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Template:Probability distribution

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.

We say X follows an inverse Wishart distribution, denoted as X𝒲1(Ψ,ν), if its inverse X1 has a Wishart distribution 𝒲(Ψ1,ν).

Density

The probability density function of the inverse Wishart is:

|Ψ|ν22νp2Γp(ν2)|X|ν+p+12e12tr(ΨX1)

where X and Ψ are p×p positive definite matrices, and Γp(·) is the multivariate gamma function.

Theorems

Distribution of the inverse of a Wishart-distributed matrix

If A𝒲(Σ,ν) and Σ is of size p×p, then X=A1 has an inverse Wishart distribution X𝒲1(Σ1,ν) .[1]

Marginal and conditional distributions from an inverse Wishart-distributed matrix

Suppose A𝒲1(Ψ,ν) has an inverse Wishart distribution. Partition the matrices A and Ψ conformably with each other

A=[A11A12A21A22],Ψ=[Ψ11Ψ12Ψ21Ψ22]

where Aij and Ψij are pi×pj matrices, then we have

i) A11 is independent of A111A12 and A221, where A221=A22A21A111A12 is the Schur complement of A11 in A;

ii) A11𝒲1(Ψ11,νp2);

iii) A111A12|A221MNp1×p2(Ψ111Ψ12,A221Ψ111), where MNp×q(,) is a matrix normal distribution;

iv) A221𝒲1(Ψ221,ν), where Ψ221=Ψ22Ψ21Ψ111Ψ12;

Conjugate distribution

Suppose we wish to make inference about a covariance matrix Σ whose prior p(Σ) has a 𝒲1(Ψ,ν) distribution. If the observations X=[x1,,xn] are independent p-variate Gaussian variables drawn from a N(0,Σ) distribution, then the conditional distribution p(Σ|X) has a 𝒲1(A+Ψ,n+ν) distribution, where A=XXT is n1 times the sample 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 Σ.

P(X|Ψ,ν)=P(X|Σ)P(Σ|Ψ,ν)dΣ=|Ψ|ν2Γp(ν+n2)πnp2|Ψ+A|ν+n2Γp(ν2)

(this is useful because the variance matrix Σ is not known in practice, but because Ψ is known a priori, and A can be obtained from the data, the right hand side can be evaluated directly).

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.

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E(X)=Ψνp1.

The variance of each element of X:

Var(xij)=(νp+1)ψij2+(νp1)ψiiψjj(νp)(νp1)2(νp3)

The variance of the diagonal uses the same formula as above with i=j, which simplifies to:

Var(xii)=2ψii2(νp1)2(νp3).

The covariance of elements of X are given by:

Cov(xij,xkl)=2ψijψkl+(νp1)(ψikψjl+ψilψkj)(νp)(νp1)2(νp3)

Related distributions

A univariate specialization of the inverse-Wishart distribution is the inverse-gamma distribution. With p=1 (i.e. univariate) and α=ν/2, β=Ψ/2 and x=X the probability density function of the inverse-Wishart distribution becomes

p(x|α,β)=βαxα1exp(β/x)Γ1(α).

i.e., the inverse-gamma distribution, where Γ1() is the ordinary Gamma function.

A generalization is the inverse multivariate gamma distribution.

Another generalization has been termed the generalized inverse Wishart distribution, 𝒢𝒲1. A p×p positive definite matrix X is said to be distributed as 𝒢𝒲1(Ψ,ν,S) if Y=X1/2S1X1/2 is distributed as 𝒲1(Ψ,ν). Here X1/2 denotes the symmetric matrix square root of X, the parameters Ψ,S are p×p positive definite matrices, and the parameter ν is a positive scalar larger than 2p. Note that when S is equal to an identity matrix, 𝒢𝒲1(Ψ,ν,S)=𝒲1(Ψ,ν). This generalized inverse Wishart distribution has been applied to estimating the distributions of multivariate autoregressive processes.[2]

A different type of generalization is the normal-inverse-Wishart distribution, essentially the product of a multivariate normal distribution with an inverse Wishart distribution.

See also

References

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