## 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.