**G**eorge Deligiannidis, Arnaud Doucet and Sylvain Rubenthaler have constructed a form of Rao-Blackwellised estimate out of a regular rejection sampler. Doubly surprisingly as turning importance sampling into regular sampling plus gaining over the standard accept-reject estimate. They call their approach ensemble rejection sampling. This is done by seeing the N-sample created from the proposal as an importance sampler, exploiting the importance weights towards estimating the (intractable) normalising constant of the target density, and creating an upper bound on this estimate Ẑ. That depends on the current value X from the N-sample under consideration for acceptance as

Z⁺=Ẑ+{max(w)-w(X)}/N

with a probability Ẑ/Z⁺ to accept X. The amazing result is that the X thus marginaly produced is distributed from the target! Meaning that this is a case for a self-normalised importance sampling distribution producing an exact simulation from the target. While this cannot produce an iid sample, it can be exploited to produce unbiased estimators of expectations under the target. Without even resampling and at a linear cost in the sample size N.

The method can be extended to the dynamic (state-space) case. At a cost of O(N²T) as first observed by Radford Neal. However, the importance sample seems to be distributed from a product of proposals that do not account for the previous particles. But maybe accounting for the observations. While the result involves upper bounds on the dynamic importance weights, the capacity to deliver exact simulations remains a major achievement, in my opinion.

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