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. threshold model / Bayesian analysis / Gibbs sampling / dog Résumé-Inférence bayésienne dans les modèles à seuil avec échantillonnage de Gibbs. Une analyse bayésienne du modèle à seuil avec des catégories multiples ordonnées est présentée ici. Les marginalisations nécessaires sont obtenues par échantillonnage de Gibbs. On montre que l'utilisation de données augmentées-la variable continue sousjacente non observée étant alors considérée comme une inconnue dans le modèleconduit à des distributions conditionnelles a posteriori faciles à échantillonner. Celles-ci sont des distributions uniformes indépendantes pour les seuils et des distributions normales tronquées indépendantes pour les sensibilités (les variables sous-jacentes). Les paramètres restants du modèle ont des distributions conditionneLles a posteriori identiques à celles qu'on trouve en modèle linéaire gaussien. La méthodologie est illustrée sur un modèle paternel appliquée à une dysplasie de la hanche chez le chien, et les résultats sont comparés à ceux d'une étude précédente basée sur un maximum de vraisemblance approché. Deux séquences de Gibbs indépendantes, longues chacune de 620 000 échantillons, ont été réalisées. Les erreurs d'échantillonnage de type Monte Carlo des moments des densités a posteriori ont été obtenues par des méthodes de séries temporelles. Les résultats obtenus avec les 2 séquences indépendantes sont dans la limite des erreurs d'échantillonnage de Monte-Carlo. À l'exception de la variance paternelle et de l'héritabilité, les distributions marginales a posteriori semblent normales. De ce fait, les inférences basées sur la présente méthode sont en bon accord avec ...

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