Re-considering the variance parameterization in multiple precision models

Abstract

Recent developments in Bayesian computing allow accurate estimation of integrals, making advanced Bayesian analysis feasible. However, some problems remain difficult, such as estimating posterior distributions for variance parameters. For models with three or more variances, this paper proposes a simplex parameterization for the variance structure, which has appealing properties and eases the related burden of specifying a reference prior. This parameterization can be profitably used in several multiple-precision models, including crossed random-effect models, many linear mixed models, smoothed ANOVA, and the conditionally autoregressive (CAR) model with two classes of neighbor relations, often useful for spatial data. The simplex parameterization has at least two attractive features. First, it typically leads to simple MCMC algorithms with good mixing properties regardless of the parameterization used to specify the model's reference prior. Thus, a Bayesian analysis can take computational advantage of the simplex parameterization even if its prior was specified using another parameterization. Second, the simplex parameterization suggests a natural reference prior that is proper, invariant under multiplication of the data by a constant, and which appears to reduce the posterior correlation of smoothing parameters with the error precision. We use simulations to compare the simplex parameterization, with its reference prior, to other parameterizations with their reference priors, according to bias and mean-squared error of point estimates and coverage of posterior 95% credible intervals. The results suggest advantages for the simplex approach, particularly when the error precision is small. We offer results in the context of two real data sets from the fields of periodontics and prosthodontics.