Analysis of competing hypothesis, a method for evaluating explanations of observed evidence, is used in numerous fields, including counterterrorism, psychology, and intelligence analysis. We propose a Bayesian extension of the methodology, posing the problem in terms of a multinomial-Dirichlet hierarchical model. The yet-to-be observed true hypothesis is regarded as a multinomial random variable and the evaluation of the evidence is treated as a structured elicitation of a prior distribution on the probabilities of the hypotheses. This model provides the user with measures of uncertainty for the probabilities of the hypotheses. We discuss inference, such as point and interval estimates of hypothesis probabilities, ratios of hypothesis probabilities, and Bayes factors. A simple example involving the stadium relocation of the San Diego Chargers is used to illustrate the method. We also present several extensions of the model that enable it to handle special types of evidence, including evidence that is irrelevant to one or more hypotheses, evidence against hypotheses, and evidence that is subject to deception.