Discussion
Which of the following statements about the Bayes factor is true ?
(I) Bayes factor involves marginal likelihood.
(II) Bayes factor expresses the support given by the data for one or other of the models.
(III) Bayes factor involves the conditional likelihood
Answer: B
The key accepts (I) and (II) as true and (III) as false.
(I) The Bayes factor for models \(M_{1}\) and \(M_{2}\) is the ratio of their marginal likelihoods, \(B_{12} = p(D \mid M_{1}) / p(D \mid M_{2})\), where \(p(D \mid M) = \int p(D \mid \theta, M)\,\pi(\theta \mid M)\,d\theta\). (II) It measures how strongly the data support one model over the other, and it turns prior odds into posterior odds.
(III) concerns the likelihood conditional on the parameters, \(p(D \mid \theta, M)\). That quantity is integrated over the prior inside each marginal likelihood, so the Bayes factor is built from the averaged form and not from the conditional one. The wording of (III) is loose: the conditional likelihood does enter the integral, so a reader could call it involved. We follow the key's reading, which makes options A, C and D false.
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