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I want to calculate

$ f_{\mathbf x}(x_1,\ldots,x_k)\, = \frac{1}{(2\pi)^{k/2}|\boldsymbol\Sigma|^{1/2}} \exp\left(-\frac{1}{2}({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu}) \right), $

for a huge ($k$ up 100.000) multivariate Gaussian distribution where $\boldsymbol\Sigma$ is sparse. Cholesky decomposition works fine if the $k$ is smallish and has the advantage that the determinant can be calculated easily from the factors. For larger $k$ I would like to use CG (conjugate gradient optimization), which is really fast to compute $({\mathbf x}-{\boldsymbol\mu})^T{\boldsymbol\Sigma}^{-1}({\mathbf x}-{\boldsymbol\mu})$, however I still need the determinant and have no idea how to (efficiently) compute or approximate it.

Are there any algorithms designed for this problem?

Thank you!


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since the determinant is just for normalization, you should not need it if you use this distrubution to compute expectation values; and by the way, what is CG? – Carlo Beenakker Dec 15 '12 at 17:58
@Carlos CG: Conjugate gradient optimization (I modified my question accordingly). You are right, the mean, median, variance, covariance are trivial to "compute". Unfortunately I need the density. – Manuel Schmidt Dec 17 '12 at 5:57
For what do you need the density? If this is for maximum likelihood estimation, did you consider to use composite likelihood instead? – kjetil b halvorsen Dec 17 '12 at 17:08

Here are a few references that should help you get started in the area of approximating determinants:

Approximation of the determinant of large sparse symmetric positive definite matrices

Determinant Approximations

Matrices, Moments and Quadrature

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Thank you for the links. They might be helpful. – Manuel Schmidt Dec 20 '12 at 6:17

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