# Help with an integral

1. May 27, 2014

### pamparana

This is not a homework question but something that has come up during work and I just need some guidance on what is the best way to solve this.

I am computing a quantity where one of the terms to compute is a derivative of an integral of the form

$\nabla_{\mu}log\left(Z(\mu, \Sigma)\right)$ where $Z(\mu, \Sigma)$ is of the form $\int t(w) \, q(w) \,dw$. My plan was to first get an expression for Z and then take the derivative.

Now, $q(w)$ is a Gaussian distribution on w and is given by $K_1 \exp\left(-0.5 (w-\mu)^{T} \Lambda (w-\mu)\right)$ where $K_1$ is a constant and $T$ denotes a transpose operation, $\Lambda$ is the inverse of the covariance matrix associated with the Gaussian.

Similarly, $t(w)$ is also a Gaussian but not on $w$. t(w) can be written as:

$t(w) = K_2 \exp\left(-0.5 (y-f(x, w))^{T} \Sigma (y-f(x, w)\right)$

where $y$ and $x$ are two observed quantities and can be treated as constants. $f$ is some non-linear function of $w$.

So, ultimately I need to compute Z as an integral of this complicated expression given by:

$Z = \int \left[K_2 \exp\left(-0.5 (y-f(x, w))^{T} \Sigma (y-f(x, w)\right)\right] \; \left[K_1 \exp\left(-0.5 (w-\mu)^{T} \Lambda (w-\mu)\right)\right] dw$

I must admit my upper level calculus skills are not great. I was wondering if someone can comment on how to go about or if there is some other simplifications I can do to solve this.

Thanks a lot!

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