Gabriel Cardoso and coauthors (among whom Éric Moulines and Achille Thin, with whom I collaborated on the inFINE/NEO algorithm), have arXived a nice entry on a cheap way to reduce bias in the famously biased self-normalised importance sampling estimator. Which is a standard solution when the target density is not normalised. They reconsider a 2004 technical paper by Tjemeland—I remember reading at the time—, which constructs a sampling resampling algorithm by creating a Markov chain and choosing between the current value and a pool of M proposed values (from the importance function), according to the importance weight, which, thanks to Tjemeland’s reformulation with two copies of the current state, constitutes a Gibbs sampler with the correct target. As in Tjemeland (2004), they propose to recycle all proposed values into the integral estimate, which then turned being unbiased under stationarity, rather unexpectedly. The paper then proceeds to analyse convergence towards this expectation, linearly in the size of the pool and exponentially in the number of Markov iterations.
