# Monte Carlo Method in Statistical Physics - Importance Sampling

Importance Sampling

The estimative, under Monte Carlo integration, of an integral defined as

is

where are uniformly obtained from all the phase space (PS) and N is the number of sampling points (or function evaluations).

From all the phase space, some zones of it are generally more important to the mean of the variable A than others. In particular, those that have the value of sufficiently high when compared to the rest of the energy spectra are the most relevant for the integral. Using this fact, the natural question to ask is: is it possible to choose, with more frequency, the states that are known to be more relevant to the integral? The answer is yes, using the Importance sampling technique.

Lets assume is a distribution that chooses the states that are known to be more relevant to the integral.

The mean value of can be rewritten as

,

where are the sampled values taking into account the importance probability . This integral can be estimated by

where are now randomly generated using the distribution. Since most of the times it is not easy to find a way of generating states with a given distribution, the Metropolis algorithm must be used.