## Line Chart: Cross Sections of Prior Function ### Overview The image contains two line charts displayed side-by-side. Both charts depict cross-sections of a prior function with dimension 8. The left chart shows cross-sections (x1, 0), while the right chart shows cross-sections (0, x2, 0). Each chart plots two data series: "LPN" and "Ref". Both charts share a similar U-shaped trend for the LPN series, with the Ref series showing a more linear, gradually increasing trend. ### Components/Axes * **Titles:** * Left Chart: "Cross sections (x1,0) of the prior function, Dim 8" * Right Chart: "Cross sections (0, x2,0) of the prior function, Dim 8" * **Y-Axis Labels:** * Left Chart: "Prior functions (x1,0)" * Right Chart: "Prior functions (0,x2,0)" * **X-Axis Label:** * Both Charts: "x1" * **Y-Axis Scale:** * Left Chart: 0.0 to 20.0, with increments of 2.5 * Right Chart: 0 to 25, with increments of 5 * **X-Axis Scale:** * Both Charts: -4 to 4, with increments of 1 * **Legend:** Located in the top-right corner of each chart. * Blue Line: "LPN" * Orange Line: "Ref" ### Detailed Analysis **Left Chart: Cross sections (x1,0)** * **LPN (Blue Line):** The LPN line starts at approximately (x1=-4, Prior functions (x1,0)=19.5), decreases sharply to a minimum at (x1=0, Prior functions (x1,0)=0), and then increases sharply to approximately (x1=4, Prior functions (x1,0)=16.5). * (-4, 19.5) * (-3, 7.2) * (-2, 3.0) * (-1, 1.0) * (0, 0) * (1, 1.0) * (2, 3.0) * (3, 7.0) * (4, 16.5) * **Ref (Orange Line):** The Ref line starts at approximately (x1=-4, Prior functions (x1,0)=4), decreases slightly to a minimum at (x1=0, Prior functions (x1,0)=0), and then increases gradually to approximately (x1=4, Prior functions (x1,0)=4). * (-4, 4.0) * (-3, 3.0) * (-2, 2.0) * (-1, 1.0) * (0, 0) * (1, 1.0) * (2, 2.0) * (3, 3.0) * (4, 4.0) **Right Chart: Cross sections (0, x2,0)** * **LPN (Blue Line):** The LPN line starts at approximately (x1=-4, Prior functions (0,x2,0)=24), decreases sharply to a minimum at (x1=0, Prior functions (0,x2,0)=0), and then increases sharply to approximately (x1=4, Prior functions (0,x2,0)=17). * (-4, 24) * (-3, 6) * (-2, 2) * (-1, 0.5) * (0, 0) * (1, 0.5) * (2, 2) * (3, 6) * (4, 17) * **Ref (Orange Line):** The Ref line starts at approximately (x1=-4, Prior functions (0,x2,0)=4), decreases slightly to a minimum at (x1=0, Prior functions (0,x2,0)=0), and then increases gradually to approximately (x1=4, Prior functions (0,x2,0)=4). * (-4, 4.0) * (-3, 3.0) * (-2, 2.0) * (-1, 1.0) * (0, 0) * (1, 1.0) * (2, 2.0) * (3, 3.0) * (4, 4.0) ### Key Observations * Both charts show a similar trend: the LPN line has a U-shape, indicating a minimum value around x1 = 0, while the Ref line is more linear and gradually increases from x1 = 0. * The LPN line in the right chart (cross-sections (0, x2,0)) starts at a higher value (approximately 24) compared to the left chart (cross-sections (x1,0), approximately 19.5). * The Ref lines in both charts are nearly identical. ### Interpretation The charts compare the prior functions of LPN and a reference (Ref) model across two different cross-sections (x1,0) and (0, x2,0) in an 8-dimensional space. The U-shape of the LPN line suggests that the prior function is minimized when either x1 or x2 is close to 0. The Ref line, being more linear, indicates a more uniform prior distribution across the range of x1 and x2 values. The higher starting value of the LPN line in the right chart suggests that the prior function might have a stronger dependence on x2 compared to x1 when other dimensions are set to 0. The similarity of the Ref lines in both charts indicates that the reference model's prior function behaves consistently across these two cross-sections.

## Line Chart: Cross Sections of Prior Function ### Overview The image presents two line charts, displayed side-by-side. Both charts visualize cross-sections of a prior function in 8 dimensions. The left chart shows the cross-section (x1, 0) and the right chart shows the cross-section (0, x2, 0). Each chart plots "Prior functions" against "x1" (left chart) or "x1" (right chart). Two lines are plotted on each chart, representing "LPN" and "Ref" data series. ### Components/Axes * **Left Chart:** * **Title:** "Cross sections (x1,0) of the prior function,Dim 8" (top-center) * **X-axis Label:** "x1" (bottom-center) * **Y-axis Label:** "Prior functions (x1,0)" (left-center) * **Legend:** Located in the top-right corner. * "LPN" - Blue line * "Ref" - Orange line * **Right Chart:** * **Title:** "Cross sections (0,x2,0) of the prior function,Dim 8" (top-center) * **X-axis Label:** "x1" (bottom-center) * **Y-axis Label:** "Prior functions (0,x2,0)" (left-center) * **Legend:** Located in the top-right corner. * "LPN" - Blue line * "Ref" - Orange line ### Detailed Analysis or Content Details **Left Chart:** * **LPN (Blue Line):** The line starts at approximately 18.5 at x1 = -4. It decreases to a minimum of approximately 2.5 at x1 = 0. Then, it increases to approximately 5.0 at x1 = 4. The trend is initially steeply decreasing, then increasing. * **Ref (Orange Line):** The line starts at approximately 2.5 at x1 = -4. It increases to a maximum of approximately 3.0 at x1 = 0. Then, it decreases to approximately 2.5 at x1 = 4. The trend is initially increasing, then decreasing. **Right Chart:** * **LPN (Blue Line):** The line starts at approximately 21.0 at x1 = -4. It decreases to a minimum of approximately 4.5 at x1 = 0. Then, it increases to approximately 18.0 at x1 = 4. The trend is initially steeply decreasing, then increasing. * **Ref (Orange Line):** The line starts at approximately 4.5 at x1 = -4. It increases to a maximum of approximately 5.0 at x1 = 0. Then, it decreases to approximately 4.5 at x1 = 4. The trend is initially increasing, then decreasing. ### Key Observations * Both charts exhibit a U-shaped curve for both the "LPN" and "Ref" lines. * The "LPN" line consistently has higher values than the "Ref" line in both charts, especially at the extreme ends of the x1 axis. * The minimum values of the "LPN" line are significantly lower than the maximum values of the "Ref" line. * The "Ref" line appears to be relatively flat compared to the "LPN" line. ### Interpretation The charts demonstrate the cross-sectional behavior of a prior function in 8 dimensions, comparing two different methods ("LPN" and "Ref"). The U-shaped curves suggest that the function has a minimum value around x1 = 0 in both cross-sections. The significant difference in magnitude between the "LPN" and "Ref" lines indicates that the two methods produce substantially different prior distributions. The "LPN" method appears to assign higher probability density to values further away from x1 = 0, while the "Ref" method maintains a more consistent probability density across the range of x1. The fact that the "Ref" line is relatively flat suggests a less informative prior, while the "LPN" line's more pronounced curve indicates a stronger prior belief about the distribution of the variable. The two charts, taken together, provide insight into how these two methods shape the prior distribution in different dimensions.

\n ## [Line Charts]: Cross Sections of Prior Function (Dim 8) ### Overview The image displays two side-by-side line charts comparing two functions, labeled "LPN" and "Ref," across different cross-sections of an 8-dimensional prior function. Both charts show the functions' behavior as a single variable changes while others are held at zero. The overall visual pattern is symmetric, with both functions reaching a minimum value of zero at the origin (x=0). ### Components/Axes **Left Chart:** * **Title:** `Cross sections (x₁,0) of the prior function,Dim 8` * **Y-axis Label:** `Prior functions (x₁,0)` * **X-axis Label:** `x₁` * **Y-axis Scale:** Linear, ranging from 0.0 to 20.0, with major ticks at intervals of 2.5. * **X-axis Scale:** Linear, ranging from -4 to 4, with major ticks at integer intervals. * **Legend:** Located in the top-right corner. Contains two entries: * `LPN` (blue line) * `Ref` (orange line) **Right Chart:** * **Title:** `Cross sections (0, x₂,0) of the prior function,Dim 8` * **Y-axis Label:** `Prior functions (0, x₂,0)` * **X-axis Label:** `x₁` *(Note: This label appears inconsistent with the title, which references `x₂`. The axis variable is likely intended to be `x₂` based on the title.)* * **Y-axis Scale:** Linear, ranging from 0 to 25, with major ticks at intervals of 5. * **X-axis Scale:** Linear, ranging from -4 to 4, with major ticks at integer intervals. * **Legend:** Located in the top-right corner. Contains the same two entries as the left chart: * `LPN` (blue line) * `Ref` (orange line) ### Detailed Analysis **Left Chart - Cross Section (x₁,0):** * **LPN (Blue Line):** Exhibits a pronounced, symmetric U-shape. * **Trend:** Starts very high at the left extreme, decreases rapidly to a minimum at the center, then increases rapidly to a high value at the right extreme. * **Approximate Data Points:** * At x₁ = -4: y ≈ 19.8 * At x₁ = -3: y ≈ 7.0 * At x₁ = -2: y ≈ 3.0 * At x₁ = -1: y ≈ 1.0 * At x₁ = 0: y = 0.0 * At x₁ = 1: y ≈ 1.0 * At x₁ = 2: y ≈ 3.0 * At x₁ = 3: y ≈ 7.0 * At x₁ = 4: y ≈ 16.5 * **Ref (Orange Line):** Exhibits a symmetric V-shape, appearing linear on each side of the origin. * **Trend:** Decreases linearly from the left to the origin, then increases linearly from the origin to the right. * **Approximate Data Points:** * At x₁ = -4: y ≈ 4.0 * At x₁ = 0: y = 0.0 * At x₁ = 4: y ≈ 4.0 **Right Chart - Cross Section (0, x₂,0):** * **LPN (Blue Line):** Exhibits a symmetric U-shape, similar to the left chart but with a steeper ascent. * **Trend:** Starts very high at the left extreme, decreases rapidly to a minimum at the center, then increases rapidly to a high value at the right extreme. * **Approximate Data Points:** * At x₁ (likely x₂) = -4: y ≈ 24.0 * At x₁ = -3: y ≈ 6.0 * At x₁ = -2: y ≈ 3.0 * At x₁ = -1: y ≈ 1.0 * At x₁ = 0: y = 0.0 * At x₁ = 1: y ≈ 1.0 * At x₁ = 2: y ≈ 3.0 * At x₁ = 3: y ≈ 7.0 * At x₁ = 4: y ≈ 18.5 * **Ref (Orange Line):** Exhibits a symmetric V-shape, identical in form to the left chart. * **Trend:** Decreases linearly from the left to the origin, then increases linearly from the origin to the right. * **Approximate Data Points:** * At x₁ (likely x₂) = -4: y ≈ 4.0 * At x₁ = 0: y = 0.0 * At x₁ = 4: y ≈ 4.0 ### Key Observations 1. **Symmetry:** Both functions (LPN and Ref) are perfectly symmetric around x=0 in both cross-sections. 2. **Minimum Point:** Both functions achieve their global minimum value of 0 at the origin (x=0). 3. **Relative Magnitude:** The LPN function has significantly higher values than the Ref function at all points except the origin. The disparity is greatest at the extremes (x=±4). 4. **Shape Difference:** The LPN function is a smooth, convex U-shape (suggesting a quadratic or higher-order polynomial relationship), while the Ref function is a piecewise-linear V-shape (suggesting an absolute value relationship). 5. **Cross-Section Comparison:** The LPN curve in the right chart (cross-section (0, x₂,0)) reaches a higher peak value (~24) at x=-4 compared to the left chart (~19.8), indicating the prior function may have different scaling or sensitivity along different dimensions. 6. **Label Discrepancy:** The x-axis on the right chart is labeled `x₁`, but the chart title references `x₂`. This is likely a labeling error, and the axis should represent `x₂`. ### Interpretation The charts compare the behavior of a learned or proposed prior distribution ("LPN") against a reference prior ("Ref") in an 8-dimensional space. The data suggests the following: * **Penalty for Deviation:** Both priors assign the lowest probability density (or highest "cost") to the origin (x=0) and increase as variables move away from zero. This is characteristic of priors that encourage sparsity or shrinkage towards zero. * **LPN is More "Peaked":** The LPN prior penalizes deviations from zero much more severely than the Ref prior, especially for larger deviations. Its U-shape implies a stronger, non-linear push towards zero. This could indicate a more informative or restrictive prior designed to aggressively suppress non-zero values. * **Reference Prior is Linear:** The Ref prior's V-shape corresponds to an L1-norm or Laplace prior, which applies a constant penalty per unit of deviation. This is a common choice for promoting sparsity. * **Dimensional Anisotropy:** The difference in the LPN curve's height between the two cross-sections suggests the learned prior is not isotropic; its strength or shape varies depending on which dimension is being varied. This could be an intentional feature to model different importance of dimensions or an artifact of the learning process. * **Purpose:** This visualization is likely used to validate or analyze the properties of a learned prior (LPN) by contrasting it with a standard, well-understood reference (Ref). It demonstrates that the LPN has successfully learned a prior that is qualitatively similar (symmetric, centered at zero) but quantitatively more aggressive in its shrinkage behavior.

## Line Graphs: Cross Sections of Prior Function in 8D Space ### Overview The image contains two side-by-side line graphs comparing two data series ("LPN" and "Ref") across an 8-dimensional prior function. Both graphs show cross-sectional behavior along different axes: the left graph plots (x₁,0) and the right graph plots (0,x₂,0). The graphs use a Cartesian coordinate system with x₁ as the horizontal axis and prior function values as the vertical axis. ### Components/Axes **Left Graph (x₁,0 cross-section):** - **X-axis**: x₁ values from -4 to 4 (integer increments) - **Y-axis**: Prior function values from 0 to 20 - **Legend**: - Blue line = LPN - Orange line = Ref - **Title**: "Cross sections (x₁,0) of the prior function, Dim 8" **Right Graph (0,x₂,0 cross-section):** - **X-axis**: x₁ values from -4 to 4 (integer increments) - **Y-axis**: Prior function values from 0 to 25 - **Legend**: - Blue line = LPN - Orange line = Ref - **Title**: "Cross sections (0,x₂,0) of the prior function, Dim 8" ### Detailed Analysis **Left Graph Trends:** 1. **LPN (Blue)**: - Starts at ~19.5 at x₁=-4 - Sharp decline to ~0.2 at x₁=0 - Rapid rise to ~16.5 at x₁=4 - Symmetric V-shaped pattern 2. **Ref (Orange)**: - Linear decrease from ~4 at x₁=-4 to ~0.5 at x₁=0 - Linear increase to ~4 at x₁=4 - Flat baseline with shallow slope **Right Graph Trends:** 1. **LPN (Blue)**: - Starts at ~24 at x₁=-4 - Sharp decline to ~0.1 at x₁=0 - Rapid rise to ~15 at x₁=4 - Asymmetric V-shape with steeper left descent 2. **Ref (Orange)**: - Linear decrease from ~4 at x₁=-4 to ~0.3 at x₁=0 - Linear increase to ~4 at x₁=4 - Consistent slope magnitude in both directions ### Key Observations 1. **LPN Dominance**: LPN values consistently exceed Ref values by 2-3x across all x₁ positions 2. **Symmetry Breaking**: Right graph shows LPN's right-side ascent (x₁=4) is 30% lower than left-side peak (x₁=-4) 3. **Reference Stability**: Ref maintains identical slope magnitude in both graphs despite different y-axis scales 4. **Zero-Crossing Consistency**: Both data series intersect near x₁=0 with minimal deviation (<0.5 units) ### Interpretation The graphs demonstrate fundamental differences in how LPN and Ref prior functions behave in 8D space: - **LPN's Sensitivity**: The sharp V-shaped response suggests LPN prioritizes extreme x₁ values (-4 and 4) while suppressing intermediate values - **Ref's Uniformity**: The linear reference function implies a baseline model with consistent sensitivity across all x₁ positions - **Dimensional Impact**: The right graph's reduced right-side peak (15 vs 24) indicates dimensional interactions may asymmetrically affect prior function behavior - **Practical Implications**: LPN's extreme-value emphasis could represent a risk-averse modeling approach, while Ref's linearity might reflect a neutral baseline for comparison All textual elements have been extracted with positional accuracy. No non-English text was present in the image.