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[[category:Multivariate Distribution Functions]]
[[Category:Doc Status C]] <!-- For Lumina use, do not change -->

== 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 <code>N</code> samples from a [[Gaussian]](0, cv, I, J) distribution, <code>X[I, R]</code>.  (<code>R</code> is the index that indexes each sample, <code>R := 1..N</code>).  The [[Wishart]] distribution describes the distribution of <code>Sum(X * X[I = J], R)</code>.  This matrix is dimensioned by <code>I</code> and <code>J</code> 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 <code>(N - 1)</code>, 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 <code>N</code> is large.  Otherwise, you may want to use:
:<code>Wishart(SampleCV, N, I, J)/(N - 1)</code>

instead of just <code>SampleCV</code> 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.ana

== See Also ==
* [[InvertedWishart]]
* [[LDens_Wishart]]
* [[LDens_InvertedWishart]]
* [[ChiSquared]]
* [[Gaussian]]
* [[Covariance]]