Integration of exponential matrix and determinant?

Question

Is it possible to prove $$\int \exp\{-\frac{1}{2}(\beta-\hat\beta)^T(X^TH^{-1}X)(\beta-\hat\beta)\}\text{d}\beta=\{\det(X^TH^{-1}X)\}^{-1/2},$$ where $\hat\beta,X,H$ are all known?

What additional conditions are required for it to hold?

Answer 1

Perform a change of variables $x = (X^T H^{-1} X)^{1/2} (\beta - \hat\beta)$. From the usual formula involving the Jacobian of the determinant, note that $dx = \det((X^T H^{-1} X)^{1/2}) d\beta$. So you have reduced the problem to finding $\int \exp(-\tfrac12 x^T x) \, dx$. And this is well known to be $(2\pi)^{-n/2}$. The only condition needed is that $\det((X^T H^{-1} X)^{1/2}) \ne 0$.