The joint distribution of breeding values and of records usually depends on unknown parameters such as means,variances and covariances in the case of the multivariate normal distribution. If the objective of the analysis is to make selection decisions,these parameters should be considered as "nuisances". If the values of the parameters are unknown,the state of uncertainty can be represented by a prior probability distribution.This is then combined with the information contributed by the data to form a posterior distribution from which the needed predictors are calculated after integrating out the "nuisances".Prediction under alternative states of knowledge is discussed in this paper and the corresponding solutions presented.lt is shown that when the dispersion structure is unknown,modal estimators of variance parameters should be considered. Because a Bayesian framework is adopted,the estimates so obtained are necessarily non-negative. If prior knowledge about means and variances is completely vague and the distribution is multivariate normal,the "optimal" predictors in the sense of maximizing the expected merit of the selected candidates are those obtained by using the "mixed model equations" with the unknown variances replaced by restricted maximum likelihood estimates. This leads to empirical Bayes predictors of breeding values.
Proceedings of the World Congress on Genetics Applied to Livestock Production, Volume XII. Biotechnology, selection experiments, parameter estimation, design of breeding systems, management of genetic resources., , 356–370, 1986
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