Marginalizing multivariate normal over defined interval

Hello everyone,

I am trying to obtain an analytic expression for the following Gaussian integral

$$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} \; e^{-\frac{1}{2} \mathbf{x}^T \Sigma^{-1} \mathbf{x} } \; \theta (\mathbf{\alpha} \cdot \mathbf{x})$$

where $\theta$ is the Heaviside step-function and $d\mathbf{x}_{\sim i}$ indicates that I'm integrating over all components of the vector $\mathbf{x}$ except for the $i$-th one. It's as if I were marginalizing the Gaussian distribution, except that I'm integrating only over values of $\mathbf{x}$ satisfying $\mathbf{\alpha} \cdot \mathbf{x} > 0$ ($\alpha$ is fixed).

Also, the covariance matrix is diagonal, i.e., the diferent components are not correlated.

Does anyone has some intution over this, or perhaps some reference? Would it be possible to obtain an analytic expression, for instance in terms of error functions and distributions' first moments, or it's hopeless?

Thanks!

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if the covariance is diagonal, that is the Gaussian functions are independent, then the above just reduces to a nested set of integrals which can be solved numerically via Gaussian-hermite quadrature. – user2350366 Aug 8 '13 at 16:47