The Inverse-Wishart Density

Why Matrix Inversion Changes the Density

In multivariate sampling theory, a Wishart matrix often represents accumulated scatter or information. Its inverse appears when a covariance-like quantity is inverted, so its distribution matters in its own right. The transformation is not obtained by merely replacing a matrix with its inverse in the Wishart formula: the density must also account for how inversion changes volume in the space of symmetric matrices.

We use the Wishart scale convention \(W\sim W_p(\nu,\Psi)\), where \(\Psi\) is a positive-definite scale matrix and \(\nu\) is the degrees-of-freedom parameter. The central Wishart density, with respect to Lebesgue measure on the distinct entries of a symmetric matrix, is the starting point. The key calculation is the Jacobian of the map \(W\mapsto W^{-1}\).

The scale convention is important: if \(W\sim W_p(\nu,\Psi)\), then \(W^{-1}\sim IW_p(\nu,\Psi)\). In the exponential term, \(\Psi^{-1}\) multiplies \(X^{-1}\); it is not \(\Psi\). Also, \(\nu>p-1\) is the condition for the standard Wishart density on the full positive-definite cone. In the Gaussian-sample construction, the degrees of freedom are often an integer and the sample scatter has full rank when \(\nu\ge p\).

The Jacobian of Matrix Inversion

A symmetric \(p\times p\) matrix has \(p(p+1)/2\) distinct coordinates: its diagonal entries and its entries above the diagonal. Densities on symmetric matrices are understood relative to Lebesgue measure in those coordinates. The following result gives the volume change needed for inversion.

To prove this, first consider the linear map on symmetric matrices \(H\mapsto AHA^\mathsf{T}\), for nonsingular \(A\). Its absolute determinant is \(|A|^{p+1}\). Indeed, write a singular value decomposition \(A=UDV^\mathsf{T}\), where \(U,V\) are orthogonal and \(D=\operatorname{diag}(d_1,\ldots,d_p)\) with \(d_i>0\). The orthogonal congruence maps \(H\mapsto UHU^\mathsf{T}\) and \(H\mapsto VHV^\mathsf{T}\) preserve volume on the space of symmetric matrices, so it is enough to calculate for \(D\).

For diagonal \(D\), the coordinate \(h_{ii}\) is multiplied by \(d_i^2\), while each off-diagonal coordinate \(h_{ij}\), \(i<j\), is multiplied by \(d_i d_j\). Thus the absolute determinant is $$ \left(\prod_{i=1}^p d_i^2\right) \left(\prod_{i<j}d_i d_j\right) =\prod_{i=1}^p d_i^{p+1} =|A|^{p+1}. $$ For inversion, differentiating \(X X^{-1}=I\) in a symmetric direction \(H\) gives $$ D(X\mapsto X^{-1})[H]=-X^{-1}HX^{-1}. $$ This is the negative of a congruence map with \(A=X^{-1}\). Multiplication by \(-1\) has absolute determinant one on the symmetric-matrix space. The congruence result therefore gives the absolute Jacobian \(|X^{-1}|^{p+1}=|X|^{-(p+1)}\). \(\square\)

Deriving the Inverse-Wishart Density

For \(\nu>p-1\) and \(\Psi\succ0\), the central Wishart density is $$ f_W(W)= \frac{|W|^{(\nu-p-1)/2}} {2^{\nu p/2}|\Psi|^{\nu/2}\Gamma_p(\nu/2)} \exp\!\left[-\frac12\operatorname{tr}(\Psi^{-1}W)\right], \qquad W\succ0. $$ Set \(X=W^{-1}\), equivalently \(W=X^{-1}\). Inversion is a one-to-one differentiable map from the positive-definite cone onto itself, and its absolute Jacobian is \(|X|^{-(p+1)}\). The change-of-variables formula gives $$ \begin{aligned} f_X(X) &=f_W(X^{-1})|X|^{-(p+1)}\\ &=\frac{|X^{-1}|^{(\nu-p-1)/2}} {2^{\nu p/2}|\Psi|^{\nu/2}\Gamma_p(\nu/2)} \exp\!\left[-\frac12\operatorname{tr}(\Psi^{-1}X^{-1})\right] |X|^{-(p+1)}\\ &=\frac{|\Psi|^{-\nu/2}} {2^{\nu p/2}\Gamma_p(\nu/2)} |X|^{-(\nu+p+1)/2} \exp\!\left[-\frac12\operatorname{tr}(\Psi^{-1}X^{-1})\right], \qquad X\succ0. \end{aligned} $$ This is the inverse-Wishart density in the scale convention used here. The determinant exponent follows because \(|X^{-1}|=|X|^{-1}\), so the two powers of \(|X|\) combine as $$ -\frac{\nu-p-1}{2}-(p+1)=-\frac{\nu+p+1}{2}. $$ The scale determinant remains \(|\Psi|^{-\nu/2}\): inversion changes the variable-dependent factor and contributes its Jacobian, but it does not change the Wishart normalizing factor involving \(\Psi\). Since the transformation is bijective and the Wishart density integrates to one, the resulting inverse-Wishart density also integrates to one.

Worked Densities

Support, Parameterization, and a Common Pitfall

The inverse-Wishart support is the open cone \(X\succ0\), just as the Wishart support is \(W\succ0\). Inversion preserves positive definiteness: if \(X\succ0\), then \(X^{-1}\succ0\); conversely, the inverse of a positive-definite matrix is positive definite. The degrees-of-freedom condition in the density is \(\nu>p-1\), inherited from the Wishart density being transformed. This condition should not be confused with the stronger integer rank condition that arises in one particular construction from a finite Gaussian sample.

A frequent error is to invert the argument of the density but leave out the Jacobian. That would give a determinant power \(-(\nu-p-1)/2\), rather than the correct \(-(\nu+p+1)/2\). The missing factor is precisely \(|X|^{-(p+1)}\). Another common error is to change the sign of the scale determinant exponent while transforming. The Wishart density contains \(|\Psi|^{-\nu/2}\); because the Jacobian depends on \(X\), not \(\Psi\), this factor remains unchanged.