Which statement about Bayes' theorem best describes how the posterior probability is updated when new evidence B is observed?

• A. Posterior ∝ likelihood × prior; P(A|B) = P(B|A)P(A)/P(B).
• B. Posterior ∝ prior × data.
• C. Posterior ∝ likelihood × evidence.
• D. Posterior ∝ likelihood × posterior.

Answer: (A)

When new evidence arrives, you update your belief about a hypothesis by weighing how plausible the evidence is under that hypothesis (the likelihood) and then adjusting by how plausible the hypothesis was before seeing the data (the prior). This gives you the posterior probability P(A|B), which is proportional to P(B|A) P(A). To get the actual probability, you normalize by P(B): P(A|B) = [P(B|A) P(A)] / P(B).

This matches the statement that the posterior is proportional to the likelihood times the prior, with the explicit update P(A|B) = P(B|A) P(A) / P(B). The normalization by P(B) ensures the probabilities across all hypotheses sum to one.

The other formulations miss important pieces: multiplying the prior by data omits the explicit likelihood that tells you how compatible the data are with the hypothesis; using likelihood times evidence would treat P(B) as a factor that changes with the data but doesn’t reflect how plausible A is given B, and multiplying by the posterior would be circular, since you’d be updating using a quantity that already depends on the update.