## Line Chart: Log Probability Difference vs. Number of Layers ### Overview The image is a line chart that plots the difference between the absolute value of the log probability of AR data and the average log probability of AR data, normalized by N, against the number of layers, l. There are four data series, each representing a different value of L and N. The chart includes a legend, axis labels, and gridlines. The data series all show a decreasing trend as the number of layers increases. ### Components/Axes * **Title:** None explicitly present, but the chart is labeled "(b)" in the top-center. * **X-axis:** "Number of layers, l". The axis ranges from 2 to 10, with tick marks at every increment of 2 (2, 4, 6, 8, 10). * **Y-axis:** "(|⟨logP_AR^l⟩_data - ⟨logP_AR¹⁰⟩_data|)/N". The axis ranges from 0.000 to 0.006, with tick marks at every increment of 0.001 (0.000, 0.001, 0.002, 0.003, 0.004, 0.005, 0.006). * **Legend:** Located in the top-right corner. It identifies the four data series: * Dark Blue: L = 12, N = 144 * Dark Gray: L = 16, N = 256 * Olive Green: L = 20, N = 400 * Yellow: L = 24, N = 576 ### Detailed Analysis * **Dark Blue Line (L = 12, N = 144):** This line starts at approximately (2, 0.006) and decreases to approximately (10, 0.000). * (2, 0.006) * (4, 0.003) * (6, 0.0008) * (8, 0.0003) * (10, 0.000) * **Dark Gray Line (L = 16, N = 256):** This line starts at approximately (2, 0.005) and decreases to approximately (10, 0.000). * (2, 0.005) * (4, 0.0018) * (6, 0.0007) * (8, 0.0002) * (10, 0.000) * **Olive Green Line (L = 20, N = 400):** This line starts at approximately (2, 0.0047) and decreases to approximately (10, 0.000). * (2, 0.0047) * (4, 0.0013) * (6, 0.0005) * (8, 0.0001) * (10, 0.000) * **Yellow Line (L = 24, N = 576):** This line starts at approximately (2, 0.0045) and decreases to approximately (10, 0.000). * (2, 0.0045) * (4, 0.0009) * (6, 0.0004) * (8, 0.0001) * (10, 0.000) Each line has a shaded region around it, indicating some form of uncertainty or standard deviation. ### Key Observations * All four lines exhibit a decreasing trend. As the number of layers (l) increases, the value of (|⟨logP_AR^l⟩_data - ⟨logP_AR¹⁰⟩_data|)/N decreases. * The dark blue line (L = 12, N = 144) consistently has the highest values across the range of the x-axis. * The yellow line (L = 24, N = 576) consistently has the lowest values across the range of the x-axis. * The lines converge towards 0 as the number of layers approaches 10. ### Interpretation The chart suggests that as the number of layers (l) increases, the difference between the log probability of the AR data at layer l and the log probability at layer 10 decreases. This implies that with more layers, the model's performance converges towards a stable state, represented by the 10th layer. The different values of L and N seem to influence the initial difference, with lower L and N resulting in a higher initial difference. The convergence towards zero suggests that increasing the number of layers beyond a certain point (likely around 10) may not significantly improve the model's performance, as the difference becomes negligible. The shaded regions around each line indicate the variability or uncertainty in the data, which decreases as the number of layers increases, further supporting the idea of convergence.

## Chart: Difference in Log Probabilities ### Overview The image presents a chart illustrating the difference in log probabilities as a function of the number of layers. The chart displays four data series, each representing a different set of parameters (L and N values). Shaded regions represent confidence intervals around each data series. ### Components/Axes * **X-axis:** Number of layers, denoted as 'ℓ' (ell). Scale ranges from approximately 2 to 10. * **Y-axis:** ⟨⟨logP^ℓ_AR⟩_data - ⟨logP_AR⟩_data⟩/N. Scale ranges from approximately 0.000 to 0.006. * **Legend:** Located in the top-right corner. Contains the following data series labels: * L = 12, N = 144 (Dark Blue) * L = 16, N = 256 (Dark Grey) * L = 20, N = 400 (Light Grey) * L = 24, N = 576 (Yellow) * **Title:** "(b)" located in the top-left corner. ### Detailed Analysis The chart shows a decreasing trend for all data series as the number of layers increases. Each series is represented by a set of data points connected by a line, with a shaded area indicating the uncertainty or confidence interval. * **L = 12, N = 144 (Dark Blue):** The line slopes downward, starting at approximately 0.0055 at ℓ = 2 and decreasing to approximately 0.0002 at ℓ = 9. Data points are located at approximately: (2, 0.0055), (4, 0.003), (6, 0.001), (8, 0.0005), (9, 0.0002). * **L = 16, N = 256 (Dark Grey):** The line also slopes downward, starting at approximately 0.005 at ℓ = 2 and decreasing to approximately 0.0001 at ℓ = 9. Data points are located at approximately: (2, 0.005), (4, 0.0025), (6, 0.0008), (8, 0.0003), (9, 0.0001). * **L = 20, N = 400 (Light Grey):** This line exhibits a similar downward trend, beginning at approximately 0.0045 at ℓ = 2 and reaching approximately 0.00005 at ℓ = 9. Data points are located at approximately: (2, 0.0045), (4, 0.002), (6, 0.0006), (8, 0.0002), (9, 0.00005). * **L = 24, N = 576 (Yellow):** The line shows a downward trend, starting at approximately 0.004 at ℓ = 2 and decreasing to approximately 0.00002 at ℓ = 9. Data points are located at approximately: (2, 0.004), (4, 0.0018), (6, 0.0005), (8, 0.00015), (9, 0.00002). The shaded regions around each line are wider at lower values of ℓ and become narrower as ℓ increases, indicating decreasing uncertainty with more layers. ### Key Observations * All data series demonstrate a consistent downward trend. * The differences between the series are more pronounced at lower values of ℓ. * The confidence intervals narrow as the number of layers increases, suggesting greater certainty in the results with more layers. * The series with L=12, N=144 consistently shows the highest values, while the series with L=24, N=576 consistently shows the lowest values. ### Interpretation The chart suggests that as the number of layers increases, the difference in log probabilities decreases. This could indicate that the model is converging or becoming more stable with more layers. The varying values for different L and N combinations suggest that the optimal number of layers may depend on the specific parameters used. The narrowing confidence intervals with increasing layers imply that the model's behavior becomes more predictable as the number of layers grows. The consistent ordering of the series (L=12 being highest, L=24 being lowest) suggests a relationship between the L and N parameters and the magnitude of the log probability difference. The "(b)" label suggests this is part of a larger figure with other related data. The y-axis represents a normalized difference, implying that the raw log probability values are being compared relative to the number of data points (N).

## Line Chart with Shaded Regions: Autoregressive Probability Deviation vs. Number of Layers ### Overview The image is a scientific line chart, labeled "(b)" in the top-left corner, plotting a normalized difference in log-probability against the number of layers (`ℓ`). It compares four different system configurations, each represented by a distinct color and shaded region. The chart demonstrates a clear decreasing trend for all series as the number of layers increases. ### Components/Axes * **Chart Label:** "(b)" - positioned in the upper-left quadrant of the plot area. * **X-Axis:** * **Title:** "Number of layers, ℓ" * **Scale:** Linear, with major tick marks and labels at 2, 4, 6, 8, and 10. * **Y-Axis:** * **Title:** `( (⟨logP_AR^ℓ⟩_data - ⟨logP_AR^10⟩_data) ) / N` * **Scale:** Linear, ranging from 0.000 to 0.006, with major tick marks at intervals of 0.001. * **Legend:** Positioned in the top-right corner of the plot area. It contains four entries, each with a colored circle marker and a label: 1. **Dark Blue Circle:** `L = 12, N = 144` 2. **Gray Circle:** `L = 16, N = 256` 3. **Olive/Brown Circle:** `L = 20, N = 400` 4. **Yellow Circle:** `L = 24, N = 576` * **Data Series:** Each series consists of a line connecting circular data points at integer `ℓ` values (2, 3, 4, 5, 6, 8, 10) and a semi-transparent shaded region of the same color surrounding the line, likely representing a confidence interval or standard deviation. ### Detailed Analysis **Trend Verification:** All four data series exhibit a consistent, monotonic downward trend. The lines slope steeply downward from `ℓ=2` to `ℓ=4` and then continue to decrease at a diminishing rate, approaching zero as `ℓ` approaches 10. **Data Point Extraction (Approximate Y-values):** * **`L=12, N=144` (Dark Blue):** * `ℓ=2`: ~0.0058 * `ℓ=3`: ~0.0030 * `ℓ=4`: ~0.0017 * `ℓ=5`: ~0.0011 * `ℓ=6`: ~0.0008 * `ℓ=8`: ~0.0003 * `ℓ=10`: ~0.0000 * **`L=16, N=256` (Gray):** * `ℓ=2`: ~0.0050 * `ℓ=3`: ~0.0022 * `ℓ=4`: ~0.0012 * `ℓ=5`: ~0.0007 * `ℓ=6`: ~0.0004 * `ℓ=8`: ~0.0001 * `ℓ=10`: ~0.0000 * **`L=20, N=400` (Olive):** * `ℓ=2`: ~0.0048 * `ℓ=3`: ~0.0022 * `ℓ=4`: ~0.0012 * `ℓ=5`: ~0.0006 * `ℓ=6`: ~0.0004 * `ℓ=8`: ~0.0001 * `ℓ=10`: ~0.0000 * **`L=24, N=576` (Yellow):** * `ℓ=2`: ~0.0045 * `ℓ=3`: ~0.0019 * `ℓ=4`: ~0.0010 * `ℓ=5`: ~0.0005 * `ℓ=6`: ~0.0003 * `ℓ=8`: ~0.0001 * `ℓ=10`: ~0.0000 **Shaded Regions:** The width of the shaded regions (uncertainty/variance) is largest at low `ℓ` values and narrows significantly as `ℓ` increases. The dark blue series (`L=12`) has the widest shaded region, while the yellow series (`L=24`) has the narrowest. ### Key Observations 1. **Universal Decay:** The quantity on the y-axis decays towards zero for all configurations as the number of layers (`ℓ`) increases towards 10. 2. **Initial Hierarchy:** At `ℓ=2`, the series are ordered from highest to lowest y-value as: Dark Blue (`L=12`) > Gray (`L=16`) ≈ Olive (`L=20`) > Yellow (`L=24`). This suggests that smaller systems (lower `L` and `N`) exhibit a larger initial deviation. 3. **Convergence:** By `ℓ=10`, all series converge to approximately the same value (zero). The differences between the series become negligible after `ℓ=6`. 4. **Uncertainty Correlation:** The magnitude of the y-value and the width of the shaded uncertainty region are positively correlated. The series with the highest values also shows the greatest variance. ### Interpretation This chart likely illustrates the convergence behavior of an autoregressive model's probability estimates as a function of its depth (number of layers `ℓ`). The y-axis represents a normalized difference between the average log-probability at layer `ℓ` and at a reference layer (10), scaled by system size `N`. The data suggests that: * **Layer-wise Refinement:** The model's internal probability estimates undergo significant adjustment in the early layers (steep drop from `ℓ=2` to `4`), with diminishing changes in deeper layers. This implies the core "computation" or "refinement" of the probability distribution happens early in the network. * **System Size Effect:** Larger systems (higher `L` and `N`, e.g., yellow line) start with a smaller deviation from the final (layer 10) probability estimate. This could indicate that larger models are more stable or require less adjustment across layers. * **Convergence to a Stable State:** The convergence of all lines to zero at `ℓ=10` confirms that layer 10 is being used as the reference point. The narrowing shaded regions indicate that the model's behavior becomes more deterministic and consistent across different samples or runs in the deeper layers. * **Peircean Reading:** The chart demonstrates a **law of diminishing returns** in the context of neural network depth. Adding more layers beyond a certain point (here, around `ℓ=6`) yields minimal change in this specific probability metric. The initial conditions (system size `L, N`) affect the starting point of this decay curve but not its ultimate convergence point, highlighting a fundamental property of the model's architecture or training.

## Line Graph: Convergence of ⟨logPℓ_AR⟩ with Number of Layers ### Overview The graph illustrates the convergence behavior of the average absolute difference between ⟨logPℓ_AR⟩_data and ⟨logPℓ_AR⟩_data across varying numbers of layers (ℓ). Four data series are plotted, each corresponding to different combinations of model depth (L) and sample size (N). The y-axis represents the normalized absolute difference, while the x-axis shows the number of layers. ### Components/Axes - **X-axis**: "Number of layers, ℓ" (integer values from 2 to 10) - **Y-axis**: "(|⟨logPℓ_AR⟩_data − ⟨logPℓ_AR⟩_data|)/N" (normalized absolute difference, logarithmic scale from 0.000 to 0.006) - **Legend**: Located in the top-right corner, with four entries: - Dark blue: L = 12, N = 144 - Gray: L = 16, N = 256 - Olive: L = 20, N = 400 - Yellow: L = 24, N = 576 - **Shaded Regions**: Confidence intervals or error margins around each data series. ### Detailed Analysis 1. **L = 12, N = 144 (Dark Blue)**: - Data points at ℓ = 2, 3, 4, 5, 6, 7, 8. - Values: 0.006 (ℓ=2), 0.003 (ℓ=3), 0.0018 (ℓ=4), 0.0012 (ℓ=5), 0.0009 (ℓ=6), 0.0007 (ℓ=7), 0.0002 (ℓ=8). - Trend: Steepest decline, with values dropping by ~50% between ℓ=2 and ℓ=3, then gradually flattening. 2. **L = 16, N = 256 (Gray)**: - Data points at ℓ = 2, 3, 4, 5, 6, 7, 8, 9, 10. - Values: 0.005 (ℓ=2), 0.0025 (ℓ=3), 0.0015 (ℓ=4), 0.001 (ℓ=5), 0.0008 (ℓ=6), 0.0006 (ℓ=7), 0.0004 (ℓ=8), 0.0003 (ℓ=9), 0.0002 (ℓ=10). - Trend: Smoother decline, with ~50% reduction between ℓ=2 and ℓ=3, followed by slower convergence. 3. **L = 20, N = 400 (Olive)**: - Data points at ℓ = 2, 3, 4, 5, 6, 7, 8, 9, 10. - Values: 0.0045 (ℓ=2), 0.002 (ℓ=3), 0.001 (ℓ=4), 0.0008 (ℓ=5), 0.0006 (ℓ=6), 0.0005 (ℓ=7), 0.0004 (ℓ=8), 0.0003 (ℓ=9), 0.0002 (ℓ=10). - Trend: Moderate decline, with ~55% reduction between ℓ=2 and ℓ=3, then gradual flattening. 4. **L = 24, N = 576 (Yellow)**: - Data points at ℓ = 2, 3, 4, 5, 6, 7, 8, 9, 10. - Values: 0.004 (ℓ=2), 0.0018 (ℓ=3), 0.001 (ℓ=4), 0.0007 (ℓ=5), 0.0005 (ℓ=6), 0.0004 (ℓ=7), 0.0003 (ℓ=8), 0.0002 (ℓ=9), 0.0001 (ℓ=10). - Trend: Fastest initial decline (~55% between ℓ=2 and ℓ=3), followed by steady convergence. ### Key Observations - **Universal Decline**: All series show a decreasing trend as ℓ increases, indicating improved convergence with more layers. - **Rate of Convergence**: Larger L and N values (e.g., L=24, N=576) achieve lower absolute differences faster than smaller configurations. - **Shaded Regions**: Narrower error margins at higher ℓ values suggest increased precision with deeper models. - **ℓ=10 Asymptote**: All series approach ~0.0001–0.0002 at ℓ=10, implying diminishing returns beyond this point. ### Interpretation The graph demonstrates that increasing the number of layers (ℓ) reduces the discrepancy between ⟨logPℓ_AR⟩_data and ⟨logPℓ_AR⟩_data, with larger models (higher L and N) achieving faster convergence. The shaded regions highlight that uncertainty decreases as ℓ grows, reinforcing the reliability of deeper models. The rapid initial decline (ℓ=2 to ℓ=3) across all series suggests that early layers have the most significant impact on convergence. The flattening trend at higher ℓ values implies diminishing returns, where additional layers contribute minimally to further improvement. This could inform model design choices, balancing computational cost (larger L/N) against performance gains.