Improved marginal likelihood estimation via power posteriors and importance sampling - LI, Yong (Institutional Knowledge at Singapore Management University, 2019)
The power-posterior method of Friel and Pettitt (2008) has been used to estimate the marginal likelihoods of competing Bayesian models. In this paper it is shown that the Bernstein-von Mises (BvM) theorem holds for the power posteriors under regularity conditions. Due to the BvM theorem, the power posteriors, when adjusted by the square root of the corresponding grid points, converge to the same normal distribution as the original posterior distribution, facilitating the implementation of importance sampling for the purpose of estimating the marginal likelihood. Unlike the power-posterior method that requires repeated posterior sampling from the power posteriors, the new method only requires the posterior output from the original posterior. Hence, it is computationally more eﬃcient to implement. Moreover, it completely avoids the coding eﬀorts associated with drawing samples from the power posteriors. Numerical eﬃciency of the proposed method is illustrated using two models in economics and ﬁnance.