Improving power posterior estimation of statistical evidence

By Nial Friel, Merrilee Hurn, Jason Wyse. 2012

68 citations recorded by the source corpus.

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. A key practical point is that both of these innovations incur virtually no extra cost.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Control functionals for Monte Carlo integration (2014, methodology): “Here we use the second-order quadrature that was recommended by Friel et al. (2014): log{p.y/}≈ m−1∑ i=0 ti+1 − ti 2 .μ̂i + μ̂i+1/− .ti+1 − ti/2 12 .ν̂i+1 − ν̂i/, where μ̂i and ν̂i are MC estimates of the posterior mean and variance respectively of log{p.y|θ/} when θ arises from p.θ|y, ti/.”
Towards Automatic Model Comparison: An Adaptive Sequential Monte Carlo Approach (2013, methodology): “As noted by Friel et al. (2012) we can use more sophisticated numerical integration strategies to reduce the path sampling estimator bias.”
The Controlled Thermodynamic Integral for Bayesian Model Evidence Evaluation (2016, methodology): “Following Friel, Hurn, and Wyse (2014a), we consider Bayesian model comparison between a pair of competing models: Model 1: Model 2: Here ¯ x and ¯ z are the sample means of the x i and z i , respectively.”
An adaptive scheduling scheme for calculating Bayes factors with thermodynamic integration using Simpson’s rule (2015, methodology): “It can then be shown that Integrating both sides with respect to τ and using (5) leads to the thermodynamic integral (Lartillot and Philippe 2006), The integrand E p τ [ log p ( Y | θ , M ) ] is called the expected log deviance (Friel et al. 2013).”
The Controlled Thermodynamic Integral for Bayesian Model Comparison (2014, methodology): “Following Friel et al. (2014) we consider Bayesian model comparison between a pair of competing models: Model 1: yi = α+ β(xi − x̄) + i, i ∼ N(0, τ−1) (21) Model 2: yi = γ + δ(zi − z̄) + ηi, ηi ∼ N(0, λ−1) (22) Here x̄ and z̄ are the sample means of the xi and zi respectively.”