Index: The Book of Statistical ProofsModel Selection ▷ Bayesian model selection ▷ Log model evidence ▷ Derivation

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

1) as

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

2) or

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

Proof:

1) The first expression is a simple consequence of the law of marginal probability for continuous variables according to which

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

which, when logarithmized, gives

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

2) The second expression can be derived from Bayes’ theorem which makes a statement about the posterior distribution:

\[\label{eq:BT} p(\theta|y,m) = \frac{p(y|\theta,m) \, p(\theta|m)}{p(y|m)} \; .\]

Rearranging for $p(y \vert m)$ and logarithmizing, we have:

\[\label{eq:LME-bayes-qed} \begin{split} \mathrm{LME}(m) = \log p(y|m) & = \log \frac{p(y|\theta,m) \, p(\theta|m)}{p(\theta|y,m)} \\ &= \log p(y|\theta,m) + \log p(\theta|m) - \log p(\theta|y,m) \; . \end{split}\]
Sources:

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