Hit-and-run sampler

The hit-and-run sampler moves within a full-dimensional compact convex set by choosing a uniform direction, then a uniform point on the chord through the current point in that direction. Its invariant law is the uniform distribution on the set. Starting in the interior keeps the chain in the interior almost surely. To define absolutely continuous transitions also at boundary points, redraw directions whose chords have zero length. Without this convention, a corner can have a positive holding atom.

For a planar hit-and-run sampler, let $\ell (x,u)$ be the chord length and $a(x)$ the fraction of directions with positive chord length. With zero-length chords redrawn, the probability density function isfor almost every $y$ in the body. Interior points have $a(x)=1$. The two signed representations of a direction contribute the factor two in the polar-coordinate calculation. If the body has diameter $D$, a version of this density is bounded below by $1/(\pi D^2)$. This is a global Doeblin condition after normalization by the body's area.