Posterior odds in Bayesian testing represent the ratio of posterior probabilities of which hypotheses?

Posterior odds in Bayesian testing represent the ratio of posterior probabilities of which hypotheses?

• A. The ratio of the prior probabilities of the hypotheses.
• B. The ratio of the likelihoods of the data under the two hypotheses.
• C. The ratio of the posterior probabilities of the null to the alternative hypotheses.
• D. The ratio of the posterior probability of the alternative hypothesis to the null hypothesis after data.

Answer: (D)

Posterior odds measure how the data shift our belief from the null to the alternative. They are the ratio of the posterior probability that the alternative hypothesis is true to the posterior probability that the null hypothesis is true, after observing the data. In symbols, this is P(H1|data) divided by P(H0|data). This quantity can be broken down as the product of the Bayes factor and the prior odds: (P(data|H1)/P(data|H0)) × (P(H1)/P(H0)). That unpacking shows how the data (through the likelihoods) and our starting beliefs (through the priors) combine to form the updated belief about which hypothesis is true. If the posterior odds exceed 1, the data favor the alternative; if they are below 1, the data favor the null. Other options correspond to related but different ideas: a ratio of priors is the prior odds, a ratio of likelihoods is the Bayes factor, and the ratio of the null to the alternative posterior probabilities is the reciprocal of the standard posterior odds.

Posterior odds measure how the data shift our belief from the null to the alternative. They are the ratio of the posterior probability that the alternative hypothesis is true to the posterior probability that the null hypothesis is true, after observing the data. In symbols, this is P(H1|data) divided by P(H0|data).

This quantity can be broken down as the product of the Bayes factor and the prior odds: (P(data|H1)/P(data|H0)) × (P(H1)/P(H0)). That unpacking shows how the data (through the likelihoods) and our starting beliefs (through the priors) combine to form the updated belief about which hypothesis is true. If the posterior odds exceed 1, the data favor the alternative; if they are below 1, the data favor the null.

Other options correspond to related but different ideas: a ratio of priors is the prior odds, a ratio of likelihoods is the Bayes factor, and the ratio of the null to the alternative posterior probabilities is the reciprocal of the standard posterior odds.