Extrapolation of Tempered Posteriors

Tempering is a popular tool in Bayesian computation, being used to transform a posterior distribution p₁ into a reference distribution p₀ that is more easily approximated. Several algorithms exist that start by approximating p₀ and proceed through a sequence of intermediate distributions pₜ until an approximation to p₁ is obtained. Our contribution reveals that high-quality approximation of terms up to p₁ is not essential, as knowledge of the intermediate distributions enables posterior quantities of interest to be extrapolated. Specifically, we establish conditions under which posterior expectations are determined by their associated tempered expectations on any non-empty t interval. Harnessing this result, we propose novel methodology for approximating posterior expectations based on extrapolation and smoothing of tempered expectations, which we implement as a post-processing variance-reduction tool for sequential Monte Carlo.