Inverse Wishart Distribution

Definition

The probability density function of the d-dimensional Inverse Wishart distribution is given by

$y = f (Χ, Σ, ν) = \frac{{| T |}^{(ν / 2)} e^{(- \frac{1}{2} trace (T X^{- 1}))}}{2^{(ν d) / 2} π^{(d (d - 1)) / 4} {| X |}^{(ν + d + 1) / 2} Γ (ν / 2) ... Γ (ν - (d - 1)) / 2},$

where X and T are d-by-d symmetric positive definite matrices, and ν is a scalar greater than or equal to d. While it is possible to define the Inverse Wishart for singular Τ, the density cannot be written as above.

If a random matrix has a Wishart distribution with parameters T^–1 and ν, then the inverse of that random matrix has an inverse Wishart distribution with parameters Τ and ν. The mean of the distribution is given by

$\frac{1}{ν - d - 1} T$

where d is the number of rows and columns in T.

Only random matrix generation is supported for the inverse Wishart, including both singular and nonsingular T.

Background

The inverse Wishart distribution is based on the Wishart distribution. In Bayesian statistics it is used as the conjugate prior for the covariance matrix of a multivariate normal distribution.

Example

Notice that the sampling variability is quite large when the degrees of freedom is small.

Tau = [1 .5; .5 2];
df = 10; S1 = iwishrnd(Tau,df)*(df-2-1)

S1 =
       1.7959      0.64107
      0.64107       1.5496

df = 1000; S2 = iwishrnd(Tau,df)*(df-2-1)

S2 =
       0.9842      0.50158
      0.50158       2.1682