Conditional Distributions Under Matrix-T Models

Why Conditioning Changes the Matrix-T Law

For a matrix-normal model, conditioning on one block gives another matrix-normal law whose covariance parameters are determined by the original covariance structure. Under a matrix-t model, the corresponding conditional law is still matrix-t, but its degrees of freedom increase and its row scale depends on the block that was observed. This dependence is a consequence of the common scale uncertainty in the joint matrix-t distribution.

We consider a split into column blocks. The same calculations can be adapted to row blocks, but the block dimensions and scale matrices must be transposed consistently. Throughout, the row-scale matrix describes the row dimension and the column-scale matrix describes the column dimension; neither should be silently interchanged.

Parameterization and Block Notation

Let $X$ be a $p\times q$ random matrix with location matrix $M$, positive-definite row-scale matrix $\Sigma$ of size $p\times p$, positive-definite column-scale matrix $\Omega$ of size $q\times q$, and degrees of freedom $\nu>p-1$. In this tutorial, $X\sim MT_{p,q}(\nu,M,\Sigma,\Omega)$ means the density below. This specifies the scale convention explicitly.

Definition: The matrix-t distribution $MT_{p,q}(\nu,M,\Sigma,\Omega)$ has density, with respect to the $pq$ entries of $X$, $$ f(X)= \frac{\Gamma_p\!\left((\nu+q)/2\right)} {\pi^{pq/2}\Gamma_p(\nu/2)|\Sigma|^{q/2}|\Omega|^{p/2}} \left|I_p+\Sigma^{-1}(X-M)\Omega^{-1}(X-M)'\right|^{-(\nu+q)/2}, $$ where $\Gamma_p(a)=\pi^{p(p-1)/4}\prod_{i=1}^{p}\Gamma\!\left(a-(i-1)/2\right)$ is the multivariate gamma function. Here $\Sigma$ and $\Omega$ are scale matrices in this density, not automatically covariance matrices.

Split the columns into $q_1$ columns to be conditioned on and $q_2$ columns to be observed, so $q=q_1+q_2$, with $q_1,q_2\geq 1$. Write $$ X=[X_1\ X_2],\qquad M=[M_1\ M_2],\qquad \Omega= \begin{pmatrix} \Omega_{11}&\Omega_{12}\\ \Omega_{21}&\Omega_{22} \end{pmatrix}. $$ The block $\Omega_{22}$ is positive definite. Define the Schur complement and the centered observed block by $$ C=\Omega_{11}-\Omega_{12}\Omega_{22}^{-1}\Omega_{21}, \qquad Y_2=X_2-M_2. $$ Because $\Omega$ is positive definite, $C$ is positive definite as well.

Worked Example: A One-Column Conditional Block

Suppose $p=2$, $q_1=q_2=1$, $\nu=5$, $M=0$, $\Sigma=I_2$, and $$ \Omega=\begin{pmatrix}2&1\\1&2\end{pmatrix}. $$ Condition on $X_2=(1,0)'$.

The blocks are $\Omega_{11}=2$, $\Omega_{12}=\Omega_{21}=1$, and $\Omega_{22}=2$. Therefore $$ C=2-(1)(1/2)(1)=3/2, \qquad \Omega_{22}^{-1}\Omega_{21}=1/2. $$ The conditional location for $X_1$ is $M_1+Y_2/2=(1/2,0)'$. The observed-block contribution to the row scale is $$ Y_2\Omega_{22}^{-1}Y_2' =\frac12\begin{pmatrix}1&0\\0&0\end{pmatrix}. $$ Thus the conditional row scale is $\begin{pmatrix}3/2&0\\0&1\end{pmatrix}$. The conditional degrees of freedom are $5+1=6$, so the conditional law is $MT_{2,1}$ with degrees of freedom $6$, location $(1/2,0)'$, row scale $\operatorname{diag}(3/2,1)$, and column scale $3/2$.

The Conditional Distribution

For a particular observed value $X_2=x_2$, use $Y_2=x_2-M_2$ and define $$ U=X_1-M_1-Y_2\Omega_{22}^{-1}\Omega_{21}, \qquad S=\Sigma+Y_2\Omega_{22}^{-1}Y_2'. $$ The matrix $S$ is positive definite: $\Sigma$ is positive definite and the added term is positive semidefinite.

Theorem: If $X\sim MT_{p,q}(\nu,M,\Sigma,\Omega)$ under the conditions above, then, for every $x_2$, the conditional distribution of $X_1$ given $X_2=x_2$ is $$ X_1\mid X_2=x_2 \sim MT_{p,q_1}\!\left( \nu+q_2,\ M_1+(x_2-M_2)\Omega_{22}^{-1}\Omega_{21},\ \Sigma+(x_2-M_2)\Omega_{22}^{-1}(x_2-M_2)',\ \Omega_{11}-\Omega_{12}\Omega_{22}^{-1}\Omega_{21} \right). $$ In particular, conditioning increases the degrees of freedom by the number $q_2$ of observed columns. The conditional location uses the column-scale regression coefficient, and the conditional row scale depends on the observed value.

Proof by Dividing the Joint Density by the Marginal Density

The marginal law of $X_2$ is the matrix-t marginal with row scale $\Sigma$, column scale $\Omega_{22}$, and degrees of freedom $\nu$. This is the marginal property of the matrix-t law. We now derive the conditional density by taking the ratio, keeping track of the determinant and normalizing constants.

The block inverse identity for $\Omega$ gives the quadratic-form decomposition $$ (X-M)\Omega^{-1}(X-M)' = U C^{-1}U'+Y_2\Omega_{22}^{-1}Y_2'. $$ Set $R=Y_2\Omega_{22}^{-1}Y_2'$ and $S=\Sigma+R$. Then $$ I_p+\Sigma^{-1}(X-M)\Omega^{-1}(X-M)' =\Sigma^{-1}\left(S+UC^{-1}U'\right). $$ Taking determinants and using $|I+AB|=|I+BA|$ yields $$ \left|I_p+\Sigma^{-1}(X-M)\Omega^{-1}(X-M)'\right| = \frac{|S|}{|\Sigma|} \left|I_{q_1}+C^{-1}U'S^{-1}U\right|. $$ For the marginal density of $X_2$, the corresponding determinant is $|S|/|\Sigma|$. Dividing the joint density by this marginal density leaves the factor $$ \left|I_{q_1}+C^{-1}U'S^{-1}U\right|^{-(\nu+q)/2}. $$ Since $q=q_1+q_2$, the exponent is $-(\nu+q_2+q_1)/2$, which is the matrix-t exponent for $q_1$ columns and degrees of freedom $\nu+q_2$.

It remains to verify the conditional normalizing constant. The determinant identity for the block scale is $|\Omega|=|\Omega_{22}||C|$. Thus the ratio of the joint and marginal constants is $$ \frac{\Gamma_p((\nu+q)/2)} {\pi^{pq_1/2}\Gamma_p((\nu+q_2)/2)} |\Sigma|^{-q_1/2}|C|^{-p/2}. $$ The determinant factor remaining from dividing the joint and marginal densities contributes $(|S|/|\Sigma|)^{-q_1/2}$. Multiplying the $|\Sigma|$ factors gives $|S|^{-q_1/2}$. The resulting constant is exactly the matrix-t normalizer with degrees of freedom $\nu+q_2$, row scale $S$, and column scale $C$. This proves the stated conditional law. $\square$

Worked Example: Two Conditional Columns

Let $p=1$, $q_1=2$, $q_2=1$, $\nu=5$, $M=0$, and $\Sigma=4$. Take $$ \Omega= \begin{pmatrix} 2&0&1\\ 0&1&0\\ 1&0&1 \end{pmatrix}, \qquad x_2=2. $$ This scale matrix is positive definite: its lower-right block is $1$, and its Schur complement is $I_2$.

Here $\Omega_{22}=1$, $\Omega_{21}=(1,0)$, and $\Omega_{11}=\operatorname{diag}(2,1)$, so $$ C=\Omega_{11}-\Omega_{12}\Omega_{22}^{-1}\Omega_{21} =\begin{pmatrix}1&0\\0&1\end{pmatrix}. $$ The conditional location is $x_2\Omega_{22}^{-1}\Omega_{21}'=(2,0)$, and the conditional row scale is $4+2(1)^{-1}2=8$. The degrees of freedom increase from $5$ to $6$, because one column is observed. Therefore $X_1\mid X_2=2$ has matrix-t law $MT_{1,2}$ with degrees of freedom $6$, location $(2,0)$, row scale $8$, and column scale $I_2$.

Worked Example: An Observation Alters the Row Scale

Let $p=2$, $q_1=q_2=1$, $\nu=4$, $M=0$, $\Sigma=\operatorname{diag}(2,1)$, and $\Omega=I_2$. Condition on $X_2=(1,1)'$.

Since the column scales are independent, $\Omega_{22}=1$, $\Omega_{21}=0$, and $C=1$. The conditional location stays at zero. But the observed value changes the row scale: $$ S=\begin{pmatrix}2&0\\0&1\end{pmatrix} +\begin{pmatrix}1\\1\end{pmatrix} \begin{pmatrix}1&1\end{pmatrix} =\begin{pmatrix}3&1\\1&2\end{pmatrix}. $$ Its determinant is $3(2)-1(1)=5>0$, as required for a positive-definite scale matrix. The conditional degrees of freedom are $4+1=5$. Hence $X_1\mid X_2=(1,1)'$ has law $MT_{2,1}$ with degrees of freedom $5$, location zero, row scale $S$, and column scale $1$. If instead the observed block were at its location, $x_2=0$, the added outer product would be zero and the conditional row scale would remain $\Sigma$.

Interpreting the Updates

The conditional location is a linear adjustment of $M_1$ based on the observed block. Its coefficient, $\Omega_{22}^{-1}\Omega_{21}$, reflects dependence between the two column blocks under the column-scale matrix. If the off-diagonal block $\Omega_{21}$ is zero, the conditional location is unchanged, as in the last example.

The row-scale update has a different role. It adds the positive-semidefinite matrix $Y_2\Omega_{22}^{-1}Y_2'$ to $\Sigma$. Thus the observed value can alter both the size and the orientation of the conditional scale. This is a scale parameter in the conditional density; it should not be called a covariance without separately checking the moment conditions and the parameterization.

The degrees-of-freedom update is tied to the dimension of the observed block, not to the size of the observed values. Changing $x_2$ changes $S$ and the conditional location, but it does not change $\nu+q_2$. This separation is useful when evaluating conditional densities or comparing conditional distributions across possible observations.

Common Mistake

A frequent error is to condition a matrix-t variable as though it were matrix-normal: keep the original degrees of freedom and scales, and merely substitute a regression-adjusted location. That misses the uncertainty shared across the full matrix. The correction is to use all three updates: add $q_2$ to the degrees of freedom, replace the column scale by its Schur complement $C$, and add the observed-block quadratic term to the row scale. Even when the location adjustment is zero, the scale update need not be zero.

Check Your Understanding

Use the stated column-block convention for each question.

Which block dimension determines the increase in conditional degrees of freedom, and by how much does it increase them?
Write the conditional location in terms of $M_1$, $x_2-M_2$, and the blocks of $\Omega$.
Why is the Schur complement $C$ positive definite when $\Omega$ is positive definite?
If $\Omega_{21}=0$, which conditional parameter remains unchanged from its location value, and which scale update may still occur?
For $Y_2=0$, what is the conditional row scale? Does this remove the degrees-of-freedom update?