Matrix-T Laws From Gaussian-Wishart Mixing

From Gaussian Variation to Polynomial Tails

A matrix-normal model describes variation conditional on its covariance structure. In some models, however, that structure is itself uncertain. If the unknown precision is assigned a Wishart distribution and then integrated out, the resulting matrix distribution is no longer Gaussian: it has a determinant-shaped density with polynomial rather than exponential decay. The Wishart degrees of freedom govern how strongly the precision is concentrated and, consequently, how much tail weight remains after mixing.

We derive this law for a matrix-normal observation whose row precision is random. Being explicit about which matrix is mixed matters: the calculation below mixes a precision matrix, not a covariance matrix. The choice determines both the determinant in the answer and its exponent.

Definition: Let \(Y\) be a \(p\times q\) random matrix, let \(M\) be a \(p\times q\) location matrix, and let \(V\) be a positive-definite \(q\times q\) column covariance matrix. Conditional on a positive-definite \(p\times p\) row precision matrix \(\Omega\), suppose \(Y\mid\Omega\sim MN_{p,q}(M,\Omega^{-1},V)\). Independently draw \(\Omega\sim W_p(\nu,\Psi)\), where \(\Psi\) is positive definite and \(\nu>p-1\). The marginal law of \(Y\), obtained by integrating out \(\Omega\), is the matrix-variate \(t\) law generated by this Gaussian-Wishart mixture.

The Two Conditional Densities

We use the convention that \(W_p(\nu,\Psi)\) has scale matrix \(\Psi\), so its density at a positive-definite \(\Omega\) is

f_{\Omega}(\Omega)= \frac{|\Omega|^{(\nu-p-1)/2}} {2^{\nu p/2}|\Psi|^{\nu/2}\Gamma_p(\nu/2)} \exp\left\{-\frac12\operatorname{tr}(\Psi^{-1}\Omega)\right\}.

Here \(\Gamma_p(a)=\pi^{p(p-1)/4}\prod_{i=1}^p\Gamma(a-(i-1)/2)\), defined for \(a>(p-1)/2\). Under the stated matrix-normal convention, writing \(E=Y-M\), the conditional density is

f_{Y\mid\Omega}(Y\mid\Omega)= (2\pi)^{-pq/2}|\Omega|^{q/2}|V|^{-p/2} \exp\left\{-\frac12\operatorname{tr} \left(\Omega E V^{-1}E^{T}\right)\right\}.

The factor \(|\Omega|^{q/2}\) appears because there are \(q\) columns, each with row precision \(\Omega\). This factor will increase the determinant power in the Wishart integral. That is why the marginal exponent involves \(\nu+q\), rather than just \(\nu\).

Deriving the Marginal Density

Theorem: Under the conditions above, the marginal density of \(Y\), with respect to its \(pq\) real entries, is

f_Y(Y)= \frac{\Gamma_p((\nu+q)/2)} {\pi^{pq/2}\Gamma_p(\nu/2)} |V|^{-p/2}|\Psi|^{q/2} \left|I_p+\Psi (Y-M)V^{-1}(Y-M)^T\right|^{-(\nu+q)/2}.

The matrix inside the determinant is positive definite: it is similar, for determinant purposes, to \(I_p+\Psi^{1/2}EV^{-1}E^T\Psi^{1/2}\), which is positive definite. To derive the density, multiply the conditional and mixing densities and integrate over all positive-definite \(\Omega\). Set \(S=EV^{-1}E^T\) and \(A=\Psi^{-1}+S\). The factors depending on \(\Omega\) become

|\Omega|^{(\nu+q-p-1)/2} \exp\left\{-\frac12\operatorname{tr}(A\Omega)\right\}.

For any positive-definite \(A\) and \(a>(p-1)/2\), the Wishart normalizing integral is

\int_{\Omega>0}|\Omega|^{a-(p+1)/2} e^{-\operatorname{tr}(A\Omega)/2}\,d\Omega =2^{ap}\Gamma_p(a)|A|^{-a}.

This identity is the Wishart density normalizer with degrees of freedom \(2a\) and scale \(A^{-1}\), rearranged so that the integral is isolated. Here \(a=(\nu+q)/2\), which satisfies the condition because \(\nu>p-1\) and \(q\geq1\). Substitution gives a complete expression for the marginal:

\begin{aligned} f_Y(Y) ={}&(2\pi)^{-pq/2}|V|^{-p/2} \frac{1}{2^{\nu p/2}|\Psi|^{\nu/2}\Gamma_p(\nu/2)} \,2^{(\nu+q)p/2}\Gamma_p((\nu+q)/2)|A|^{-(\nu+q)/2}\\ ={}&\frac{\Gamma_p((\nu+q)/2)} {\pi^{pq/2}\Gamma_p(\nu/2)} |V|^{-p/2}|\Psi|^{-\nu/2}|A|^{-(\nu+q)/2}. \end{aligned}

The powers of \(2\) combine to \(2^{pq/2}\), which cancels the \(2^{-pq/2}\) in \((2\pi)^{-pq/2}\). Finally, \(|A|=|\Psi|^{-1}|I_p+\Psi S|\). Replacing \(|A|\) in the last expression supplies the factor \(|\Psi|^{(\nu+q)/2}\), which combines with \(|\Psi|^{-\nu/2}\) to give \(|\Psi|^{q/2}\). These steps yield the displayed density. The integration is valid because the joint density is nonnegative, so Tonelli's theorem permits integrating its factors over \(\Omega\). \(\square\)

Worked Example: Checking the Exponent and Normalizer

Take \(p=2\), \(q=1\), \(\nu=5\), \(M=0\), \(V=[1]\), and \(\Psi=\tfrac14 I_2\). Evaluate the marginal density at \(Y=(1,2)^T\).

Here \(E=Y\), so \(S=EE^T=\begin{pmatrix}1&2\\2&4\end{pmatrix}\). The determinant in the density is

\left|I_2+\Psi S\right| = \left|\begin{pmatrix}5/4&1/2\\1/2&2\end{pmatrix}\right| =\frac54\cdot2-\frac12\cdot\frac12 =\frac94.

The determinant exponent is \(-(\nu+q)/2=-(5+1)/2=-3\). Also, \(|V|=1\) and \(|\Psi|=(1/4)^2=1/16\), so \(|\Psi|^{q/2}=1/4\). The multivariate gamma ratio is

\frac{\Gamma_2(3)}{\Gamma_2(5/2)} = \frac{\pi^{1/2}\Gamma(3)\Gamma(5/2)} {\pi^{1/2}\Gamma(5/2)\Gamma(2)} =2.

Since \(pq/2=1\), the density is therefore

f_Y(Y)=\frac{2}{\pi}\cdot\frac14\left(\frac94\right)^{-3} =\frac{1}{2\pi}\left(\frac94\right)^{-3}.

The determinant exponent and gamma ratio use the same updated value \((\nu+q)/2=3\); replacing it with \(3.5\) would contradict both the integrated power of \(|\Omega|\) and the gamma ratio.

Worked Consequences of the Mixture

Worked Example: A Scalar Student t Distribution

Let \(p=q=1\), \(M=0\), \(V=[9]\), \(\nu=4\), and \(\Psi=[1/4]\). In one dimension, \(\Omega\sim W_1(4,1/4)\) is a chi-square variable divided by \(4\): \(\Omega\sim\chi^2_4/4\). Conditional on \(\Omega\), \(Y\mid\Omega\sim N(0,9/\Omega)\). This is the usual scale-mixture construction of a Student \(t\) variable with four degrees of freedom and scale \(3\).

At \(y=3\), the marginal formula has gamma ratio \(\Gamma(5/2)/\Gamma(2)=3\sqrt{\pi}/4\), determinant factor \(|\Psi|^{1/2}=1/2\), and determinant term \(1+\Psi y^2/V=1+(1/4)(9/9)=5/4\). Thus

f_Y(3) =\frac{1}{\sqrt{\pi}}\frac{\Gamma(5/2)}{\Gamma(2)} \frac{1}{3}\frac12\left(\frac54\right)^{-5/2} =\frac18\left(\frac54\right)^{-5/2}.

The mixing degrees of freedom agree with the ordinary Student \(t\) degrees of freedom in this scalar case. The scale is \(3\), not the conditional standard deviation at a particular \(\Omega\), because that standard deviation varies across the mixture.

Worked Example: How the Degrees of Freedom Change Decay

For \(p=q=1\), set \(M=0\), \(V=[1]\), and \(\Psi=[1]\). For a fixed \(\nu>0\), the density is a constant times \((1+y^2)^{-(\nu+1)/2}\). Along \(y=t\), for \(t>0\),

(1+t^2)^{-(\nu+1)/2} =t^{-(\nu+1)}(1+t^{-2})^{-(\nu+1)/2} \sim t^{-(\nu+1)} \quad\text{as }t\to\infty.

For example, \(\nu=3\) gives density decay proportional to \(t^{-4}\), whereas \(\nu=7\) gives decay proportional to \(t^{-8}\). Thus the smaller mixing degrees of freedom produce the slower-decaying, heavier-tailed density in this comparison. This calculation concerns density decay; it does not by itself give the tail probability.

Why the Mixing Degrees of Freedom Matter

The same relationship holds in full matrix dimensions along a full-row-rank direction. Let \(A_0\) be a fixed \(p\times q\) matrix of row rank \(p\), and consider \(Y=M+tA_0\). Then \(B=\Psi A_0V^{-1}A_0^T\) is similar to a positive-definite matrix and has positive determinant. The determinant in the marginal density satisfies

|I_p+t^2B| =t^{2p}|B|\,|I_p+t^{-2}B^{-1}| \sim t^{2p}|B|.

The gamma and scale factors in the density do not depend on \(t\), so this proves the density decay along that direction:

Result: Along every fixed full-row-rank direction \(A_0\), the matrix density is asymptotic to a positive constant times \(t^{-p(\nu+q)}\) as \(t\to\infty\). Increasing \(\nu\) increases the decay exponent; decreasing \(\nu\) gives more slowly decaying density along these directions.

The full-row-rank condition matters: if \(A_0\) has lower rank, not all eigenvalues of \(I_p+t^2B\) grow at the same rate, so the displayed power need not apply. This result describes the density along specified directions, not a general probability statement about all possible tail regions.

Common Mistake: Mixing the Wrong Matrix

A frequent error is to see a Wishart matrix in a Gaussian model and treat it as though it were a covariance matrix without changing the conditional density. Here \(\Omega\) is a precision, so the conditional row covariance is \(\Omega^{-1}\). Its contribution to the matrix-normal density is \(|\Omega|^{q/2}\). Combining that with the Wishart power gives the exponent \((\nu+q-p-1)/2\) inside the integral, and hence the marginal determinant exponent \(-(\nu+q)/2\).

If one instead mixes a covariance matrix, the conditional density and the integral are different; the formula derived here cannot be carried over by simply renaming \(\Omega\). Another common arithmetic pitfall is to miscompute \((\nu+q)/2\): for \(\nu=5\) and \(q=1\), it is \(3\), not \(3.5\). The worked calculation above checks the corresponding gamma ratio, determinant power, and resulting density together.

Key takeaway: Integrating a matrix-normal model over a Wishart row precision yields a matrix-t density with determinant exponent \(-(\nu+q)/2\). The degrees of freedom enter through the Wishart mixing law: smaller values give slower polynomial density decay along full-row-rank directions.

Check Your Understanding

Use the parameterization and row-precision convention stated above.

Which determinant factor in the conditional matrix-normal density changes the power of \(|\Omega|\) during mixing?
For \(p=3\), \(q=2\), and \(\nu=8\), what is the exponent on the marginal determinant?
Why does the Wishart integral exist under the stated condition \(\nu>p-1\) and \(q\geq1\)?
In the worked \(p=q=1\) case, what are the Student \(t\) degrees of freedom and scale?
What additional condition on a direction matrix \(A_0\) was used to obtain the stated full matrix density-decay rate?