Using MCMC chain outputs to efficiently estimate Bayes factors

One of the most important methodological problems in psychological research is assessing the reasonableness of null models, which typically constrain a parameter to a specific value such as zero. Bayes factor has been recently advocated in the statistical and psychological literature as a principled means of measuring the evidence in data for various models, including those where parameters are set to specific values. Yet, it is rarely adopted in substantive research, perhaps because of the difficulties in computation. Fortunately, for this problem, the Savage–Dickey density ratio (Dickey & Lientz, 1970) provides a conceptually simple approach to computing Bayes factor. Here, we review methods for computing the Savage–Dickey density ratio, and highlight an improved method, originally suggested by Gelfand and Smith (1990) and advocated by Chib (1995), that outperforms those currently discussed in the psychological literature. The improved method is based on conditional quantities, which may be integrated by Markov chain Monte Carlo sampling to estimate Bayes factors. These conditional quantities efficiently utilize all the information in the MCMC chains, leading to accurate estimation of Bayes factors. We demonstrate the method by computing Bayes factors in one-sample and one-way designs, and show how it may be implemented in WinBUGS.