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Consider the same situation as in importance sampling, where we have an unnormalized measure (X) from which it is hard to sample, and a proposal distribution Q which is (hopefully) close to the normalized distribution P ∝ , from which we can draw independent samples. Consider a Markov chain where we define

for x’ ≠ x, where w(x) is as defined; we define  Intuitively, the transition from x to x 0 selects an independent sample x 0 from Q and then moves toward it, depending on whether its importance weight is better than that of our current point x.

a. Show that T defines a legal transition model.

b. Show that P is the stationary distribution of T.