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Large-Sample Joint Posterior Approximations When Full Conditionals Are Approximately Normal

Modern Bayesian statistical methods, such as Gibbs and Metropolis–Hastings sampling, were developed to liberate statisticians from the necessity of making large-sample assumptions and to facilitate the numerical approximation of problems that had previously been analytically intractable. Counter to this trend, we develop a method for constructing asymptotic joint posterior approximations based on models with k blocks of parameters and where the corresponding properly normalized full conditionals are themselves asymptotically normal. We illustrate these techniques by applying them to particular linear and generalized linear mixed models (GLMMs). We also consider the relevance of different parameterizations with regard to our asymptotics. Recent work has indicated that Gibbs samplers based on so-called “centering parameterizations” result in better convergence properties for the resulting Markov chains. Our results for the one-way random-effects model shed some light on this issue. For this example, we also consider the distinction between letting the within-group sample size, n, tend to infinity versus letting the number of groups K (as defined by the random-effects part of the model) tend to infinity. Letting n grow results in a proper limiting normal distribution only when the weight on the prior for the variance component grows at a rate comparable to n. With large K, on the other hand, proper limits are obtained without this assumption, and thus it is seen that the information in the data will ultimately swamp standard prior information. We compare results based on simulated data when n and K are large. A dataset involving the effect of smoking on hormone function is analyzed using our asymptotics and compared with results based on Gibbs sampling.