Index: The Book of Statistical ProofsProbability Distributions ▷ Matrix-variate continuous distributions ▷ Normal-Wishart distribution ▷ Definition

Definition: Let $X$ be an $n \times p$ random matrix and let $Y$ be a $p \times p$ positive-definite symmetric matrix. Then, $X$ and $Y$ are said to follow a normal-Wishart distribution

\[\label{eq:nw} X,Y \sim \mathrm{NW}(M, U, V, \nu) \; ,\]

if the distribution of $X$ conditional on $Y$ is a matrix-normal distribution with mean $M$, covariance across rows $U$, covariance across columns $Y^{-1}$ and $Y$ follows a Wishart distribution with scale matrix $V$ and degrees of freedom $\nu$:

\[\label{eq:matn-wish} \begin{split} X \vert Y &\sim \mathcal{MN}(M, U, Y^{-1}) \\ Y &\sim \mathcal{W}(V, \nu) \; . \end{split}\]

The $p \times p$ matrix $Y$ can be seen as the precision matrix across the columns of the $n \times p$ matrix $X$.


Metadata: ID: D175 | shortcut: nw | author: JoramSoch | date: 2022-05-14, 23:06.