Notation W − 1 ( Ψ , ν ) {displaystyle {mathcal {W}}^{-1}({mathbf {Psi } },u )} Parameters ν > p − 1 {displaystyle u >p-1} degrees of freedom (real) Ψ > 0 {displaystyle mathbf {Psi } >0} scale matrix (pos. def) Support X {displaystyle mathbf {X} } is positive definite PDF | Ψ | ν 2 2 ν p 2 Γ p ( ν 2 ) | X | − ν + p + 1 2 e − 1 2 tr ( Ψ X − 1 ) {displaystyle {rac {left|{mathbf {Psi } }ight|^{rac {u }{2}}}{2^{rac {u p}{2}}Gamma _{p}({rac {u }{2}})}}left|mathbf {X} ight|^{-{rac {u +p+1}{2}}}e^{-{rac {1}{2}}operatorname {tr} ({mathbf {Psi } }mathbf {X} ^{-1})}} Γ p {displaystyle Gamma _{p}} is the multivariate gamma function t r {displaystyle mathrm {tr} } is the trace function Mean Ψ ν − p − 1 {displaystyle {rac {mathbf {Psi } }{u -p-1}}} For ν > p + 1 {displaystyle u >p+1} Mode Ψ ν + p + 1 {displaystyle {rac {mathbf {Psi } }{u +p+1}}} |
In statistics, the inverse Wishart distribution, also called the inverted Wishart distribution, is a probability distribution defined on real-valued positive-definite matrices. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution.
Contents
- Density
- Distribution of the inverse of a Wishart distributed matrix
- Marginal and conditional distributions from an inverse Wishart distributed matrix
- Conjugate distribution
- Moments
- Related distributions
- References
We say
Density
The probability density function of the inverse Wishart is:
where
Distribution of the inverse of a Wishart-distributed matrix
If
Marginal and conditional distributions from an inverse Wishart-distributed matrix
Suppose
where
i)
ii)
iii)
iv)
Conjugate distribution
Suppose we wish to make inference about a covariance matrix
Because the prior and posterior distributions are the same family, we say the inverse Wishart distribution is conjugate to the multivariate Gaussian.
Due to its conjugacy to the multivariate Gaussian, it is possible to marginalize out (integrate out) the Gaussian's parameter
(this is useful because the variance matrix
Moments
The following is based on Press, S. J. (1982) "Applied Multivariate Analysis", 2nd ed. (Dover Publications, New York), after reparameterizing the degree of freedom to be consistent with the p.d.f. definition above.
The mean:
The variance of each element of
The variance of the diagonal uses the same formula as above with
The covariance of elements of
Related distributions
A univariate specialization of the inverse-Wishart distribution is the inverse-gamma distribution. With
i.e., the inverse-gamma distribution, where
A generalization is the inverse multivariate gamma distribution.
Another generalization has been termed the generalized inverse Wishart distribution,
A different type of generalization is the normal-inverse-Wishart distribution, essentially the product of a multivariate normal distribution with an inverse Wishart distribution.