Normal-gamma distribution

Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Normal-gamma distribution ▷ Definition

Definition: Let $X$ be an $n \times 1$ random vector and let $Y$ be a positive random variable. Then, $X$ and $Y$ are said to follow a normal-gamma distribution

\[\label{eq:ng} X,Y \sim \mathrm{NG}(\mu, \Lambda, a, b) \; ,\]

if the distribution of $X$ conditional on $Y$ is a multivariate normal distribution with mean vector $\mu$ and covariance matrix $(y \Lambda)^{-1}$ and $Y$ follows a gamma distribution with shape parameter $a$ and rate parameter $b$:

\[\label{eq:mvn-gam} \begin{split} X \vert Y &\sim \mathcal{N}(\mu, (Y \Lambda)^{-1}) \\ Y &\sim \mathrm{Gam}(a, b) \; . \end{split}\]

The $n \times n$ matrix $\Lambda$ is referred to as the precision matrix of the normal-gamma distribution.

Sources:

Koch KR (2007): "Normal-Gamma Distribution"; in: Introduction to Bayesian Statistics, ch. 2.5.3, pp. 55-56, eq. 2.212; URL: https://www.springer.com/gp/book/9783540727231; DOI: 10.1007/978-3-540-72726-2.

Metadata: ID: D5 | shortcut: ng | author: JoramSoch | date: 2020-01-27, 14:28.