Improving power posterior estimation of statistical evidence

The statistical evidence (or marginal likelihood) is a key quantity in Bayesian statistics, allowing one to assess the probability of the data given the model under investigation. This paper focuses on refining the power posterior approach to improve estimation of the evidence. The power posterior method involves transitioning from the prior to the posterior by powering the likelihood by a temperature variable. In common with other tempering algorithms, the power posterior involves some degree of tuning, and this paper addresses this issue. The main contributions of this article are twofold -- we present a result from the numerical analysis literature which can reduce the bias in the estimate of the evidence by addressing the error arising from numerically integrating across the temperature. We also address the choice of temperature ladder, and present an adaptive algorithm which gives excellent performance in the examples considered here. A key practical point is that both of these innovations incur virtually no extra cost.