(Independent Metropolis-Hastings sampler). We consider the Independent Metropolis-Hastings algorithm (see Example 5.38) Let µ be a σ-finite measure on (X, X ). Let π denote the density with respect to...

(Independent Metropolis-Hastings sampler). We consider the Independent Metropolis-Hastings algorithm (see Example 5.38) Let µ be a σ-finite measure on (X, X ). Let π denote the density with respect to µ of the target distribution and q the proposal density. Assume that supx∈X π(x)/q(x)

(a) Show that the transition kernel P is reversible with respect to π and that π is a stationary distribution for P.

(b) Assume now that there exists η > 0 such that for all x ∈ X q (x) ≥ ηπ (x). Show that the kernel is uniformly ergodic.