I'm after a reference for an integral. In particular, I am looking a way to approximate or calculate the following:

$$ \int \limits_{\| \theta \|_2 = 1} e^{(-(\theta - \mu)^T \Sigma (\theta - \mu))} d\theta $$.

I know how to do this when say $\Sigma$ is the Identity matrix. To see this, $$ \int \limits_{\| \theta \|_2 = 1} e^{(-(\theta - \mu)^T I (\theta - \mu))} d\theta = \int \limits_{\| \theta \|_2 = 1} e^{- \| \theta - \mu \|_2^2} d\theta = e^{-1} e^{-\|{\mu}\|_2^2} \underbrace{\int \limits_{\| \theta \|_2 = 1} e^{2\|{\mu}\|_2 \theta^T(\frac{\mu}{\|{\mu}\|_2})} d\theta}_{T1} $$

Now, T1 is essentially the normalization constant for the Von-Mises Fisher Distribution.

How do I do it for positive definite $\Sigma$? Is there some simple trick that I am missing?

  • 1
    $\begingroup$ You surely do not want $\Sigma$ to be a general matrix. Perhaps it should be positive definite? $\endgroup$ – KConrad May 15 at 4:13
  • $\begingroup$ Yes, thanks. I updated the question. $\endgroup$ – user550008 May 15 at 4:18
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    $\begingroup$ Every positive definite matrix is the square of a real symmetric matrix: $\Sigma = S^2$ with $S^\top = S$. Then, using dot products, $$\mathbf v^\top \Sigma \mathbf v = \mathbf v \cdot \Sigma \mathbf v = \mathbf v \cdot SS\mathbf v = S^\top \mathbf v \cdot S\mathbf v = S\mathbf v \cdot S\mathbf v = ||S\mathbf v||_2^2$$. $\endgroup$ – KConrad May 15 at 4:45

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