  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 an inverse temperature. In common with other tempering
algorithms, the power posterior involves some degree of tuning. 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 inverse temperatures. We also tackle the selection of the inverse
temperature ladder, applying this approach additionally to the Stepping Stone
sampler estimation of evidence.
