Bayesian conjugate analysis using a generalized inverted Wishart distribution accounts for differential uncertainty among the genetic parameters--an application to the maternal animal model

Consider the estimation of genetic (co)variance components from a maternal animal model (MAM) using a conjugated Bayesian approach. Usually, more uncertainty is expected a priori on the value of the maternal additive variance than on the value of the direct additive variance. However, it is not possible to model such differential uncertainty when assuming an inverted Wishart (IW) distribution for the genetic covariance matrix. Instead, consider the use of a generalized inverted Wishart (GIW) distribution. The GIW is essentially an extension of the IW distribution with a larger set of distinct parameters. In this study, the GIW distribution in its full generality is introduced and theoretical results regarding its use as the prior distribution for the genetic covariance matrix of the MAM are derived. In particular, we prove that the conditional conjugacy property holds so that parameter estimation can be accomplished via the Gibbs sampler. A sampling algorithm is also sketched. Furthermore, we describe how to specify the hyperparameters to account for differential prior opinion on the (co)variance components. A recursive strategy to elicit these parameters is then presented and tested using field records and simulated data. The procedure returned accurate estimates and reduced standard errors when compared with non-informative prior settings while improving the convergence rates. In general, faster convergence was always observed when a stronger weight was placed on the prior distributions. However, analyses based on the IW distribution have also produced biased estimates when the prior means were set to over-dispersed values.