Shrink the Gaussian Window

The marginal can have several peaks. Fix one observed z=xt, then shrink the noise step to see why the local reverse conditional can still become Gaussian-like.

1.25
0.120 · √h=0.346

Fixed setup: Z=Y+√h ε, ε∼N(0,1), with a multimodal pY. The exact conditional is p(y|z) ∝ exp(−(z−y)²/(2h)) pY(y); the dashed curve below is only the zeroth-order approximation N(y; z, h).

Whole distribution → local window → reverse conditional

Marginal Likelihood kernel p(z|y) Local window z ± 2√h Joint slice Exact conditional Zeroth-order N(y; z, h)

Drag the red observation line, or use the slider. When the chart is focused, the left and right arrow keys also move it. The first two rows are peak-scaled to reveal their shapes; the two conditional curves in the last row share one density scale.

Approximation gap 0.112 total variation distance
Only the conditional is becoming Gaussian. The marginal stays multimodal. A smaller h makes the likelihood kernel more local. The dashed N(y; z, h) captures the zeroth-order Gaussian shape but not the reverse mean shift. Numerical integral of exact posterior: 1.000000