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

Definition: Let $m$ be a generative model with model parameters $\theta$ and hyper-parameters $\lambda$ implying the likelihood function $p(y \vert \theta, \lambda, m)$ and prior distribution $p(\theta \vert \lambda, m)$. Then, the Empirical Bayesian log model evidence is the logarithm of the marginal likelihood, maximized with respect to the hyper-parameters:

\[\label{eq:ebLME} \mathrm{ebLME}(m) = \log p(y \vert \hat{\lambda}, m)\]

where

\[\label{eq:ML} p(y \vert \lambda, m) = \int p(y \vert \theta, \lambda, m) \, (\theta \vert \lambda, m) \, \mathrm{d}\theta\]

and

\[\label{eq:EB} \hat{\lambda} = \operatorname*{arg\,max}_{\lambda} \log p(y \vert \lambda, m) \; .\]
 
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Metadata: ID: D114 | shortcut: eblme | author: JoramSoch | date: 2020-11-25, 07:43.