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$\newcommand{\tr}{\operatorname{tr}}$I'm reading Matrix Algebra (2007) by Gentle, and one of the problems given was to find the normalizing constant of the Wishart distribution, with the probability density of the Wishart distribution proportional to $e^{\tr(-\Sigma^{-1}W/2)}|W|^{(n-d-1)/2}$. In this case, $\Sigma$ is a fixed symmetric positive definite matrix of size $d \times d$, and $W$ is a $d \times d$ positive semi-definite matrix. $n$ is a positive parameter, and $|W|$ is the determinant of $W$.

The normalizing constant is $2^{dn/2}\Gamma_d(\frac{1}{2})|\Sigma|^{n/2}$, where $\Gamma_d(x)$ is the multivariate gamma function. However, I'm not sure how to show that. A hint given was to use the change of variables $W = 2\Sigma^{\frac{1}{2}} Y \Sigma^{\frac{1}{2}}$.

I believe my main problem is that I'm unfamiliar with the formula to use for the change of variables in this case, where the integration variable is a matrix. Using the suggested change of variable of $W$ to $Y,$ and taking the integral over $D$, the set of all positive definite matrices, I get:

\begin{align*} & \int_D e^{\tr(-\Sigma^{-1}W/2)}|W|^{(n-d-1)/2} \, dW \\[8pt] = {} & \int_D e^{-\tr(Y)}|2\Sigma^{\frac{1}{2}}Y\Sigma^{\frac{1}{2}}|^{(n-d-1)/2} (??) \, dY\\[8pt] = {} & 2^{d(n-d-1)/2} |\Sigma|^{(n-d-1)/2} (??) \int_D e^{-\tr(Y)}|Y|^{(n-d-1)/2} \,dY\\[8pt] = {} & 2^{d(n-d-1)/2} |\Sigma|^{(n-d-1)/2} (??) \Gamma_d \left(\frac{1}{2}\right) \end{align*}

In the above, I assume the (??) is the determinant of some Jacobian? My initial thought was that it would be equal to $|2\Sigma|^d$, but it does not lead to the correct answer. I'm quite new to linear algebra and matrix calculus, so I'm not sure where I went wrong, any hints are appreciated.

Michael Hardy
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backlit_life
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1 Answers1

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What this problem boils down to is knowing that $$\begin{align} \lvert GXG^{\mathsf{T}}\rvert&=\lvert GG^{\mathsf{T}}\rvert\lvert X\rvert\\ \mathrm{d}(GXG^{\mathsf{T}})&=\lvert GG^{\mathsf{T}}\rvert^{(d+1)/2}\mathrm{d}X \end{align}$$

and thus that the measure $$\mathrm{d}_*X\stackrel{\text{def}}{=}\frac{\mathrm{d} X}{\lvert X\rvert^{(d+1)/2}}$$ does not vary under transformations $X\to GXG^{\mathsf{T}}$, whence

$$\frac{1}{\lvert K\rvert^{s}}=\frac{\int_D \mathrm{e}^{-(K,X)} \lvert X \rvert^s \, \mathrm{d}_*X}{\int_D \mathrm{e}^{-\operatorname{Tr} X} \lvert X \rvert^s \, \mathrm{d}_*X}.$$

Addendum

There are a few ways to get the cited results. You can

  • use the $LDL^{\mathsf{T}}$ Cholesky factorization, as in Gross and Richards (1987) (they call it a "bitriangular structure"), or
  • write $G=\mathrm{e}^{tA}$ and show that the rate of change of $\mathrm{d}X$ with respect to $t$ is $\tfrac{d+1}{2}(A+A^{\mathsf{T}})\mathrm{d}X$, or
  • use results from the theory of functions on a Euclidean Jordan algebra. This is the modern, "highbrow" approach.
K B Dave
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