## independent Metropolis-Hastings

“In this paper we have demonstrated the potential benefits, both theoretical and practical, of the independence sampler over the random walk Metropolis algorithm.”

**P**eter Neal and Tsun Man Clement Lee arXived a paper on optimising the independent Metropolis-Hastings algorithm. I was a bit surprised at this “return” of the independent sampler, which I hardly mention in my lectures, so I had a look at the paper. The goal is to produce an equivalent to what Gelman, Gilks and Wild (1996) obtained for random walk samplers. In the formal setting when the target is a product of n identical densities f, the optimal number k of components to update in one Metropolis-Hastings (within Gibbs) round is approximately 2.835/I, where I is the symmetrised Kullback-Leibler distance between the (univariate) target f and the independent proposal q. When I is finite. The most surprising part is that the optimal acceptance rate is again 0.234, as in the random walk case. This is surprising in that I usually associate the independent Metropolis-Hastings algorithm with high acceptance rates. But this is of course when calibrating the proposal q, not the block size k of the Gibbs part. Hence, while this calibration of the independent Metropolis-within-Gibbs sampler is worth the study and almost automatically applicable, it remains that it only applies to a certain category of problems where blocking can take place. As in the disease models illustrating the paper. And requires an adequate choice of proposal distribution for, otherwise, the above quote becomes inappropriate.

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