Feed-Forward:** An orange block labeled "Feed-Forward" follows the second "Norm" block. 10. **Addition (3rd):** The output of the "Feed-Forward" block is added to the input of the second "Norm" block. 11. **Norm (3rd):** A dark blue block labeled "Norm" follows the third addition operation. 12. **Linear:** A pink block labeled "Linear" follows the third "Norm" block. 13. **Softmax:** A red block labeled "Softmax" follows the "Linear" block. ### Key Observations * The diagram illustrates a sequential flow of data through the encoder block. * Residual connections are used to add the input of certain layers to their output, which helps to mitigate the vanishing gradient problem. * The "N multi-head attention sub-blocks" section contains the core attention mechanism. ### Interpretation The diagram represents a single encoder block in a Transformer model. The encoder block processes input embeddings by adding positional information, normalizing the data, applying masked multi-head attention, feeding the data through a feed-forward network, and applying another normalization. The residual connections help to improve the training process. The final linear and softmax layers likely produce the output probabilities for each token in the sequence. The "N" in "N multi-head attention sub-blocks" indicates that this block can be repeated multiple times in the encoder. </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 ## Diagram: Textual Entailment Relationships ### Overview The image illustrates textual entailment relationships between pairs of sentences. It shows three distinct relationships: Contradiction, Neutral, and Entailment. Each relationship is represented by a colored circle (red, yellow, and green, respectively) connecting two text boxes via chain links. The text boxes contain sentences, and the color of the circle indicates the relationship between the sentences. ### Components/Axes * **Relationship Types (Labels):** * Contradiction (top) * Neutral (middle) * Entailment (bottom) * **Sentence Pairs:** Each relationship type has a pair of sentences associated with it. The first sentence in each pair is in a light blue box, and the second sentence is in a light pink box. * **Connectors:** Chain links connect the sentence boxes to the colored circles. * **Relationship Indicators (Colored Circles):** * Red: Contradiction * Yellow: Neutral * Green: Entailment ### Detailed Analysis or ### Content Details * **Contradiction:** * Sentence 1 (Light Blue Box): "A student presented a research paper at the conference." * Connector: Chain links * Relationship Indicator: Red circle labeled "Contradiction" * Connector: Chain links * Sentence 2 (Light Pink Box): "The student skipped the conference." * **Neutral:** * Sentence 1 (Light Blue Box): "The orchestra enjoyed the concert." * Connector: Chain links * Relationship Indicator: Yellow circle labeled "Neutral" * Connector: Chain links * Sentence 2 (Light Pink Box): "The orchestra performed at the concert." * **Entailment:** * Sentence 1 (Light Blue Box): "The team was out on vacation yesterday." * Connector: Chain links * Relationship Indicator: Green circle labeled "Entailment" * Connector: Chain links * Sentence 2 (Light Pink Box): "The team was not in the office yesterday." ### Key Observations * The diagram clearly distinguishes between three types of relationships between sentences: contradiction, neutrality, and entailment. * The use of color-coding (red, yellow, green) provides a visual cue for each relationship type. * The chain links visually connect the sentences and the relationship indicators. ### Interpretation The diagram effectively illustrates the concept of textual entailment by providing concrete examples of sentence pairs that exhibit contradiction, neutrality, and entailment. The use of color-coding and visual connectors makes the relationships easy to understand. The diagram demonstrates how one sentence can contradict, be neutral to, or entail another sentence. * **Contradiction:** The first sentence implies the student attended the conference, while the second sentence states the student skipped it, creating a contradiction. * **Neutral:** The first sentence expresses enjoyment, while the second simply states the orchestra performed. There is no direct contradiction or entailment. * **Entailment:** If the team was out on vacation, it logically follows that they were not in the office. The first sentence entails the second. </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 LLM’s 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 accessible—hence the term white-box —taking 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 ## Chat Conversation: Volleyball Olympic Debut ### Overview The image depicts a chat conversation about when volleyball debuted as an Olympic sport. The initial question is "When did volleyball become an Olympic Sport?". The response states that volleyball debuted at the Tokyo Olympic Games in 1964. A subsequent data list provides percentage values associated with different cities: Tokyo, Osaka, Kyoto, Beijing, and Mumbai. ### Components/Axes * **Chat Bubbles:** Two blue chat bubbles containing the conversation. * **Profile Icons:** A circular icon with a knot-like design on the left and a silhouette of a person on the right. * **Data List:** A list of cities with associated percentage values, enclosed in a box. ### Detailed Analysis or ### Content Details * **Question:** "When did volleyball become an Olympic Sport?" * **Answer:** "Volleyball debuted at the Tokyo Olympic Games in 1964." * **Data List:** * Tokyo = 87.28% * Osaka = 7.49% * Kyoto = 4.16% * Beijing = 1.04% * Mumbai = 0.03% ### Key Observations * The answer to the question is explicitly stated. * The data list provides percentage values for different cities, but the context or meaning of these percentages is not immediately clear from the image itself. * Tokyo has a significantly higher percentage (87.28%) compared to the other cities. ### Interpretation The conversation establishes that volleyball debuted at the Tokyo Olympics in 1964. The data list, however, lacks context. Without additional information, it's impossible to determine what these percentages represent. Possible interpretations could include: * **Popularity:** The percentages could represent the popularity of volleyball in each city. * **Participation:** The percentages could represent the proportion of athletes from each city participating in volleyball at the Olympics. * **Search Interest:** The percentages could represent the search interest in volleyball in each city. The high percentage for Tokyo likely relates to the fact that the city hosted the Olympic Games when volleyball was introduced. The other cities may be ranked by some other metric. Further information is needed to fully understand the meaning of the data. </details> Figure 8. White-box uncertainty quantification methods utilize an LLM’s internal information, e.g., the model’s probabilities for the token associated with each output. \Description [White-box uncertainty quantification methods utilize an LLM’s internal information, e.g., the model’s probabilities for the token associated with each output.]White-box uncertainty quantification methods utilize an LLM’s internal information, e.g., the model’s probabilities for the token associated with each output. <details> <summary>extracted/6588127/images/background/uq_techniques/black_box.png Details</summary>  ### Visual Description ## Diagram: LLM Response Generation ### Overview The image is a diagram illustrating the process of generating responses to the question "Who was Abraham Lincoln?" using a Large Language Model (LLM). It shows the flow from the initial question, through the LLM, to randomly-generated responses, and finally to a confidence estimate. ### Components/Axes * **Top:** Blue rectangle containing the question "Who was Abraham Lincoln?" with a silhouette of a person's head in the top right corner. * **Middle-Top:** Green square labeled "LLM" with a symbol resembling interconnected nodes inside. * **Middle:** Light blue rectangle labeled "Randomly-Generated Responses" containing two example responses. * **Bottom:** Pink rectangle labeled "Confidence Estimate: 75%". * **Arrows:** Black arrows indicating the flow of information from top to bottom. ### Detailed Analysis * **Question:** The initial question is "Who was Abraham Lincoln?". * **LLM:** The LLM processes the question. * **Randomly-Generated Responses:** * Response 1: "Abraham Lincoln was the fifteenth president of the U.S., serving from 1861 to 1865." * Response 2: "Abraham Lincoln was the sixteenth president of the U.S., serving from 1861 to 1864." * There is an ellipsis "..." between the two responses, indicating that there are more randomly generated responses. * **Confidence Estimate:** The confidence estimate for the responses is 75%. ### Key Observations * The LLM generates multiple responses to the same question. * The responses contain factual inaccuracies (Abraham Lincoln was the 16th president, not the 15th). * The confidence estimate is relatively high (75%) despite the inaccuracies in the responses. ### Interpretation The diagram illustrates a potential issue with LLMs: they can generate responses that sound plausible but contain factual errors. The high confidence estimate despite the inaccuracies highlights the importance of verifying the information provided by LLMs. The diagram suggests that while LLMs can be useful for generating information, they should not be relied upon without critical evaluation and fact-checking. The presence of multiple responses indicates the LLM's attempt to provide a comprehensive answer, but the variability in accuracy raises concerns about the reliability of the information. </details> Figure 9. Black-box uncertainty quantification methods do not access the internal states or probabilities computed by the model, quantifying the model’s 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 model’s 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 model’s 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 model’s confidence. The average token probability is related to the product of the token probabilities. 1. Perplexity. The perplexity of a model’s 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: ${\mathcal{H}(Y\mid x)=-∈t p(y\mid x)\ln p(y\mid x)dy.}$ In the discrete case, the entropy associated with the output distribution of token $j$ in sentence $i$ is defined by: ${\mathcal{H}_{ij}=-\sum_{w∈\mathcal{D}}p_{ij}(w)\log p_{ij}(w),}$ 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 model’s 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 model’s 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 OpenAI’s 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 model’s 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 LLM’s 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 LLM’s 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 LLM’s 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 LLM’s 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 model’s 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 model’s 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 LLM’s 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 LLM’s 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 LLM’s 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 ## Diagram: LLM Response with and without Epistemic Markers ### Overview The image is a diagram illustrating how a Large Language Model (LLM) responds to the question "Which team won the 2022 NBA Finals?" with and without epistemic markers. The diagram shows the question being fed into the LLM, and then two different responses are generated: one with epistemic markers (indicating uncertainty) and one without. ### Components/Axes * **Top:** Blue rounded rectangle containing the question "Which team won the 2022 NBA Finals?" and a silhouette of a person. * **Middle:** Green rounded square containing a symbol resembling two intertwined ampersands. The label "LLM" is to the left of the square. An arrow points from the top rectangle to this square. * **Bottom Left:** Light blue rounded rectangle labeled "With Epistemic Markers" in a yellow box. Inside this rectangle is a black-bordered box containing the text "I think the Milwaukee Bucks won the 2022 NBA Finals, but I am not sure." The words "I think" and "but I am not sure" are in red. * **Bottom Right:** Light blue rounded rectangle labeled "Without Epistemic Markers" in a yellow box. Inside this rectangle is a black-bordered box containing the text "The Milwaukee Bucks won the 2022 NBA Finals." * **Arrows:** Two downward-pointing arrows, one connecting the top rectangle to the middle square, and another connecting the middle square to the two bottom rectangles. ### Detailed Analysis or ### Content Details * **Question:** "Which team won the 2022 NBA Finals?" * **LLM Representation:** The symbol in the green square represents the LLM. * **Response with Epistemic Markers:** "I think the Milwaukee Bucks won the 2022 NBA Finals, but I am not sure." * **Response without Epistemic Markers:** "The Milwaukee Bucks won the 2022 NBA Finals." ### Key Observations * The LLM provides two different responses to the same question, differing in the inclusion of epistemic markers. * The response with epistemic markers expresses uncertainty ("I think," "but I am not sure"). * The response without epistemic markers presents the information as a definitive statement. ### Interpretation The diagram demonstrates how LLMs can be programmed to provide responses with varying degrees of certainty. The inclusion of epistemic markers can be used to indicate the LLM's confidence in its answer, which can be important for users who need to assess the reliability of the information. The absence of epistemic markers can make the LLM's response sound more authoritative, but it can also be misleading if the LLM is not actually certain about the answer. The diagram highlights the importance of considering the context and purpose of the LLM's response when interpreting its output. </details> Figure 10. The LLM provides an incorrect response, but communicates its uncertainty using epistemic markers, e.g., “I think.” \Description [The LLM provides an incorrect response, but communicates its uncertainty using epistemic markers, e.g., “I think.”]The LLM provides an incorrect response, but communicates its uncertainty using epistemic markers, e.g., “I think.” <details> <summary>extracted/6588127/images/self_verbalized_uq/self_verbalized_uncertainty.png Details</summary>  ### Visual Description ## Chat Log: Capital of Luxembourg ### Overview The image is a screenshot of a chat log. The user asks "What is the capital of Luxembourg?". The response from the other party is "Luxembourg is the capital of Luxembourg." followed by "Confidence: I am 98% confident.". ### Components/Axes * **Chat Bubbles:** The messages are contained within rounded chat bubbles. * The user's message bubble is blue and located on the top-right. It includes a silhouette of a person. * The bot's message bubbles are blue and purple, located on the left. They include a green icon with a black design. * **Text Content:** The messages themselves. ### Detailed Analysis or ### Content Details 1. **User Question (Top-Right):** * Text: "What is the capital of Luxembourg?" * Bubble Color: Blue * Icon: Silhouette of a person 2. **Bot Response 1 (Middle-Left):** * Text: "Luxembourg is the capital of Luxembourg." * Bubble Color: Blue * Icon: Green circle with a black knot-like design 3. **Bot Response 2 (Bottom-Left):** * Text: "Confidence: I am 98% confident." * Bubble Color: Purple * Icon: Green circle with a black knot-like design ### Key Observations * The bot correctly identifies the capital of Luxembourg. * The bot expresses a high level of confidence (98%) in its answer. ### Interpretation The chat log demonstrates a successful query to a bot regarding the capital of Luxembourg. The bot provides the correct answer and indicates a high degree of certainty. The use of different colored bubbles and icons distinguishes the user's input from the bot's response. </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 LLM’s response to a given prompt is correct, given the input prompt, its response, and the LLM’s representations of the prompt and its response. In addition, the output of the calibrator and the LLM’s 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 LLM’s 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 model’s confidence, e.g., in Figure 11, or a scaled characterization of the model’s 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 listener’s perceived confidence of the speaker’s response. In essence, LACIE aims to fine-tune an LLM to produce a response composed of epistemic markers that are aligned with the model’s 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 ## Chatbot Response Comparison ### Overview The image shows two examples of a chatbot's response to the question "How deep is the Pacific Ocean?". The first example, labeled "Default", provides a specific answer. The second example, labeled "Abstain when Uncertain", indicates that the chatbot is unsure and declines to answer. ### Components/Axes * **Top Section:** * Label: "Default" (in red) * User Query: "How deep is the Pacific Ocean?" (in a blue speech bubble) * Chatbot Response: "The Pacific Ocean is 3500 meters (11483 feet) deep." (in a blue speech bubble) * Icon: A green icon with a white intertwined symbol is present to the left of the chatbot response. * User Icon: A black silhouette of a person is present to the right of the user query. * **Bottom Section:** * Label: "Abstain when Uncertain" (in green) * User Query: "How deep is the Pacific ocean?" (in a blue speech bubble) * Chatbot Response: "Sorry, I am unsure about the correct answer, so I humbly decline to provide an answer." (in a blue speech bubble) * Icon: A green icon with a white intertwined symbol is present to the left of the chatbot response. * User Icon: A black silhouette of a person is present to the right of the user query. ### Detailed Analysis or ### Content Details * **Default Response:** The chatbot provides a specific answer, stating the depth of the Pacific Ocean as 3500 meters, with a conversion to 11483 feet. * **Abstain Response:** The chatbot acknowledges the question but indicates its inability to provide a correct answer, choosing to abstain. ### Key Observations * The chatbot demonstrates two distinct behaviors: providing a direct answer when confident and abstaining when uncertain. * The difference in capitalization ("Ocean" vs. "ocean") in the user query does not seem to affect the chatbot's response in the "Abstain" case. ### Interpretation The image illustrates a chatbot's ability to handle uncertainty. The "Default" response shows the chatbot's knowledge base, while the "Abstain when Uncertain" response demonstrates its ability to recognize and avoid providing potentially incorrect information. This is a crucial aspect of responsible AI design, as it prevents the chatbot from spreading misinformation. The example highlights the importance of incorporating mechanisms for uncertainty detection and graceful failure in chatbot systems. </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 ## Diagram: LLM Question and Response Generation ### Overview The image is a diagram illustrating the process of a Large Language Model (LLM) generating semantically similar responses to a question. It shows a user query, the LLM processing the query, and the resulting generated responses. ### Components/Axes * **Top:** A blue rounded rectangle containing the question: "Where is Buckingham Palace in the United Kingdom?". A black silhouette of a person's head is located to the right of the question box. * **Middle:** A green square containing a symbol resembling intertwined links, labeled "LLM" to the left of the square. An arrow points downwards from the question box to this square. * **Bottom:** A light blue rounded rectangle labeled "Randomly-Generated Semantically-Similar Responses". An arrow points downwards from the green square to this rectangle. Inside this rectangle are two black-bordered boxes. The left box contains the text "Buckingham Palace is located in London." The right box contains the text "London is home to Buckingham Palace." Between the two boxes is an ellipsis (...). ### Detailed Analysis or ### Content Details * **Question:** "Where is Buckingham Palace in the United Kingdom?" * **LLM Processing:** The LLM processes the question. The symbol inside the green square is an abstract representation of this processing. * **Responses:** * "Buckingham Palace is located in London." * "London is home to Buckingham Palace." * The ellipsis indicates that there may be more responses. ### Key Observations * The diagram illustrates a question-answering process using an LLM. * The LLM generates multiple semantically similar responses to the same question. * The responses are factually correct and relevant to the question. ### Interpretation The diagram demonstrates how an LLM can take a user's question and generate multiple, slightly different, but semantically equivalent answers. This highlights the LLM's ability to understand the underlying meaning of the question and provide varied responses that convey the same information. The use of "Randomly-Generated Semantically-Similar Responses" suggests that the LLM can produce a range of phrasings while maintaining accuracy and relevance. </details> Figure 13. When prompted to answer a question, e.g., “Where is Buckingham Palace in the United Kingdom?”, 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., “Where is Buckingham Palace in the United Kingdom?”, 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., “Where is Buckingham Palace in the United Kingdom?”, 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 (Martín 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 statement’s 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 LLM’s generation and the tool LLM’s 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 predecessor’s preference between two generations. A calibration dataset is used to learn a threshold that determines each judge’s 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 ## Flow Diagram: Hypothesis, Fundamental Objects, and Methods ### Overview The image is a flow diagram illustrating the relationships between hypotheses, fundamental objects, and methods. It shows how hypotheses relate to fundamental objects (features and circuits), and how these objects are analyzed using different methods. ### Components/Axes * **Titles:** * Hypothesis (left) * Fundamental Objects (center) * Methods (right) * **Hypothesis:** * Superposition (light blue box) * Universality (light blue box) * **Fundamental Objects:** * Features (light green box) * Circuits (light purple box) * **Methods:** * SAEs (dark blue box) * Probing (dark blue box) * Logit Lens (dark blue box) * **Arrows:** * Black arrows indicate a two-way relationship between Features and Circuits, and a one-way relationship from Circuits and Features to Superposition and Universality. * Light green arrows indicate a one-way relationship from Features to SAEs and Probing. * Light purple arrow indicates a one-way relationship from Circuits to Logit Lens. ### Detailed Analysis or ### Content Details * **Hypothesis:** * Superposition: Located in the top-left, connected to "Features" by a black arrow pointing from "Superposition" to "Features". * Universality: Located below "Superposition", connected to "Circuits" by a black arrow pointing from "Universality" to "Circuits". * **Fundamental Objects:** * Features: Located in the center, connected to "Superposition" and "SAEs" and "Probing". * Circuits: Located below "Features", connected to "Universality" and "Logit Lens". * **Methods:** * SAEs: Located in the top-right, connected to "Features" by a light green arrow pointing from "Features" to "SAEs". * Probing: Located below "SAEs", connected to "Features" by a light green arrow pointing from "Features" to "Probing". * Logit Lens: Located in the bottom-right, connected to "Circuits" by a light purple arrow pointing from "Circuits" to "Logit Lens". Also connected to "Features" by a light green arrow pointing from "Features" to "Logit Lens". ### Key Observations * "Features" and "Circuits" are interconnected, suggesting a reciprocal relationship. * "Superposition" is linked to "Features", while "Universality" is linked to "Circuits". * "Features" are analyzed using "SAEs" and "Probing". * "Circuits" are analyzed using "Logit Lens". * "Logit Lens" is also connected to "Features". ### Interpretation The diagram illustrates a conceptual framework for understanding how hypotheses (Superposition, Universality) relate to fundamental objects (Features, Circuits) and how these objects are analyzed using different methods (SAEs, Probing, Logit Lens). The interconnection between "Features" and "Circuits" suggests that they are interdependent. The diagram suggests that "SAEs" and "Probing" are used to analyze "Features", while "Logit Lens" is used to analyze "Circuits". The additional connection from "Features" to "Logit Lens" suggests that "Logit Lens" may also be used to analyze "Features" or that there is an indirect relationship between "Features" and "Circuits" that is captured by "Logit Lens". </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 “Golden Gate Bridge” 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 model’s 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 “apple” 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 “codebook” 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 LLM’s 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 LLM’s 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 LLM’s 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, ${∀ p∈[0,1]}$ : $$ \mathbb{P}[Y=\hat{Y}\mid\hat{P}=p]=p, \tag{2} $$ 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 model’s confidence estimates. <details> <summary>extracted/6588127/images/calibration/calibrated_confidence.png Details</summary>  ### Visual Description ## Chart Type: Calibration Comparison Chart ### Overview The image presents two bar charts side-by-side, comparing the accuracy of a model before and after calibration. The left chart displays the accuracy before calibration, while the right chart shows the accuracy after calibration. Both charts plot accuracy against confidence, with a dashed line representing perfect calibration. ### Components/Axes * **Titles:** * Left Chart: "Before Calibration" (top-left) * Right Chart: "After Calibration" (top-left) * **X-axis (Confidence):** * Label: "Confidence" (bottom) * Scale: 0 to 1, with markers at 0, 0.5, and 1. * **Y-axis (Accuracy):** * Label: "Accuracy" (left side) * Scale: 0 to 1, with markers at 0, 0.2, 0.4, 0.6, 0.8, and 1. * **Data Representation:** * Bars: Represent the accuracy at different confidence levels. * Left Chart: Blue bars * Right Chart: Green bars * Dashed Line: Represents perfect calibration (where accuracy equals confidence). * Left Chart: Red dashed line * Right Chart: Purple dashed line ### Detailed Analysis **Left Chart: Before Calibration** * **Bars (Blue):** * Confidence ~0: Accuracy ~0.1 * Confidence ~0.1: Accuracy ~0.3 * Confidence ~0.2: Accuracy ~0.4 * Confidence ~0.9: Accuracy ~0.5 * Confidence ~1: Accuracy ~0.8 * **Dashed Line (Red):** * Starts at (0,0) and ends at (1,1), indicating perfect calibration. **Right Chart: After Calibration** * **Bars (Green):** * Confidence ~0.2: Accuracy ~0.1 * Confidence ~0.4: Accuracy ~0.3 * Confidence ~0.6: Accuracy ~0.5 * Confidence ~0.9: Accuracy ~0.8 * **Dashed Line (Purple):** * Starts at (0,0) and ends at (1,1), indicating perfect calibration. ### Key Observations * **Before Calibration:** The model appears to be overconfident, especially at higher confidence levels. The blue bars deviate significantly from the red dashed line. * **After Calibration:** The green bars are closer to the purple dashed line, indicating improved calibration. The model's confidence is now more aligned with its actual accuracy. * The bars in the "After Calibration" chart are generally closer to the dashed line than the bars in the "Before Calibration" chart. ### Interpretation The charts demonstrate the effect of calibration on a model's confidence scores. Before calibration, the model's confidence was not well-aligned with its accuracy, leading to overconfidence. After calibration, the model's confidence scores are better aligned with its accuracy, resulting in a more reliable and trustworthy model. The calibration process has successfully adjusted the model's confidence to better reflect its true performance. </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 i’th 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: ${\max_{p∈[0,1]}\left\lvert\mathbb{P}[Y=\hat{Y}\mid\hat{P}=p]-p\right\rvert,}$ which is also challenging to compute exactly, like the ECE. As a result, we can estimate an upper bound, given by: ${\mathrm{MCE}=\max_{m∈\{1,...,M\}}\left\lvert\mathrm{acc}(B_{m})-\mathrm{% conf}(B_{m})\right\rvert,}$ 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 network’s 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 model’s 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 $f(x_{i})$ —such as the confidence estimate provided by the aforementioned UQ methods—aligns 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: ${\mathbb{P}(x_{n+1}∈ C(x_{n+1})|\mathcal{D}_{\text{cal}})≥ 1-\varepsilon(% \delta),}$ 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 model’s 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 LLM’s 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 model’s 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 LLM’s 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., “the cat was lost after leaving the house” 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 LLM’s 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 LLM’s outputs for the presence of epistemic linguistic uncertainty markers and the consistency of the LLM’s 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 LLM’s tokens increases when an LLM is given jailbreak prompts. However, the LLM’s 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 LLM’s 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 LLM’s 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 ## Diagram: AI Response Flow ### Overview The image depicts a flow diagram representing an AI's response to a question, including confidence and hallucination scores. The diagram consists of three rectangular blocks connected by downward-pointing arrows. The first block contains the initial question, the second block contains the AI's answer, the third block contains the token-level confidence estimate, and the fourth block contains the hallucination score. ### Components/Axes * **Block 1 (Top-Right):** Blue rectangle containing the question "What is the most smallest country in Asia, by land area?". A silhouette of a person's head is present to the right of the block. * **Block 2 (Top-Left):** Blue rectangle containing the AI's response "Nepal is the smallest country in Asia, by land area.". A green icon with a white "88" is present to the left of the block. * **Arrow 1:** Black downward-pointing arrow connecting Block 2 to Block 3. * **Block 3 (Center):** Pink rectangle containing "Token-Level Confidence Estimate: 13%". * **Arrow 2:** Black downward-pointing arrow connecting Block 3 to Block 4. * **Block 4 (Bottom):** Peach rectangle containing "Hallucination Score: 80%". ### Detailed Analysis * **Question:** The initial question posed to the AI is "What is the most smallest country in Asia, by land area?". * **AI Response:** The AI responds with "Nepal is the smallest country in Asia, by land area.". * **Token-Level Confidence Estimate:** The AI's token-level confidence estimate for its response is 13%. * **Hallucination Score:** The AI's hallucination score for its response is 80%. ### Key Observations * The AI provides a definitive answer to the question. * The token-level confidence estimate is relatively low (13%). * The hallucination score is high (80%). ### Interpretation The diagram illustrates a scenario where an AI provides an answer to a question but expresses low confidence and a high likelihood of hallucination. This suggests that while the AI provides an answer, it may not be accurate or reliable. The high hallucination score indicates that the AI may be generating information that is not based on factual data or evidence. The low confidence score further reinforces the unreliability of the response. The AI's response that "Nepal is the smallest country in Asia" is factually incorrect. The smallest country in Asia by land area is the Maldives. This confirms the high hallucination score. </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 ## Task Execution Diagram: Robotic Table Clearing ### Overview The image depicts a task execution diagram for a robot clearing a table. It shows a robotic arm interacting with objects on a table, followed by a question prompt and a list of possible actions with associated confidence scores. ### Components/Axes * **Image:** A photograph showing a robotic arm positioned above a table with bowls. The table has three drawers. * **Arrow:** A downward-pointing arrow indicating a progression or flow. * **Question Prompt:** A blue speech bubble containing the text "How do I clear the table?". A silhouette of a person is present to the right of the speech bubble. * **Action List:** A blue rounded rectangle containing a list of actions, each preceded by a confidence score (a decimal number). * **Icon:** A green circle with a white infinity-like symbol inside. ### Detailed Analysis * **Image Details:** The robotic arm is white and appears to be holding an object. The table is gray with three drawers. There are two bowls on the table, one metal and one ceramic. * **Question Prompt:** The question "How do I clear the table?" suggests the diagram is part of an interactive system where a user is seeking guidance. * **Action List:** The list provides possible actions the robot could take, along with associated confidence scores. The actions are: * **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):** Move the table to the living room. * **0.06 (Red):** Remove the top drawer. ### Key Observations * The confidence scores vary significantly, with moving the ceramic bowl into the top drawer having the highest confidence (0.51). * The actions involve manipulating objects on the table and potentially moving the table itself. * The color of the confidence score is different for each action. ### Interpretation The diagram represents a decision-making process for a robot tasked with clearing a table. The robot is presented with a question, "How do I clear the table?", and the diagram provides a set of possible actions with associated confidence scores. The confidence scores likely reflect the robot's assessment of the feasibility and appropriateness of each action based on its perception of the environment and its programming. The higher the confidence score, the more likely the robot is to perform that action. The diagram suggests a system where the robot can provide multiple options, allowing for flexibility in task execution. The user's silhouette suggests that this is an interactive system where a human is involved in the decision-making process. </details> Figure 17. Robotics applications utilize UQ methods to estimate the LLM’s 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 LLM’s confidence in the sub-tasks proposed by the LLM, to determine when human assistance is required.]Robotics applications utilize UQ methods to estimate the LLM’s 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 LLM’s 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: “What happened to Google in June 2007, in a single sentence?” 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 model’s 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 LLM’s 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 ## Diagram: LLM Fact-Checking Flow ### Overview The image is a diagram illustrating the process of fact-checking the response of a Large Language Model (LLM) to a specific question. The diagram shows the flow from the initial question to the LLM, the generated responses, a consistency estimate, and the final fact-check result. ### Components/Axes * **Top:** Blue rounded rectangle containing the question: "What happened to Google in June 2007?" A silhouette of a person's head is present to the right of the question. * **Arrow 1:** A downward-pointing arrow connects the question to the LLM. * **LLM:** A light green square labeled "LLM" with a knot-like symbol inside. * **Arrow 2:** A downward-pointing arrow connects the LLM to the randomly-generated responses. * **Randomly-Generated Responses:** A light blue rounded rectangle labeled "Randomly-Generated Responses" containing two example responses: * "In June 2007, Google introduced Android, its mobile operating system." * "Google launched its open-source mobile operating system Android in June 2007." * An ellipsis "..." is present between the two responses. * **Arrow 3:** A downward-pointing arrow connects the randomly-generated responses to the consistency estimate. * **Consistency Estimate:** A pink rounded rectangle labeled "Consistency Estimate: 99%". * **Arrow 4:** A downward-pointing arrow connects the consistency estimate to the fact-check result. * **Fact-Check:** A red "X" symbol and the text "Fact-Check: False". ### Detailed Analysis The diagram depicts a sequence of steps: 1. A question is posed: "What happened to Google in June 2007?". 2. The question is fed to an LLM. 3. The LLM generates multiple responses. Two example responses are provided, both stating that Google introduced or launched Android in June 2007. 4. A consistency estimate is calculated, resulting in 99%. 5. A fact-check is performed, and the result is "False". ### Key Observations * The LLM provides consistent responses (99% consistency). * Despite the high consistency, the fact-check indicates that the responses are false. ### Interpretation The diagram illustrates a scenario where an LLM can generate consistent but incorrect information. The high consistency estimate suggests that the LLM is confident in its answer, but the fact-check reveals that the information is inaccurate. This highlights the importance of fact-checking LLM outputs, even when the model exhibits high consistency in its responses. The diagram suggests that consistency alone is not a reliable indicator of accuracy. </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 ## Chatbot Response Analysis ### Overview The image shows a chatbot-like interaction where a question is posed, a response is given, a confidence estimate is provided, and a fact-check result is displayed. The question is "What is the most populous country in the world in 2024?". The response is "The United States of America with a population of 345,426,571." The confidence estimate is 91%, and the fact-check result is "False". ### Components/Axes * **Question:** "What is the most populous country in the world in 2024?" (Contained in a blue speech bubble at the top) * **Response:** "The United States of America with a population of 345,426,571." (Contained in a blue speech bubble in the middle) * Visual representation: There are 10 blocks above the response. 9 are green and 1 is yellow. * **Confidence Estimate:** "Confidence Estimate from Token Probability: 91%" (Contained in a pink rectangle below the response) * **Fact-Check:** "Fact-Check: False" (Located at the bottom right, accompanied by a red "X" symbol) * **Arrow:** A black arrow points from the response to the confidence estimate. * **Icons:** A person icon is in the top right corner. A "connection" icon is in the top left corner of the response. ### Detailed Analysis or ### Content Details * **Question:** The question is straightforward, asking for the most populous country in 2024. * **Response:** The response claims the United States of America is the most populous country with a population of 345,426,571. * **Confidence Estimate:** The confidence estimate is relatively high at 91%, suggesting the model is quite certain of its answer. * **Fact-Check:** The fact-check result is "False", indicating the response is incorrect. * **Visual Representation:** The 10 blocks above the response are likely a visual representation of the tokens used to generate the response. The green blocks likely represent correct tokens, while the yellow block likely represents an incorrect token. ### Key Observations * The chatbot provides a confident but incorrect answer. * The high confidence estimate contrasts with the "False" fact-check result. * The visual representation of tokens suggests a mix of correct and incorrect predictions. ### Interpretation The image highlights a potential issue with large language models: they can generate confident but incorrect answers. Even with a high confidence estimate (91%), the model's response is factually wrong. This underscores the importance of fact-checking and validation mechanisms when using these models, especially in contexts where accuracy is critical. The visual representation of tokens may provide insights into the model's reasoning process and help identify potential sources of error. The yellow block may indicate a specific token that led to the incorrect conclusion. </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 LLM’s output are not necessarily aligned with the factuality of the model’s 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 LLM’s confidence estimates (Achiam et al., 2023). Further, the conditional distribution of each token might not be indicative of the factuality of an LLM’s 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 LLM’s uncertainty at each episode without considering the history of the agent’s 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 sautéing it. Note that utilizing many existing techniques for uncertainty quantification would require the assumption that the LLM’s 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 LLM’s outputs requires the consideration of the history of the agent’s 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 LLM’s 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 model’s natural-language output, relying on consistency metrics to estimate the LLM’s confidence. In contrast, mechanistic interpretability requires access to the LLM’s 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. References - (1) - Achiam et al. (2023) Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. 2023. Gpt-4 technical report. arXiv preprint arXiv:2303.08774 (2023). - Ahdritz et al. 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