What does posterior probability represent in Bayesian testing?

• A. The probability of the data given the model before data were collected.
• B. The probability of the model given the data.
• C. The probability of the data under the null hypothesis.
• D. The p-value of the test.

Answer: (B)

In Bayesian testing, the posterior probability is the updated belief about which model or hypothesis is true after observing the data. It answers the question: given the data, how plausible is this model?

This belief is obtained by combining what you thought before seeing the data (the prior probability of the model) with how likely the observed data are under that model (the likelihood). Bayes’ rule says the posterior probability is proportional to the likelihood times the prior. In other words, strong data that fit a model well will raise that model’s posterior probability, while weak or conflicting data will lower it.

This is different from the likelihood itself, which is the probability of the data assuming a particular model is true, and from a p-value, which is a frequentist measure about how extreme the data are under a null hypothesis. The posterior probability provides a direct probability about which model is true, after seeing the data.

So the best answer is that the posterior probability represents the probability of the model given the data.