## pseudo slice sampling

**T**he workshop in Warwick last week made me aware of (yet) another arXiv posting I had missed: Pseudo-marginal slice sampling by Iain Murray and Matthew Graham. The idea is to mix the pseudo-marginal approach of Andrieu and Roberts (2009) with a noisy slice sampling scheme à la Neal (2003). The auxiliary random variable **u** used in the (pseudo-marginal) unbiased estimator of the target I(θ), Î(θ,**u**), and with distribution q(**u**) is merged with the random variable of interest so that the joint is

Î(θ,**u**)q(**u**)/C

and a Metropolis-Hastings proposal on that target simulating from k(θ,θ’)q(**u’**) *[meaning the auxiliary is simulated independently]* recovers the pseudo-marginal Metropolis-Hastings ratio

Î(θ’,**u**‘)k(θ’,θ)/Î(θ,**u**)k(θ,θ’)

(which is a nice alternative proof that the method works!). The novel idea in the paper is that the proposal on the auxiliary **u** can be of a different form, while remaining manageable. For instance, as a two-block Gibbs sampler. Or an elliptical slice sampler for the **u** component. The argument being that an independent update of **u** may lead the joint chain to get stuck. Among the illustrations in the paper, an Ising model (with no phase transition issue?) and a Gaussian process applied to the Pima Indian data set (despite a recent prohibition!). From the final discussion, I gather that the modification should be applicable to every (?) case when a pseudo-marginal approach is available, since the auxiliary distribution q(**u**) is treated as a black box. Quite an interesting read and proposal!

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