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

Bayes factors often require numerical estimation due to the unavailability of closed-form solutions. In seven simulation studies, we explored the trade-offs between variance, 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. In generalized linear mixed effect models for normally and binomially distribute 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. But Savage-Dickey density ratio requires careful consideration of posterior density estimation to mitigate bias while limiting variability of Bayes factor estimates. Weprovide practical recommendations to guide researchers in selecting the most suitable estimation method for their applications.