Index: The Book of Statistical Proofs ▷ General Theorems ▷ Bayesian statistics ▷ Prior distributions ▷ Reference 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 $p(\theta \vert y, \lambda, m)$ be the posterior distribution that is proportional to the the joint likelihood. Then, the prior distribution is called a “reference prior”, if it maximizes the expected Kullback-Leibler divergence of the posterior distribution relative to the prior distribution:

\[\label{eq:prior-ref} \lambda_{\mathrm{ref}} = \operatorname*{arg\,max}_{\lambda} \left\langle \mathrm{KL} \left[ p(\theta \vert y, \lambda, m) \, || \, p(\theta \vert \lambda, m) \right] \right\rangle \; .\]
 
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Metadata: ID: D123 | shortcut: prior-ref | author: JoramSoch | date: 2020-12-02, 18:26.