Improved marginal likelihood estimation via power posteriors and importance sampling

Power posteriors have become popular in estimating the marginal likelihood of a Bayesian model. A power posterior is referred to as the posterior distribution that is proportional to the likelihood raised to a power b∈[0,1]. Important power-posterior-based algorithms include thermodynamic integration (TI) of Friel and Pettitt (2008) and steppingstone sampling (SS) of Xie et al. (2011). In this paper, it is shown that the Bernstein–von Mises (BvM) theorem holds for power posteriors under regularity conditions. Due to the BvM theorem, power posteriors, when adjusted by the square root of the auxiliary constant, have the same limit distribution as the original posterior distribution, facilitating the implementation of the modified TI and SS methods via importance sampling. Unlike the TI and SS methods that require repeated sampling from the power posteriors, the modified methods only need the original posterior output and hence, are computationally more efficient. Moreover, they completely avoid the coding efforts associated with sampling from the power posteriors. Primitive conditions, under which the TI and modified TI algorithms can produce consistent estimators of the marginal likelihood, are provided. The numerical efficiency of the proposed methods is illustrated using two models.