Index: The Book of Statistical Proofs ▷ Model Selection ▷ Bayesian model selection ▷ Model evidence ▷ Derivation of the log model evidence

Theorem: Let $p(y \vert \theta,m)$ be a likelihood function of a generative model $m$ for making inferences on model parameters $\theta$ given measured data $y$. Moreover, let $p(\theta \vert m)$ be a prior distribution on model parameters $\theta$. Then, the log model evidence (LME), also called marginal log-likelihood,

\[\label{eq:LME-term} \mathrm{LME}(m) = \log p(y|m) \; ,\]

can be expressed in terms of likelihood and prior as

\[\label{eq:LME-marg} \mathrm{LME}(m) = \log \int p(y|\theta,m) \, p(\theta|m) \, \mathrm{d}\theta \; .\]

Proof: This a consequence of the law of marginal probability for continuous variables

\[\label{eq:prob-marg} p(y|m) = \int p(y,\theta|m) \, \mathrm{d}\theta\]

and the law of conditional probability according to which

\[\label{eq:prob-cond} p(y,\theta|m) = p(y|\theta,m) \, p(\theta|m) \; .\]

Combining \eqref{eq:prob-marg} with \eqref{eq:prob-cond} and logarithmizing, we have:

\[\label{eq:LME-marg-qed} \mathrm{LME}(m) = \log p(y|m) = \log \int p(y|\theta,m) \, p(\theta|m) \, \mathrm{d}\theta \; .\]
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Metadata: ID: P13 | shortcut: lme-der | author: JoramSoch | date: 2020-01-06, 21:27.