The Wishart distribution is a common matrix distribution on square symmetric semi-definite matrices.

The Wishart distribution is a common matrix distribution on square symmetric semi-definite matrices. It is usually denoted $\mathscr{W}_p(k,\Xi)$ where $p$ represents the dimension of the matrix, $k$ the degrees of freedom and $\Xi$ a $(p,p)$ ssd matrix, $k\Xi$ being the mean of the distribution.

Considering $k$ iid $\mathcal{N}_p(0,\Xi)$ vectors $X_i$, the $(p,p)$ matrix$$\mathbf{W}=\sum_{i=1}^k X_iX_i^\text{T}$$is a random $(p,p)$ symmetric semi-definite (ssd) matrix. The corresponding distribution is called the Wishart distribution and it is denoted $\mathscr{W}_p(k,\Xi)$ . Its density is $$p(\mathbf{w})=\dfrac{2^{-kp/2}}{\Gamma_p(k/2)|\mathbf{w}|^{k/2}}\,\exp\{-\text{tr}(\Xi^{-1}\mathbf{w}/2\}$$ where tr$(A)$ denotes the trace of the matrix $A$ and $\Gamma_p(k)$ is a generalised Gamma function. The matrix is almost surely positive definite when $k\ge p$ and $$\mathbb{E}[\mathbf{W}]=k\Xi$$ This distribution is commonly used in the Bayesian analysis of matrices and of graphical models, along with declinations like the inverse-Wishart distribution. The name comes from the physicist John Wishart.

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