Matrix-T Density And Competing Scale Conventions

Why Matrix-T Conventions Need to Be Explicit

A matrix-t distribution can arise by mixing a matrix-normal distribution over a random precision matrix. The crucial choice is which matrix dimension carries that random precision: the row dimension or the column dimension. That choice changes the degrees of freedom in the density’s exponent, the dimension of the Wishart mixing distribution, and the condition that makes the mixing distribution proper.

We will use $\Sigma$ for a positive-definite row scale and $\Omega$ for a positive-definite column scale. The matrix dimensions are $p$ rows and $q$ columns. Throughout, $\nu$ is the degrees-of-freedom parameter of the Wishart mixing distribution—not a parameter to infer from the exponent alone.

Definition: Let $\Sigma$ be a $p\times p$ positive-definite matrix, $\Omega$ a $q\times q$ positive-definite matrix, $M$ a $p\times q$ matrix, and $\nu>p-1$. The row-precision matrix-t convention is the law of $X$ generated by $$ S\sim W_p(\nu,\Sigma^{-1}), \qquad X\mid S\sim MN_{p,q}(M,S^{-1},\Omega). $$ Here $W_p(\nu,\Sigma^{-1})$ denotes a Wishart distribution with scale matrix $\Sigma^{-1}$, and the matrix-normal distribution has row covariance $S^{-1}$ and column covariance $\Omega$. The resulting density is $$ f(X)= \frac{\Gamma_p\!\left((\nu+q)/2\right)} {\pi^{pq/2}\Gamma_p(\nu/2)\,|\Sigma|^{q/2}|\Omega|^{p/2}} \left|I_p+\Sigma^{-1}(X-M)\Omega^{-1}(X-M)'\right|^{-(\nu+q)/2}. $$

The multivariate gamma function in this expression is $$ \Gamma_p(a)=\pi^{p(p-1)/4}\prod_{j=1}^{p}\Gamma\!\left(a-\frac{j-1}{2}\right), \qquad a>\frac{p-1}{2}. $$ The restriction $\nu>p-1$ is the usual condition for the $p$-dimensional Wishart mixing distribution to be proper. The density is defined for every real $p\times q$ matrix $X$; its support is not restricted to positive-definite matrices.

Deriving the Row-Precision Density

Write $D=X-M$. Under the matrix-normal convention, the conditional density of $X$ given $S$ is $$ f(X\mid S)= (2\pi)^{-pq/2}|S|^{q/2}|\Omega|^{-p/2} \exp\!\left\{-\frac12\operatorname{tr}\!\left(SD\Omega^{-1}D'\right)\right\}. $$ The Wishart density of $S$, with scale $\Sigma^{-1}$, is $$ f(S)= \frac{|\Sigma|^{\nu/2}}{2^{\nu p/2}\Gamma_p(\nu/2)} |S|^{(\nu-p-1)/2} \exp\!\left\{-\frac12\operatorname{tr}(\Sigma S)\right\}, \qquad S>0. $$ Multiplying these densities combines the two trace terms. Their matrix in the resulting exponential is $B=\Sigma+D\Omega^{-1}D'$, which is positive definite because $\Sigma$ is positive definite and $D\Omega^{-1}D'$ is positive semidefinite.

Theorem: For $a>(p-1)/2$ and positive-definite $B$, the Wishart gamma integral is $$ \int_{S>0}|S|^{a-(p+1)/2} \exp\!\left\{-\frac12\operatorname{tr}(BS)\right\}\,dS =2^{ap}\Gamma_p(a)|B|^{-a}. $$

In the product $f(X\mid S)f(S)$, the power of $|S|$ is $$ \frac{q}{2}+\frac{\nu-p-1}{2} =\frac{\nu+q-p-1}{2}. $$ Thus the gamma integral applies with $a=(\nu+q)/2$. Integrating out $S$ gives

\begin{aligned} f(X) &=(2\pi)^{-pq/2}|\Omega|^{-p/2} \frac{|\Sigma|^{\nu/2}}{2^{\nu p/2}\Gamma_p(\nu/2)} \frac{2^{p(\nu+q)/2}\Gamma_p((\nu+q)/2)} {|\,\Sigma+D\Omega^{-1}D'\,|^{(\nu+q)/2}}\\ &= \frac{\Gamma_p((\nu+q)/2)} {\pi^{pq/2}\Gamma_p(\nu/2)|\Omega|^{p/2}} \frac{|\Sigma|^{\nu/2}} {|\,\Sigma+D\Omega^{-1}D'\,|^{(\nu+q)/2}}. \end{aligned}

The power of $2$ left after integration is $2^{pq/2}$, which cancels the $2^{-pq/2}$ from $(2\pi)^{-pq/2}$, leaving $\pi^{-pq/2}$. To obtain the stated scale factors, factor $\Sigma$ from the determinant: $$ |\Sigma+D\Omega^{-1}D'| =|\Sigma|\left|I_p+\Sigma^{-1}D\Omega^{-1}D'\right|. $$ This contributes $|\Sigma|^{-(\nu+q)/2}$; combined with $|\Sigma|^{\nu/2}$, it gives $|\Sigma|^{-q/2}$. The result is the density in the definition. Since it was obtained by integrating a proper joint density, it integrates to one. $\square$

Worked Example: Evaluating Both Conventions for a Two-Row Matrix

Let $p=2$, $q=1$, $\nu=5$, $M=0$, $\Sigma=\operatorname{diag}(4,1)$, $\Omega=(9)$, and $X=(2,3)'$. First evaluate the row-precision density.

The multivariate-gamma ratio is $$ \frac{\Gamma_2(3)}{\Gamma_2(5/2)} =\frac{3\pi/2}{3\pi/4}=2. $$ Also, $|\Sigma|^{1/2}=2$, $|\Omega|^{p/2}=9$, and $$ X'\Sigma^{-1}X =(2,3) \begin{pmatrix}1/4&0\\0&1\end{pmatrix} \begin{pmatrix}2\\3\end{pmatrix} =1+9=10. $$ The determinant inside the density is the scalar $1+10/9=19/9$. Therefore $$ f(X)=\frac{2}{\pi\cdot2\cdot9} \left(\frac{19}{9}\right)^{-3} =\frac{1}{9\pi}\left(\frac{9}{19}\right)^3 =\frac{81}{6859\pi}. $$

For comparison, mixing column precision uses a scalar Wishart distribution with $\nu=5$. Its gamma ratio is $\Gamma(7/2)/\Gamma(5/2)=5/2$, and its exponent is $(\nu+p)/2=7/2$. The determinant factor involving the scales is still $2\cdot9$, so this convention gives $$ f_{\mathrm{column}}(X) =\frac{5}{36\pi}\left(\frac{19}{9}\right)^{-7/2}. $$ The two evaluated densities differ. The row-precision density uses a $p$-dimensional Wishart and exponent $(\nu+q)/2$; the column-precision density uses a $q$-dimensional Wishart and exponent $(\nu+p)/2$.

The Alternative: Mixing Column Precision

There is another coherent construction. Let $U$ be a random precision on the column dimension, rather than the row dimension. The conditional matrix-normal distribution then has row covariance $\Sigma$ and column covariance $U^{-1}$.

Definition: For $\nu>q-1$, the column-precision matrix-t convention is generated by $$ U\sim W_q(\nu,\Omega^{-1}), \qquad X\mid U\sim MN_{p,q}(M,\Sigma,U^{-1}). $$ Its density is $$ f_{\mathrm{column}}(X)= \frac{\Gamma_q\!\left((\nu+p)/2\right)} {\pi^{pq/2}\Gamma_q(\nu/2)\,|\Sigma|^{q/2}|\Omega|^{p/2}} \left|I_q+\Omega^{-1}(X-M)'\Sigma^{-1}(X-M)\right|^{-(\nu+p)/2}. $$

This formula follows by the same integration as before, now integrating over the $q\times q$ matrix $U$. In the conditional density, $|U|$ has power $p/2$, so the resulting Wishart gamma integral has parameter $(\nu+p)/2$. The scale determinant becomes $|\Omega|^{-p/2}$, while $|\Sigma|^{-q/2}$ comes from the matrix-normal density. Properness requires $\nu>q-1$, the Wishart condition in dimension $q$.

The two determinant terms are equal as scalars. For compatible rectangular matrices $A$ and $B$, the identity $|I+AB|=|I+BA|$ gives $$ \left|I_p+\Sigma^{-1}D\Omega^{-1}D'\right| = \left|I_q+\Omega^{-1}D'\Sigma^{-1}D\right|. $$ However, equal determinant kernels do not by themselves make the densities equal: the exponents and multivariate-gamma factors can differ. If $p=q$, the exponents and gamma factors match, so the two formulas agree for the same parameters. If $p\ne q$, they are generally different distributions.

Worked Example: Checking Agreement When the Matrix Is Square

Take $p=q=2$, $\nu=4$, $M=0$, $\Sigma=\Omega=I_2$, and $X=\operatorname{diag}(1,2)$. Both Wishart conditions hold because $4>2-1$. The determinant in either convention is $$ |I_2+XX'| =\left|\begin{pmatrix}2&0\\0&5\end{pmatrix}\right| =10. $$ The exponent in both formulas is $(4+2)/2=3$, and $$ \frac{\Gamma_2(3)}{\Gamma_2(2)} =\frac{3\pi/2}{\pi/2}=3. $$ The scale determinants are one, so both conventions yield $$ f(X)=\frac{3}{\pi^2}\,10^{-3} =\frac{3}{1000\pi^2}. $$ Here agreement follows from $p=q$, not from a general equivalence between the two mixing constructions.

Worked Example: Different Densities at the Center for a Rectangular Matrix

Let $p=1$, $q=2$, $\nu=3$, $M=0$, $\Sigma=(4)$, and $\Omega=\operatorname{diag}(1,4)$. Both conventions are proper: $3>1-1$ for row precision and $3>2-1$ for column precision. Evaluate each density at $X=M$. The determinant factor is one in both formulas.

For row precision, $\Gamma_1(5/2)/\Gamma_1(3/2)=3/2$, while $|\Sigma|^{q/2}=4$ and $|\Omega|^{p/2}=2$. Hence $$ f_{\mathrm{row}}(M) =\frac{3/2}{\pi\cdot4\cdot2} =\frac{3}{16\pi}. $$ For column precision, $\Gamma_2(2)=\Gamma_2(3/2)=\pi/2$, so their ratio is one. The scale determinant factors are again $4$ and $2$, giving $$ f_{\mathrm{column}}(M)=\frac{1}{8\pi}. $$ These values differ. The row-precision exponent is $(3+2)/2=5/2$, while the column-precision exponent is $(3+1)/2=2$. This rectangular case makes clear that the mixing dimension is part of the distribution’s specification.

Common Mistake and Practical Interpretation

Common mistake: “The two densities have the same determinant term, so they define the same matrix-t distribution.” This reasoning overlooks that the shared determinant is raised to different powers when $p\ne q$, and that the multivariate-gamma normalizing constants may also differ. The correction is to record the full convention: which dimension is mixed, which Wishart distribution is used, and the associated degrees-of-freedom condition.

This distinction matters when reading a formula, implementing a likelihood, or comparing software output. The labels “row scale” and “column scale” alone are not sufficient to identify a matrix-t law: one must also specify whether the random precision is on rows or columns. In particular, the Wishart dimension determines the admissible range of $\nu$, while the other matrix dimension enters the exponent.

Key takeaway: Under row-precision mixing, $S\sim W_p(\nu,\Sigma^{-1})$, the condition is $\nu>p-1$, and the exponent is $(\nu+q)/2$. Under column-precision mixing, $U\sim W_q(\nu,\Omega^{-1})$, the condition is $\nu>q-1$, and the exponent is $(\nu+p)/2$. State the convention before using a matrix-t density.

Check Your Understanding

Answer using the row-precision and column-precision conventions defined above.

For a $p\times q$ matrix under row-precision mixing, what is the dimension of the Wishart matrix and what condition must $\nu$ satisfy?
Which matrix dimension appears alongside $\nu$ in the exponent of the row-precision density?
For the column-precision convention, identify the conditional row and column covariance matrices.
Why does equality of the two determinant kernels not generally imply equality of the two matrix-t densities?
When do the two displayed conventions give the same density for the same parameters?

Matrix-T Density And Competing Scale Conventions

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