Reliability, bias, and computational cost of estimating the Bayes factor using bridge sampling and the Savage-Dickey density ratio.

Bayes factors often require numerical estimation because closed-form solutions are unavailable. In six simulation studies, we explored the reliability, bias, and computational cost of two easy-to-use and broadly applicable methods: bridge sampling and the Savage-Dickey density ratios, based on Gaussian, logspline, and spline-smoothed kernel density approximations of the posterior distribution, as well as conditional marginal density estimation. In generalized linear mixed effect models for normally and binomially distributed data, we explore the effects of the (1) number of MCMC samples from the posterior, (2) size of effects or magnitude of the Bayes factor, (3) number of participants, and (4) number of model parameters. Our findings suggest that, with enough MCMC samples, both methods yield reliable and accurate estimates across a wide range of conditions. However, with many model parameters, bridge sampling becomes computationally expensive and can be unreliable. In contrast, the Savage-Dickey density ratio scales well, remaining computationally efficient and reliable, even with many model parameters. However, Savage-Dickey density ratio requires careful consideration of posterior density estimation to mitigate bias while limiting the variability of Bayes factor estimates. We provide practical recommendations to guide researchers in selecting the most suitable estimation method for their applications.