Index: The Book of Statistical ProofsGeneral Theorems ▷ Bayesian statistics ▷ Prior distributions ▷ Empirical Bayes priors

Definition: Let $m$ be a generative model with likelihood function $p(y \vert \theta, m)$ and prior distribution $p(\theta \vert \lambda, m)$ using prior hyperparameters $\lambda$. Let $p(y \vert \lambda, m)$ be the marginal likelihood when integrating the parameters out of the joint likelihood. Then, the prior distribution is called an “Empirical Bayes prior”, if it maximizes the logarithmized marginal likelihood:

\[\label{eq:prior-eb} \lambda_{\mathrm{EB}} = \operatorname*{arg\,max}_{\lambda} \log p(y \vert \lambda, m) \; .\]

Metadata: ID: D122 | shortcut: prior-eb | author: JoramSoch | date: 2020-12-02, 18:19.