Does anyone know a simple, or at least tractable, formula for E[log|W|], where W is a real Wishart-distributed matrix? It seems like there ought to be one, but my search so far has been frustrating; most of the current research seems to be on complex Wishart matrices, which I understand is very important in signal processing (the field in which most of the papers I'm turning up appear) but is pretty much irrelevant to what I'm trying to do, and seems to complicate things unnecessarily. I have the feeling that the answer, if known, is one of those things that's been known for so long that no one bothers to spell it out explicitly in recent papers.
Expected value of logarithm of determinant of Wishart-distributed matrix
- Thread starter Daniel Dvorkin
- Start date
- Tags determinant log-determninant multivariate wishart
I just search one relevant article.
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177704251
It states that
\( n^p \frac {|W|} {|V|} \sim \chi^2(n)\chi^2(n-1)...\chi^2(n-p+1) \)
where those chi-square random variables are independent.
Hence, \( E[\ln|W|] = \sum_{k=0}^{p-1} E[\ln\chi^2(n-k)] -p\ln n + \ln|V|
= \sum_{k=0}^{p-1}\psi\left(\frac {n-k} {2}\right) + p\ln \frac {2} {n} + \ln|V|\)
where \( \psi \) is the digamma function.
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aoms/1177704251
It states that
\( n^p \frac {|W|} {|V|} \sim \chi^2(n)\chi^2(n-1)...\chi^2(n-p+1) \)
where those chi-square random variables are independent.
Hence, \( E[\ln|W|] = \sum_{k=0}^{p-1} E[\ln\chi^2(n-k)] -p\ln n + \ln|V|
= \sum_{k=0}^{p-1}\psi\left(\frac {n-k} {2}\right) + p\ln \frac {2} {n} + \ln|V|\)
where \( \psi \) is the digamma function.