Bayesian inference in threshold models using Gibbs sampling

Abstract

A Bayesian analysis of a threshold model with multiple ordered categories is presented. Marginalizations are achieved by means of the Gibbs sampler. It is shown that use of data augmentation leads to conditional posterior distributions which are easy to sample from. The conditional posterior distributions of thresholds and liabilities are independent uniforms and independent truncated normals, respectively. The remaining parameters of the model have conditional posterior distributions which are identical to those in the Gaussian linear model. The methodology is illustrated using a sire model, with an analysis of hip dysplasia in dogs, and the results are compared with those obtained in a previous study, based on approximate maximum likelihood. Two independent Gibbs chains of length 620 000 each were run, and the Monte-Carlo sampling error of moments of posterior densities were assessed using time series methods. Differences between results obtained from both chains were within the range of the Monte-Carlo sampling error. With the exception of the sire variance and heritability, marginal posterior distributions seemed normal. Hence inferences using the present method were in good agreement with those based on approximate maximum likelihood. Threshold estimates were strongly autocorrelated in the Gibbs sequence, but this can be alleviated using an alternative parameterization. © 1995 Elsevier/INRA.