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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?

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    $\begingroup$ Is there a specific context in which the question arose ? And by the way, what is the question ? $\endgroup$
    – Loïc Teyssier
    Commented Jun 27, 2014 at 14:02
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    $\begingroup$ Hi, the question is the value of this integral. I confronted the question when deriving a Bayesian framework for pattern verification. $\endgroup$
    – Yicong Liang
    Commented Jun 27, 2014 at 14:16
  • $\begingroup$ Do you also presume that $\mathbf{A}$ and $\mathbf{B}$ are symmetric? $\endgroup$
    – Igor Khavkine
    Commented Jun 27, 2014 at 15:10
  • $\begingroup$ Yes, they are symmetric, positive semidefinite. $\endgroup$
    – Yicong Liang
    Commented 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? $\endgroup$
    – Igor Khavkine
    Commented Jun 27, 2014 at 15:44

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