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Subset weighted-Tempered Gibbs Sampler (wTGS) has been recently introduced byJankowiak to reduce the computation complexity per MCMC iteration inhigh-dimensional applications where the exact calculation of the posteriorinclusion probabilities (PIP) is not essential. However, the Rao-Backwellizedestimator associated with this sampler has a high variance as the ratio betweenthe signal dimension and the number of conditional PIP estimations is large. Inthis paper, we design a new subset weighted-Tempered Gibbs Sampler (wTGS) wherethe expected number of computations of conditional PIPs per MCMC iteration canbe much smaller than the signal dimension. Different from the subset wTGS andwTGS, our sampler has a variable complexity per MCMC iteration. We provide anupper bound on the variance of an associated Rao-Blackwellized estimator forthis sampler at a finite number of iterations, $T$, and show that the varianceis $O\big(\big(\frac{P}{S}\big)^2 \frac{\log T}{T}\big)$ for a given datasetwhere $S$ is the expected number of conditional PIP computations per MCMCiteration. Experiments show that our Rao-Blackwellized estimator can have asmaller variance than its counterpart associated with the subset wTGS.

Variable-Complexity Weighted-Tempered Gibbs Samplers for Bayesian Variable Selection