Computing marginal posterior densities of genetic parameters of a multiple trait animal model using Laplace approximation or Gibbs sampling

Abstract

Two procedures for computing the marginal posterior density of heritabilities or genetic correlations, ie, Laplace's method to approximate integrals and Gibbs sampling, are compared. A multiple trait animal model is considered with one random effect, no missing observations and identical models for all traits. The Laplace approximation consists in computing the marginal posterior density for different values of the parameter of interest. This approximation requires the repeated evaluation of traces and determinants, which axe easy to compute once the eigenvalues of a matrix of dimension equal to the number of animals are determined. These eigenvalues can be efficiently computed by the Lanczos algorithm. The Gibbs sampler generates samples from the joint posterior density. These samples are used to estimate the marginal posterior density, which is exact up to a Monte-Carlo error. Both procedures were applied to a data set with semen production traits of 1957 Normande bulls. The traits analyzed were volume of the ejaculate, motility score and spermatozoa concentration. The Laplace approximation yielded very accurate approximations of the marginal posterior density for all parameters with much lower computing costs.