# How to do integration using MCMC?

I want to evaluate $I = \int_V f(\vec{x}) d\vec{x}$. The classical Monte Carlo method is to sample uniformly from within the integration volume $V$, and then compute $I \approx V \frac{1}{N} \sum_{i=1}^{N} f(\vec{x})$.

What if I sampled the $\vec{x}$'s using a MCMC approach (e.g. Metropolis-Hastings, slice sampling), how do I compute $I$? Specifically, if we use the above formular, what is $V$?

• docs.google.com/… – Steve Huntsman Oct 12 '10 at 5:23
• More generally, a search on mcmc + integration yields plenty of PDFs. – Steve Huntsman Oct 12 '10 at 5:23
• I've searched it before asking and I found the document you sent. According to that doc, I have to factorize $f(x)$ into $h(X)$ and $p(x)$ where $p$ a proper pdf. Am I right? What if I cannot do this factorization? Is the regular Monte Carlo only choice? – eakbas Oct 12 '10 at 17:04
• I knew that sooner or later I'd see pdf used with two different meanings in close proximity. – Gerry Myerson Oct 13 '10 at 5:16
• Am I missing a trivial point here? I still do not know the answer to my question :) The documents I found on the web --as I understand them-- does not answer my question. – eakbas Oct 27 '10 at 2:46

## 1 Answer

There are no significant changes when you switch from MC to MCMC. The major concept remains the same. You draw your samples according to some probability distribution (which ideally has a form of your integrand, or as close as possible to it). Then instead of drawing i.i.d. samples, you just put a Metropolis move on top of your existing MC integration routine.

The only conceptual difference is that you have a Markov chain. That means that you have the current sample (state) and each time you generate a new sample from the current one. You should compute the acceptance probability carefully though.