Posterior odds refer to the ratio of what quantities in Bayesian hypothesis testing?

• A. The ratio of the posterior probability for one hypothesis to another.
• B. The ratio of the prior probabilities of the hypotheses.
• C. The probability of the data under the null hypothesis.
• D. The likelihood of the data given the alternate hypothesis.

Answer: (A)

Posterior odds express how our belief in two competing hypotheses changes after observing data. They are the ratio of the posterior probabilities assigned to each hypothesis after seeing the data. In Bayesian updating, this ratio equals the prior odds multiplied by the Bayes factor, where the Bayes factor is the ratio of the data’s likelihood under one hypothesis to its likelihood under the other. So posterior odds combine what we believed before with how much the data support one hypothesis over the other.

The other concepts don’t capture this updating in one ratio: the prior odds reflect belief before seeing data, not after; the probability of the data under the null is just part of the evidence about the null, not a comparison of posteriors; and the likelihood under the alternate is only one side of the Bayes factor, not the full posterior comparison.