In Bayes' theorem, the posterior probability is the inverse conditional probability of which quantity?

• A. The likelihood.
• B. The prior probability.
• C. The marginal likelihood.
• D. The posterior probability of the data.

Answer: (A)

In Bayes' theorem, updating our belief about a hypothesis after seeing data involves two main pieces: how likely the observed data are under that hypothesis (the likelihood) and our prior belief about the hypothesis before seeing the data. The posterior probability of the hypothesis given the data is proportional to the likelihood times the prior: P(H|D) ∝ P(D|H) P(H). The likelihood, P(D|H), is exactly the conditional probability of observing the data given a particular hypothesis. That’s why the posterior is built from that conditional view of the data under each hypothesis and then normalized by the overall probability of the data. The prior brings in our initial belief, and the normalization factor (the marginal likelihood, P(D)) ensures the probabilities sum to one across all hypotheses. So the quantity whose conditional probability appears in the Bayes update is the likelihood.