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How can the following integral be evaluated using Mathematica,

$\int_{\beta} \frac{1}{\left(||\mathbf{y}-\mathbf{D\beta}||^2 + \gamma\right)^{\alpha}} \exp\left(-\frac{1}{2}\beta^T\Sigma^{-1}\beta\right) \mathrm{d}\beta$,

where $\mathbf{\beta}$ and $\mathbf{y}$ are vectors of sizes $M\times 1$ and $N\times1$ respectively, $\mathbf{D}$ and $\mathbf{\Sigma}$ are matrices of respective sizes that allow the combination with vectors, $\alpha$ and $\gamma$ are positive integers and $||\cdot||$ denote the $\ell_2$ norm ?

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  • $\begingroup$ I dont know if Mathematica will be able to do this independently of the dimensions $M$ and $N$. You can begin to build up your integration with M = 4; b = Array[bc, M]; int = Integrate[Exp[-b.b], b] and taking a look at Assuming. $\endgroup$ – Mauricio Fernández Mar 21 '17 at 16:23
  • $\begingroup$ Did you try to perform the usual analytic tricks like diagonalizing $\Sigma^{-1}$, transforming the integral measure and so on? $\endgroup$ – user46676 Mar 21 '17 at 16:28
  • $\begingroup$ @MauricioLobos: Indeed, building up seems to be a better approach. $\endgroup$ – Arun M Mar 21 '17 at 16:37
  • $\begingroup$ @marmot: Yes, I am trying with the simplest case of $\Sigma$ being the identity matrix, but it seems taking quite long time, and not yet converging to a closed form. However, I am a tenderfoot in making Mathematica evaluate these kind of integrals. $\endgroup$ – Arun M Mar 21 '17 at 16:37
  • $\begingroup$ @ArunM I guess this is more a maths than a mathematica problem since you can cook down the integral quite a bit by diagonalizing $\Sigma^{-1}$, i.e. $\Sigma^{-1}=U^T\cdot S\cdot U$ where $S$ is diagonal and positive (such that the integral converges), redefining $\beta$, $\beta'=\sqrt{S}\cdot U\cdot\beta$, and then` diagonalizing' $D$ in the $\beta'$ basis. I know that these steps are described in most of the literature on path integrals. $\endgroup$ – user46676 Mar 21 '17 at 17:01