Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 1 / f / Solution

Substituting the standardized deviation into the Bayes factor approximation givesThis is only a slight preference for the alternative, despite the small p-value. The two procedures measure different things: the p-value sums a tail under the null hypothesis, while the Bayes factor compares probabilities of the observed count under both models. The uniform alternative prior distribution spreads its probability over many success probabilities that fit the observations poorly. Only a narrow range around $0.515$ has large likelihood, and integrating over the whole alternative penalizes this diffuse prediction.

The prefactor makes the distinction especially clear. At a fixed standardized deviation $z$, the p-value stays approximately $2\{1-\Phi (|z|)\}$, whereas $B_{01}\approx \sqrt{2n/\pi }{e}^{-z^2/2}$ eventually grows with $n$. A fixed diffuse alternative can consequently favour the point null even while a conventional test rejects it. This is the same prior-width mechanism illustrated by the Jeffreys-Lindley paradox for a Gaussian point null.