Claude Belisle - The Hit-and-run sampler

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Claude Belisle - The Hit-and-run sampler

CLAUDE BELISLE, Département de mathématiques et statistique, Université de Laval, Montréal, Québec G1K 7P4, Canada

The Hit-and-run sampler

The hit-and-run sampler is a Markov Chain Monte Carlo method for simulating probability measures.

Let $\pi$ be an absolutely continuous probability measure on ${\bbd R}^d$ . Let $\nu$ be a full dimensional probability measure on the surface S of the d-dimensional unit ball centered at the origin. Given a current point $X_n \in {\bbd R}^d$ , the hit-and-run sampler chooses a next point X_n+1 according to the conditionalization of $\pi$ on the line through X_n and $X_n + \Theta_{n+1}$ . The directions $\Theta_1, \Theta_2, \Theta_3,\dots$ are independent and identically distributed on S, with distribution $\nu$ . Under an appropriate irreducibility condition, the Markov chain $(X_n; n \ge 0)$ converges in total variation towards the target distribution $\pi$ . In this talk, I will discuss the convergence properties of this Markov chain. Related Markov Chain Monte Carlo methods, including the Gibbs sampler, will also be discussed.

Next: David Brillinger - Some Up: Probability Theory / Théorie Previous: Siva Athreya - Existence