BAYESIAN-INFERENCE IN THRESHOLD MODELS USING GIBBS SAMPLING

A Bayesian analysis of a threshold model with multiple ordered categor ies is presented. Marginalizations are achieved by means of the Gibbs sampler. It is shown that use of data augmentation leads to conditiona l posterior distributions which are easy to sample from. The condition al posterior distributions of thresholds and liabilities are independe nt uniforms and independent truncated normals, respectively. The remai ning parameters of the model have conditional posterior distributions which are identical to those in the Gaussian linear model. The methodo logy is illustrated using a sire model, with an analysis of hip dyspla sia in dogs, and the results are compared with those obtained in a pre vious study, based on approximate maximum likelihood. Two independent Gibbs chains of length 620 000 each were. run, and the Monte-Carlo sam pling 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 exc eption of the sire variance and heritability, marginal posterior distr ibutions 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 sequenc e, but this can be alleviated using an alternative parameterization.