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### The inverse Wishart distribution

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.