# Priors on matrix determinant

I have a positive semi definite matrix (covariance matrix). Was wondering are there any distributions that can place a prior on the determinant?

Something along lines of $$\exp(-k|X|)$$

Note that the matrix gamma distribution is not what I'm looking for.

• Are you asking if their are any standard priors that incorporate a determinant? – user25658 Sep 4 '13 at 14:47
• @Bab This is a Bayesian question: the OP wants to make inferences about the distribution of a covariance matrix based on a prior distribution of its determinant. Sachin, you have already answered your question: of course you can place a prior on the determinant (although the example you wrote down is neither a CDF nor a PDF except when $k=1$). Perhaps what you are really wondering is whether there are any analytically convenient priors that will make Bayesian updating easy. Because updating depends on the presumed distribution of the matrix itself, what can you tell us about that? – whuber Sep 4 '13 at 17:46
• At Bab. Yep. Exactly what I'm asking. @whubber Basically I want to push the inverse covariance towards sparsity. Only problem with above is that I don't know what the normalising constant is. – sachinruk Sep 4 '13 at 20:42
• What is the connection between the determinant and the sparsity level of your matrix? I don't see why they would be related. – bnaul Sep 4 '13 at 21:08
• @bnaul It's related to sparse inverse covariance estimation. The higher the penalty on the determinant of inverse covariance, $-k|X|^{-1}$ the higher the sparsity. This is a frequentist approach, wondering if Bayesian approach is possible. – sachinruk Sep 5 '13 at 0:38