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Abstract:
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It is common that a statistical model has some semi-invariace struture. A popular example would be the location and scale family for a give correlated spatial covariance matrix with some unknown correlation paramaters. Other important eamples include three parameters weibull or lognormal. The common used Jeffreys prior and reference prior algorithm turn out to be problematical for the partial invariance problems, and could lead to improper posterior distributions. The first demonstration of which we are aware of this issue is Hill (1963), who showed that the two popular Jeffereys priors yielded an improper posterior distribution for the three-parameter lognormal distribution; one contribution is to solve this 50+ year-old objective Bayesian problem. In this talk, the exact reference prior for the model is given and shown to produce a proper posterior. The first is to indicate that posterior impropriety is the rule, rather than the exception, for many of the priors above and scenarios of partial invariance. Other problems with these priors are also highlighted, such as the fact that, even when yielding a proper posterior, the conditional Jeffreys prior is not unique, and can depend
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