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Inverse Wishart Distribution

The inverse Wishart distribution represents positive definite matrices.  This can be extremely useful in Bayesian analysis, as the inverse Wishart can serve as a prior for any covariance matrix \Sigma.

Basic Facts

Let \Sigma be a DxD positive definite matrix.  We say \Sigma is drawn from an inverse Wishart with parameters v, S when we have the following probability density function

    \[ p( \Sigma | v, S ) = \frac{ |S|^{\frac{v}{2}} }{ 2^{\frac{vD}{2}} \Gamma_D(\frac{v}{2} ) } |\Sigma|^{-\frac{v+D-1}{2} } e^{ -\frac{1}{2} tr( S\Sigma^{-1} ) } \]

where

  • v: scalar degrees of freedom
  • S: DxD positive definite matrix

The expected value (mean) of \Sigma under this p.d.f. is

    \[ \mathbb{E}[ \Sigma ] = \frac{ S }{ v - D - 1} \]

which is only defined for v > D-1. So for D=2, we have v \ge 2, etc.

Visualizations

To gain intuition about the influence of different parameters, here I show i.i.d. draws from an inverse Wishart with inverse scale matrix S set to the 2-dimensional identity matrix. Each plot considers a different value of v, the degrees of freedom.

Here, we use 2 degrees of freedom, the minimum permissible with a two-dimensional scale matrix.

Moving to 8 degrees of freedom… we can see the shrinking size of the covariance matrix

Finally, here’s 32 degrees of freedom.  There is very little variability here, as we’re converging to the mean (a scaled-down version of the identity matrix).

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