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I have hopefully a short/simple question regarding monte carlo estimators. The expected value of a function of a random variable can be defined as:

$$E[f(x)] = \int_{-\infty}^{\infty} f(x) p(x) dx$$

where $p(x)$ is a probability density function.

The Monte Carlo estimator is defined as:

$$\langle I\rangle = {1 \over N } \sum_{i=1}^N {f(x_i) \over p(x_i) }$$

My understanding was that the MC estimator could be used to estimate the expected value of the function of a random variable. In other words, we can use it to estimate $E[f(x)]$. So evaluating $f(x)$ for random variables $x$ and taking the average seems like an intuitive way of explaining why the MC estimator would work, but I don't understand why $f(x)$ is divided by the probability density function while in the first equation $f(x)$ is multiplied by it!?

I understand the first equation. It's quite similar to the way we compute the expected value of a discrete random variable ($E(x)=\sum x_i p_i$) so I understand the first equation but don't understand really how to go from the first equation to the MC estimator.

It would be great if someone could explain!

Thank you.

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  • $\begingroup$ Are you doing some importance sampling? $\endgroup$ – copper.hat Nov 26 '13 at 23:16
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It appears like you are trying to do something like this (see slide 15). In that case, what the monte carlo estimator $\langle I \rangle$ is doing does not make sense if you think of it as calculating the $E[f(x)]$. What that estimator is doing is calculating $\int_{-\infty}^{\infty} f(x) dx$ using a Monte Carlo method that samples the $x_i$ not uniformly but according to the density function $p(x_i)$, where $p(x_i)$ is chosen to sample the places where $f(x_i)$ is large more often than where it is small, thereby making better use of limited computer resources - this is called importance sampling (as copper.hat pointed out). To calculate the expected value of $f(x_i)$ you first need to specify the distribution of the $x_i$ and then generate Monte Carlo values of $f(x_i)$ with x's drawn from that distribution.

To summarize, the Monte Carlo estimator for the average of a random function and for the integral of a deterministic function are completely different things. You can't necessarily mix them. I think your confusion is thinking $\langle I \rangle$ is for $E[f(x_i)]$, which is it not. Your intuition on how to calculate $E[f(x_i)]$ is correct.

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