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The inverse Wishart distribution

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The inverse Wishart distribution (otherwise called the inverted Wishart distribution)  is a probability distribution that is used in the Bayesian analysis of real-valued positive definite matrices (e.g. matrices of the type that arise in risk management contexts). It is a conjugate prior for the covariance matrix of a multivariate normal distribution.

 

It has the following characteristics, where  is a  matrix,  is a positive definite matrix and  is the multivariate gamma function.

 

Parameters (and constraints on parameters):

 ( = degrees of freedom, real)

 ( = inverse scale matrix, positive definite)

Support (i.e. values that it can take)

, i.e. is positive definite,  an  matrix

Probability density function

Mean

 

If the elements of  are  and the elements of  are  then

 

 

The main use of the inverse Wishart distribution appears to arise in Bayesian statistics. Suppose we want to make an inference about a covariance matrix, , whose prior  has a  distribution. If the observation set  where the  are independent -variate Normal (i.e. Gaussian) random variables drawn from a  distribution then the conditional distribution , i.e. the probability of  given , has a  distribution, where  is the sample covariance matrix.

 

The univariate special case of the inverse Wishart distribution is the inverse gamma distribution. With , , ,  we have:

 

 

where  is the ordinary (i.e. univariate) Gamma function, see MnGamma.

 

 

For other probability distributions see here.

 


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