Definition: Empirical Bayes prior distribution

Index: The Book of Statistical Proofs ▷ General 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:

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

 

Sources:

• Wikipedia (2020): "Empirical Bayes method"; in: Wikipedia, the free encyclopedia, retrieved on 2020-12-02; URL: https://en.wikipedia.org/wiki/Empirical_Bayes_method#Introduction.

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