Importance Sampling

The importance sampling technique concentrates the distribution of sampled configurations in the parts of the integration range that are of most importance. Instead of computing the ensemble average

$$\langle A \rangle = \int d \xi P(\xi) A(\xi),$$ (344)

according to the estimator $\bar{A}$ introduced by Eq. (338), after sampling configurations $\xi$ according to the probability distribution $P(\xi)$, configurations are sampled according to a different probability distribution $\tilde{P}(\xi)$ and the ensemble average is computed according to the estimator

$$\langle A \rangle \approx \overline{\frac{g}{\tilde{P}}} \equiv \frac{1}{N} \sum_{\xi} N(\xi) \frac{g(\xi)}{\tilde{P}(\xi)},$$ (345)

where $g(\xi) \equiv P(\xi) A(\xi)$ and $\tilde{P}(\xi)$ is assumed to be normalized.

The variance of the estimator introduced by Eq. (345) is

$$\sigma^2 = \frac{1}{N} \sum_{\xi} N(\xi) \Bigg( \frac{g(\xi)}{\tilde{P}(\xi)} - <A> \Bigg)^2,$$ (346)

or

$$\sigma^2 = \frac{1}{N} \sum_{\xi} N(\xi) \frac{g(\xi)^2}{\tilde{P......- \Bigg ( \frac{1}{N} \sum_{\xi} N(\xi) \frac{g(\xi)}{\tilde{P}(\xi)} \Bigg)^2.$$ (347)

Note that according to Eq. (347), $\sigma^2 =0$, when $\tilde{P}(\xi)=g(\xi)$. Therefore, the variance can be reduced by choosing $\tilde{P}(\xi)$ similar to $\vert g(\xi)\vert$. Such choice of $\tilde{P}(\xi)$ concentrates the distribution of sampled configurations in the parts of the integration range that are of most importance. According to such distribution, the random variables $g(\xi)/\tilde{P}(\xi)$ spread over a modest range of values close to 1 and therefore the standard error of the Monte Carlo calculation is reduced.
 

The umbrella sampling technique is a particular form of importance sampling, specially designed to investigate rare events. Configurations are sampled according to the non-Boltzmann distribution P($\xi$) $\propto$ exp[-$\beta$ (E($\xi$)+W($\xi$))], where $W(\xi)$ is zero for the interesting class of configurations that defined the rare event and very large for all others.