Extrapolation of Tempered Posteriors

Tempering is a popular tool in Bayesian computation, being used to transform a posterior distribution $p_1$ into a reference distribution $p_0$ that is more easily approximated. Several algorithms exist that start by approximating $p_0$ and proceed through a sequence of intermediate distributions $p_t$ until an approximation to $p_1$ is obtained. Our contribution reveals that high-quality approximation of terms up to $p_1$ 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.