Seeking the normalizing constant (or any references) for a distribution over a subset of positive definite martrices - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T21:59:39Z http://mathoverflow.net/feeds/question/61836 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/61836/seeking-the-normalizing-constant-or-any-references-for-a-distribution-over-a-su Seeking the normalizing constant (or any references) for a distribution over a subset of positive definite martrices Jeremy 2011-04-15T16:32:06Z 2011-04-15T16:47:04Z <p>I'm interested in a probability distribution over the set of positive definite matrices with unit diagonal elements. That is, and $X$ such that:</p> <p>$X \in S^{n+}, \forall_{i}X_{ii} = 1$ where $S^{n+}$ represents the set of positive define matrices of size $n$</p> <p>This distribution is parameterized by a symmetric matrix $M$, and has the form of:</p> <p>$\textrm{P}(X ; M) = A(M) \exp(\textrm{tr}(M X))$</p> <p>where $A(M)$ is a normalizing constant that makes the integral over all $X$ of $\textrm{P}(X : M)$ equal to 1</p> <p>Specifically, I'm looking for a closed form (if it exists) for $A(M)$, or at the very least, a reference to any papers or other documentation on such a distribution or something similar. </p> <p>For those interested in why unit diagonals, my underlying distribution is a group of $n$ vectors of unit length, $V_i$, and $X_{ij} = V_i^T V_j$, thus $X_{ii} = 1$ since $|V_i| = 1$</p>