2
$\begingroup$

I am stuck in an integral problem:

$$\int_{\mathbb{R}^d}(\det(\mathbf{A}+k(\mathbf{x}-\mathbf{y}_0)(\mathbf{x}-\mathbf{y}_0)^T))^{-r_1}\prod_{i=1}^n(\det(\mathbf{B}+(\mathbf{x}-\mathbf{y}_i)(\mathbf{x}-\mathbf{y}_i)^T))^{-r_2}d\mathbf{x}$$

where $\mathbf{A}$ and $\mathbf{B}$ are positive semidefinite matrices, $k$, $r_1$ and $r_2$ are positive scalar constants, $\mathbf{x}$ and $\mathbf{y}_i, i=0,\ldots,n$ are $d$-dimensional vectors.

Any help?

$\endgroup$
6
  • 2
    $\begingroup$ Is there a specific context in which the question arose ? And by the way, what is the question ? $\endgroup$ Jun 27, 2014 at 14:02
  • 1
    $\begingroup$ Hi, the question is the value of this integral. I confronted the question when deriving a Bayesian framework for pattern verification. $\endgroup$ Jun 27, 2014 at 14:16
  • $\begingroup$ Do you also presume that $\mathbf{A}$ and $\mathbf{B}$ are symmetric? $\endgroup$ Jun 27, 2014 at 15:10
  • $\begingroup$ Yes, they are symmetric, positive semidefinite. $\endgroup$ Jun 27, 2014 at 15:12
  • $\begingroup$ One thing that would help is the en.wikipedia.org/wiki/Matrix_determinant_lemma, which lets you simplify the calculation of the determinants. Otherwise, there's little chance of an explicit solution, especially with all these parameters. You can get an explicit answer when $n=0$, maybe when $n=1$; beyond that it's doubtful. There might also be a way to approximate the integral when $r_1$ and $r_2$ are large, or when the distances between the $y_i$ vectors are large. So, which of those possibilities are important to you? $\endgroup$ Jun 27, 2014 at 15:44

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.