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Title: Recent developments in Bayesian multivariate one-way ANOVA models Authors:  Dongchu Sun - University of Nebraska-Lincoln (United States) [presenting]
Abstract: The multivariate one-way ANOVA model is important in contemporary statistical theory and application. The model has an unknown overall mean and two unknown covariance matrices, the error covariance matrix and the random effects covariance matrix. We study this problem from the Bayesian perspective. Typically, independent prior distributions are assumed for the mean and each covariance matrix; that case is considered herein, with the primary focus being the determination of when common objective priors yield proper posteriors. We study a new class of dependent priors called ``commutative priors'', motivated from three directions. First, there are problems where it is most natural to utilize a prior on the ``signal to noise ratio'' (here a matrix); second, one often tries for dimension reduction, and the commutative priors substantially reduce the dimension of the unknowns; third, the commutative priors have excellent computational features. Interestingly, the commutative prior is also a conjugate prior. Propriety and moment existence are derived for both the priors and their posteriors. Moreover, a new and computationally effective MCMC algorithm is developed for the proposed commutative priors. Simulation and real data analysis show the potential advantages of the commutative priors.