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

Theorem: Let $X \in \mathbb{R}^{n \times p}$ and $Y \in \mathbb{R}^{p \times p}$ follow a normal-Wishart distribution:

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

Then, the expected value of $X$ and $Y$ is

$\label{eq:nw-mean} \mathrm{E}[(X,Y)] = \left( M, \nu V \right) \; .$

Proof: Consider the random matrix

$\label{eq:rmat} \left[ \begin{array}{c} X \\ Y \end{array} \right] = \left[ \begin{array}{ccc} x_{11} & \ldots & x_{1p} \\ \vdots & \ddots & \vdots \\ x_{n1} & \ldots & x_{np} \\ y_{11} & \ldots & y_{1p} \\ \vdots & \ddots & \vdots \\ y_{p1} & \ldots & y_{pp} \end{array} \right] \; .$

According to the expected value of a random matrix, its expected value is

$\label{eq:mean-rmat} \mathrm{E}\left( \left[ \begin{array}{c} X \\ Y \end{array} \right] \right) = \left[ \begin{array}{ccc} \mathrm{E}(x_{11}) & \ldots & \mathrm{E}(x_{1p}) \\ \vdots & \ddots & \vdots \\ \mathrm{E}(x_{n1}) & \ldots & \mathrm{E}(x_{np}) \\ \mathrm{E}(y_{11}) & \ldots & \mathrm{E}(y_{1p}) \\ \vdots & \ddots & \vdots \\ \mathrm{E}(y_{p1}) & \ldots & \mathrm{E}(y_{pp}) \end{array} \right] = \left[ \begin{array}{c} \mathrm{E}(X) \\ \mathrm{E}(Y) \end{array} \right] \; .$

When $X$ and $Y$ are jointly normal-Wishart distributed, then by definition $X$ follows a matrix-normal distribution conditional on $Y$ and $Y$ follows a Wishart distribution:

$\label{eq:nw-def} X,Y \sim \mathrm{NW}(M, U, V, \nu) \quad \Leftrightarrow \quad X \vert Y \sim \mathcal{MN}(M, U, Y^{-1}) \quad \wedge \quad Y \sim \mathcal{W}(V, \nu) \; .$

Thus, with the expected value of the matrix-variate normal distribution and the law of conditional probability, $\mathrm{E}(X)$ becomes

$\label{eq:mean-X} \begin{split} \mathrm{E}(X) &= \iint X \cdot p(X,Y) \, \mathrm{d}X \, \mathrm{d}Y \\ &= \iint X \cdot p(X|Y) \cdot p(Y) \, \mathrm{d}X \, \mathrm{d}Y \\ &= \int p(Y) \int X \cdot p(X|Y) \, \mathrm{d}X \, \mathrm{d}Y \\ &= \int p(Y) \left\langle X \right\rangle_{\mathcal{MN}(M, U, Y^{-1})} \, \mathrm{d}Y \\ &= \int p(Y) \cdot M \, \mathrm{d}Y \\ &= M \int p(Y) \, \mathrm{d}Y \\ &= M \; , \end{split}$

and with the expected value of the Wishart distribution, $\mathrm{E}(Y)$ becomes

$\label{eq:mean-Y} \begin{split} \mathrm{E}(Y) &= \int Y \cdot p(Y) \, \mathrm{d}Y \\ &= \left\langle Y \right\rangle_{\mathcal{W}(V,\nu)} \\ &= \nu V \; . \end{split}$

Thus, the expectation of the random matrix in equations \eqref{eq:rmat} and \eqref{eq:mean-rmat} is

$\label{eq:nw-mean-qed} \mathrm{E}\left( \left[ \begin{array}{c} X \\ Y \end{array} \right] \right) = \left[ \begin{array}{c} M \\ \nu V \end{array} \right] \; ,$

as indicated by equation \eqref{eq:nw-mean}.

Sources:

Metadata: ID: P327 | shortcut: nw-mean | author: JoramSoch | date: 2022-07-14, 07:17.