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The Bayesian model evidence is the prior predictive distribution evaluated at the data $D$. For a statistical model $H_j$ with a normalized prior density ${\pi }_j$, its marginal likelihood isThis is an expected value of the likelihood function under the prior distribution; for discrete statistical parameters, replace the integral by a sum. For continuous observations, $m_j(D)$ is a probability density function value with respect to a specified data reference measure, rather than a probability of one exact observation. Comparisons must use the same data and reference measure. The same $m_j(D)$ normalizes the within-model Bayesian posterior and supplies the between-model comparison in a Bayes factor.^[ref]

A normalized prior distribution makes the Bayesian model evidence an average, so it cannot exceed the largest likelihood function value on its support. A broad prior distribution can assign little mass to values that predict the data well, even when one value gives an excellent fit. The worked uniform-alternative binomial Bayes factor below makes the difference between averaging and maximum likelihood estimation explicit.

Under the null hypothesis, $Y\sim \mathrm{Bin}(n,1/2)$, with expected value $n/2$ and variance $n/4$. Approximating its probability at an integer by the normal distribution density over a unit-width cell givesDividing by the prior predictive distribution from the preceding part gives the uniform-alternative binomial Bayes factorThe second approximation replaces $n+1$ by $n$. Both are large-sample approximations for counts in the central range, rather than exact identities at extreme counts.