# Computation of expected values

I have a $N$-dimensional (normal) random vector $\mathbf{X}$, where $N$ is large, and a function $f : \mathbb{R}^N \to \mathbb{R}$. My goal is to compute $\operatorname{E}[f(X)]$ or at least approximate it.

Things I've tried:

• Monte Carlo: I sample $\mathbf{X}$ and compute the sample mean of $f(\mathbf{X})$
• Numerical integration: the expected value is just an $N$-dimensional integral, so just evaluate it using numerical methods
• Replace $f$ by its second (or higher?) order Taylor polynomial and use the fact that I know $\mathbf{X}$'s moments. (see here)

The first two methods seem prohibitively slow, appear to grow exponentially. The third is fast enough, but I have no idea how accurate I am.

Edit: some additional info about the $f$:

$$f(\mathbf{x}) = u(\langle \mathbf{v}, e^\mathbf{x} \rangle)$$

where $\mathbf{v}$ is a fixed vector, $u : \mathbb{R} \to \mathbb{R}$ is analytic and maybe even increasing and $e^\mathbf{x}$ is done pointwise.

• What do you know about $f$?
– whuber
Jun 22 '15 at 14:12
• One specific example I'm thinking about is $f(\mathbf{x}) = u(\langle \mathbf{v}, \mathbf{x} \rangle)$ where $\mathbf{v}$ is a fixed vector and $u : \mathbb{R} \to \mathbb{R}$ is analytic. The special case of increasing $u$ is also of interest.
– user80379
Jun 22 '15 at 14:29
• Thank you! If you would edit your post to include that information, you will get answers that are much more useful than otherwise. In particular, because $\langle\mathbf{v},\mathbf{x}\rangle$ has a univariate Normal distribution you are effectively in the case $N=1$, which completely changes the nature of the question.
– whuber
Jun 22 '15 at 14:32
• I actually gave the wrong $f$, sorry. I meant to write $f(\mathbf{x}) = u (\langle \mathbf{v}, e^\mathbf{x} \rangle)$ where $e^\mathbb{x}$ is done pointwise
– user80379
Jun 22 '15 at 14:35
• My correlation matrix is non-trivial, unfortunately.
– user80379
Jun 22 '15 at 14:41