Joint genome-wide prediction in several populations accounting for randomness of genotypes: A hierarchical Bayes approach. II: Multivariate spike and slab priors for marker effects and derivation of approximate Bayes and fractional Bayes factors for the c

This study corresponds to the second part of a companion paper devoted to the development of Bayesian multiple regression models accounting for randomness of genotypes in across population genome-wide prediction. This family of models considers heterogeneous and correlated marker effects and allelic frequencies across populations, and has the ability of considering records from non-genotyped individuals and individuals with missing genotypes in any subset of loci without the need for previous imputation, taking into account uncertainty about imputed genotypes. This paper extends this family of models by considering multivariate spike and slab conditional priors for marker allele substitution effects and contains derivations of approximate Bayes factors and fractional Bayes factors to compare models from part I and those developed here with their null versions. These null versions correspond to simpler models ignoring heterogeneity of populations, but still accounting for randomness of genotypes. For each marker loci, the spike component of priors corresponded to point mass at 0 in RS, where S is the number of populations, and the slab component was a S-variate Gaussian distribution, independent conditional priors were assumed. For the Gaussian components, covariance matrices were assumed to be either the same for all markers or different for each marker. For null models, the priors were simply univariate versions of these finite mixture distributions. Approximate algebraic expressions for Bayes factors and fractional Bayes factors were found using the Laplace approximation. Using the simulated datasets described in part I, these models were implemented and compared with models derived in part I using measures of predictive performance based on squared Pearson correlations, Deviance Information Criterion, Bayes factors, and fractional Bayes factors. The extensions presented here enlarge our family of genome-wide prediction models making it more flexible in the sense that it now offers more modeling options.