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Uncertainty of prior and posterior model probability: Implications for interpreting Bayes factors
In typical applications of Bayesian model comparison or Bayesian hypothesis testing, the results are reported in terms of the Bayes factor (BF) only, not in terms of the posterior probabilities of the models, even though posterior probabilities are needed to make decisions. Posterior model probabilities are not reported because researchers are reluctant to declare prior model probabilities. The reluctance to declare prior model probabilities stems from uncertainty in the prior. Fortunately, Bayesian formalisms are designed to embrace prior uncertainty, not to ignore it. I provide a derivation of the posterior distribution of model probability, and show various examples. The posterior distribution is useful for making decisions about the model comparison or hypothesis test that go beyond merely considering the mean posterior model probability, by also taking into account the uncertainty of the posterior model probability.