# Understanding Wishart Definition

I'm trying to understand the definition of the wishart distribution. In wiki,

$X_{(i)}{=}(x_i^1,\dots,x_i^p)\sim N_p(0,V).$ What do they mean by this? Each component is drawn from a univariate $N(0,V)$? I was reading it as a multivariate, but then some lines later they write when $p=V=1$ we have a chi-squared distribution.

Any help would be appreciated.

I think that it is a multivariate, $V$ is the variance-covariance matrix for the multivariate normal distribution ($0$ describes a vector of $0$ I think). If $p = 1$ then it is not a multivariate case anymore, hence you can get to a univariate chi-squared distribution.

Yes, those are $p$-dimensional vectors and $V$ is a $p \times p$-dimensional positive definite matrix.

The article's insistence on explaining the Wishart in terms of a correspondence to the chi-squared distribution is sort of strange to me, since the chi-squared is just a special case of the gamma, and the Wishart is more properly a multivariate generalization of the gamma. You can see the correspondence pretty cleanly when you disregard normalizing constants and use a 'rate' parameterization rather than scale.

Consider $w \sim \text{gamma}(\alpha, \beta)$, which has the following density:

$$f(w) \propto w^{\alpha - 1} e^{-\beta w}$$

Compare it to the corresponding density for $\mathbf{W} \sim \text{Wishart}(\alpha, \mathbf{B})$, where $\mathbf{W} \in \mathbb{R}^{p \times p}$, and $\mathbf{B} \in \mathbb{R}^{p \times p}$ is positive-definite:

\begin{equation*} f(\mathbf{W}) \propto \det(\mathbf{W})^{\alpha - \frac{(p + 1)}{2}} e^{-\text{tr}(\mathbf{BW})} \end{equation*}

If $p = 1$ then both $\mathbf{W}$ and $\mathbf{B}$ are scalars, so $\det(\mathbf{W}) = \mathbf{W}$ and $\text{tr}(\mathbf{BW}) = \mathbf{BW}$. The result is

\begin{equation*} f(\mathbf{W}) \propto \mathbf{W}^{\alpha - 1} e^{-\mathbf{BW}} \end{equation*}

which is equivalent to the gamma density given previously.