Wishart distribution¶
Generalization of the Gamma distribution to positive definite matrices. It is used to model uncertainity in covariance matrices, and in their iverses \(\Lambda =\Sigma^{-1}\).
PDF:¶
\[ Wi(\Lambda | S,v) = \frac{1}{Z_{Wi}}|\Lambda|^{(v - D - 1)/2} \exp (-\frac{1}{2} tr(\Lambda S^{-1})) \]
Where:
v is called the degrees of freedom
S is the scale matrix.
\(Z_{Wi} = 2^{vD/2} \Gamma_D(v/2)|S|^{v/2}\) is the normalization constant
\(\Gamma_D(a)\) is the multivariate gamma function
The normalization constant exists only if \(v > D-1\).
Connection to Gamma distribution:¶
If D = 1:
\[ Wi(\lambda| s^{-1}, v)= Ga(\lambda| \frac{v}{2}, \frac{s}{2})\]
This makes marginal distributions Gamma.
Connection with Gaussian distribution:¶
Let \(x_i \sim \mathcal{N}(0,\Sigma)\), then the scatter matrix \(S = \sum_{i=1}^N x_ix_i^T\) has a Wishart distribution. \(S \sim Wi(\Sigma, 1)\). Hence \(E[S] = N \Sigma\).
