= Bayesian model evidence
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{title2=$\mathcal Z$}
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= Prior predictive distribution
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= Bayesian evidence
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= Marginal likelihood
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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> is
$$
m_j(D)=p(D\mid H_j)=\mathcal Z_j
=\int p(D\mid\theta_j,H_j)\pi_j(\theta_j\mid H_j)\,d\theta_j.
$$
This 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>.<Bayes Factors by Robert E. Kass and Adrian E. Raftery>[Kass and Raftery, Bayes Factors (1995), section 3.1, p. 776, equations (1)–(2): integrating over parameter priors]{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.
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