# Wishart

## Wishart(cv, n, I, J)

The Wishart multivariate distribution describes a probability distribution over scatter matrices arising data sampled from a Gaussian distribution. Each sample from a Wishart is a 2-dimensional square, positive-definite, symmetric matrix. Suppose you sample N samples from a Gaussian(0, cv, I, J) distribution, X[I, R]. (R is the index that indexes each sample, R := 1..N). The Wishart distribution describes the distribution of Sum(X*X[I = J], R). This matrix is dimensioned by I and J and is called the scatter matrix. The parameter «cv» must be positive-definite.

A sample drawn from the Wishart is therefore a sample scatter matrix. If you divide that sample by (N - 1), you have a sampled covariance matrix.

If you compute a sample covariance matrix from data, and then want to use this in your model, if you just use it directly, you'll be ignoring sampling error. That may be insignificant of N is large. Otherwise, you may want to use:

Wishart(SampleCV, N, I, J)/(N - 1)

instead of just SampleCV in your model. The extended variance will account for the uncertainty from the finite sample size that was used to obtain your sample CV.

If you can express a prior probability on covariances in the form of an InvertedWishart distribution, then the posterior distribution, after having computed the sample covariance matrix (assumed to be drawn, by nature, from a Wishart), is also an InvertedWishart.

The Wishart can be viewed as a generalization of ChiSquared. When Size(I) = 1, the Wishart becomes a ChiSquared distribution.

## Library

Distribution Variations library (Distribution Variations.ana)