Joint posterior p(β, σ² | y, X) — Case 5.4 Bayesian Linear Regression

Interactive 3D surface with horizontal and vertical slices. Drag the 3D plot to rotate. Use the slider to move the slice.

Fix σ² = 0.73
Horizontal slice at σ² = 0.73 Cut the 3D mountain at this σ² level. The red line on the surface IS the cross-section. The chart above shows its shape — a Normal bell for β. Smaller σ² = narrower bell (less noise = more confident about β).
Reading the plot: The 3D surface height = probability density p(β, σ² | y, X). A horizontal slice (fix σ², vary β) cuts a bell-shaped cross-section → Normal for β. A vertical slice (fix β, vary σ²) cuts a skewed cross-section → Inv-Gamma for σ². Integrating out σ² from the full surface → Student-t marginal for β.