Reference and probability-matching priors in Bayesian analysis of mixed linear models

Determination of reasonable non-informative priors in multiparameter problems is not easy; common non-informative priors, such as Jeffrey's prior, can have features that have an unexpectedly dramatic effect on the posterior. In recognition of this problem Berger and Bernardo (Bayesian Statistics IV. Oxford University Press, Oxford, UK, pp. 35-70, 1992), proposed the Reference Prior approach to the development of non-informative priors. In the present paper the reference priors of Berger and Bernardo (1992) are derived for the mixed linear model. In spite of these difficulties, there is growing evidence, mainly through examples that reference priors provide 'sensible' answers from a Bayesian point of view. We also examine whether the reference priors satisfy the probability-matching criterion. The theory and results are applied to a real problem consisting of 879 weaning weight records, from the progeny of 17 sires. These important aspects are explored via Monte Carlo simulations.