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Proper likelihoods for Bayesian analysis. (English) Zbl 0751.62012
Summary: The validity of posterior probability statements follows from probability calculus when the likelihood is the density of the observations. To investigate other cases, a second, more intuitive definition of validity is introduced, based on coverage of posterior sets. This notion of validity suggests that the likelihood must be the density of a statistic, not necessarily sufficient, for posterior probability statements to be valid. A convenient numerical method is proposed to invalidate the use of certain likelihoods for Bayesian analysis. Integrated, marginal, and conditional likelihoods, derived to avoid nuisance parameters, are also discussed.

62F15 Bayesian inference
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