Marginal And Affine Laws For Matrix-T Variables

Why Marginals Need Their Own Degrees-of-Freedom Check

For a matrix-valued random variable, selecting rows or columns is a linear operation, but the resulting distribution parameters do not always change in the same way. Under the matrix-t convention used here, row uncertainty is generated by a random row covariance matrix. Removing rows changes the degrees of freedom attached to that covariance; removing columns does not. Keeping this distinction explicit prevents a common error: copying the original degrees of freedom to every submatrix without checking which dimension was reduced.

We first specify the convention, since matrix-t distributions occur with more than one scale and degrees-of-freedom convention. All marginal and affine formulas below follow from this definition.

In this convention, the marginal density of \(X\) is

where \(\Gamma_p\) is the multivariate gamma function. The exponent involves the number of columns \(q\), while the admissibility condition \(\nu>p-1\) involves the number of rows \(p\). These features are useful checks on the chosen parameterization.

Submatrices: The Marginal Law

Let \(I\) be a set of \(r\) row indices and \(J\) a set of \(s\) column indices. Denote the corresponding submatrices by \(X_{I,J}\), \(M_{I,J}\), and the principal scale submatrices by \(\Sigma_{I,I}\) and \(\Omega_{J,J}\). The result is

The row adjustment is compatible with the existence condition for the reduced matrix: \(\nu-p+r>r-1\) is equivalent to \(\nu>p-1\). It is not an arbitrary correction. It reflects the marginal law of a principal submatrix of the inverse-Wishart row covariance.

For a row selector \(A\), this is the principal-submatrix marginal property of the inverse-Wishart distribution. One way to obtain it is to partition the inverse-Wishart density into the selected and omitted coordinates and integrate the omitted block and cross-block. The matrix gamma integral \(\int_{H>0}|H|^{a-(k+1)/2}\exp\{-\operatorname{tr}(KH)/2\}\,dH =2^{ak}\Gamma_k(a)|K|^{-a}\) evaluates the block integrals; the remaining density in the selected covariance block \(G\) is proportional to \(|G|^{-(\nu-p+r+r+1)/2}\exp\{-\operatorname{tr}(\Sigma_{I,I}G^{-1})/2\}\). This is the \(IW_r(\nu-p+r,\Sigma_{I,I})\) density. A general full-row-rank \(A\) follows by an invertible change of row coordinates. \(\square\)

We can now prove the submatrix theorem. Conditional on \(C\), selecting rows \(I\) and columns \(J\) from a matrix-normal matrix gives \(X_{I,J}\mid C_{I,I}\sim MN_{r,s}(M_{I,J},C_{I,I},\Omega_{J,J})\). The selected column covariance is the corresponding principal submatrix, and the selected row covariance has the inverse-Wishart law in the lemma. Mixing this matrix-normal law over that inverse-Wishart distribution gives precisely the defining representation of the stated \(T_{r,s}\) law. \(\square\)

Affine Transformations of Rows and Columns

Consider the transformation \(Y=AXB+D\), where \(A\) acts on rows, \(B\) acts on columns, and \(D\) is a fixed \(m\times s\) matrix. A useful full-rank case has \(A\) of size \(m\times p\) with full row rank, where \(m\le p\), and \(B\) of size \(q\times s\) with full column rank, where \(s\le q\). Then \(Y\) has a full-dimensional matrix-t law:

To prove the result, use the defining mixture. Conditional on \(C\), the transformed matrix is matrix-normal with mean \(AMB+D\), row covariance \(ACA'\), and column covariance \(B'\Omega B\). The lemma gives \(ACA'\sim IW_m(\nu-p+m,A\Sigma A')\). Mixing therefore gives the asserted matrix-t distribution. The scales are positive definite: for nonzero \(u\in\mathbb{R}^m\), \(u'A\Sigma A'u=(A'u)'\Sigma(A'u)>0\), since full row rank makes \(A'u\ne0\); the column argument is analogous. \(\square\)

Scope, Checks, and a Common Pitfall

The full-rank requirements matter. If \(A\) or \(B\) is rank deficient, the transformed matrix may lie in a lower-dimensional subspace; it then does not have a nonsingular matrix-t density of the displayed dimension. If a transformation expands a dimension rather than reducing it, its image is likewise constrained by the original matrix and should not be assigned a full-dimensional density by the formula above.

A common mistake is to say, “A marginal of a matrix-t has the same degrees of freedom, just as a marginal of an ordinary multivariate \(t\) does.” That reasoning ignores which matrix-t convention is being used. Here the row covariance is inverse-Wishart mixed, and retaining \(r\) of \(p\) rows produces degrees of freedom \(\nu-p+r\). A column-only marginal retains \(\nu\), because it does not take a lower-dimensional marginal of that row covariance. For a square nonsingular affine transformation, both dimensions are unchanged, so the original \(\nu\) is retained.

The parameterization should always be checked before comparing formulas from different sources. In this convention, row scale means the inverse-Wishart scale and the density exponent is \(-(\nu+q)/2\). Consequently, a formula using a different matrix-t scale convention cannot be substituted without converting its parameters.