# Proof: Derivation of the log model evidence

**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,

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

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:

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

**Sources:**

**Metadata:**ID: P13 | shortcut: lme-der | author: JoramSoch | date: 2020-01-06, 21:27.