Bayesian posterior distributions without Markov chains

Abstract

Bayesian posterior parameter distributions are often simulated using Markov chain Monte Carlo (MCMC) methods. However, MCMC methods are not always necessary and do not help the uninitiated understand Bayesian inference. As a bridge to understanding Bayesian inference, the authors illustrate a transparent rejection sampling method. In example 1, they illustrate rejection sampling using 36 cases and 198 controls from a case-control study (1976-1983) assessing the relation between residential exposure to magnetic fields and the development of childhood cancer. Results from rejection sampling (odds ratio (OR) = 1.69, 95% posterior interval (PI): 0.57, 5.00) were similar to MCMC results (OR = 1.69, 95% PI: 0.58, 4.95) and approximations from data-augmentation priors (OR = 1.74, 95% PI: 0.60, 5.06). In example 2, the authors apply rejection sampling to a cohort study of 315 human immunodeficiency virus seroconverters (1984-1998) to assess the relation between viral load after infection and 5-year incidence of acquired immunodeficiency syndrome, adjusting for (continuous) age at seroconversion and race. In this more complex example, rejection sampling required a notably longer run time than MCMC sampling but remained feasible and again yielded similar results. The transparency of the proposed approach comes at a price of being less broadly applicable than MCMC.

Figure 1.

Log odds ratio (OR) for childhood leukemia according to residential exposure to magnetic fields, Denver, Colorado, 1976–1983. Data were obtained from a case-control study by Savitz et al. (11). The y-axis shows the intercept (i.e., the log odds of being a leukemia case among the unexposed) (11). Panel A plots a random sample of size 20,000 from the joint prior distribution; panel B plots the 20,000 rejection-sampling draws from the joint posterior distribution; and panel C plots the 20,000 Markov chain Monte Carlo draws from the joint posterior distribution.