Slow hit-and-run sampling
Consider a probability measure \(\pi\) on \(\mathbb{R}^d\). Assume that \(\pi\) is absolutely continuous and has density \(f\) with respect to the Lebesgue measure. Let \(S_d= \{\theta\in \mathbb{R}^d:\|\theta\|= 1\}\) be the \(d\)-dimensional unit sphere centered at the origin. For every \(\theta\in S_d\), let \(P_{(\theta, \pi)}\) be the Markov kernel on \((\mathbb{R}^d,{\mathcal B}^d)\) defined by \[ P_{(\theta,\pi)}(x, B)= {\int^\infty_{-\infty} 1_B(x+ r\theta) f(x+ r\theta) dr\over \int^\infty_{-\infty} f(x+ \lambda\theta) d\lambda}, \] and for any probability measure \(\upsilon\) on \(S_d\), let \(P_{(\upsilon,\pi)}\) denote the Markov kernel on \((\mathbb{R}^d,{\mathcal B}^d)\) defined by \[ P_{(\upsilon,\pi)}(x, B)= \int_{S^d} P_{(\theta,\pi)} (x,B)\upsilon(d\theta). \] The latter is called the hit-and-run sampler with direction distribution \(\upsilon\) and target distribution \(\pi\). It is shown that the hit-and-run sampler can converge to its target distribution at an arbitrarily slow rate. It is also illustrated how the speed of convergence of the hit-and-run sampler can be affected by small perturbations of the target distribution.
scientific article; zbMATH DE number 1428604