Unbiased exponential estimators

The penalty method allows a Metropolis random walk to be correct even with the presence of noise in the probability function being sampled.

It eventually dawned on me that this same method can be used to correct any exponential estimator.

Imagine we wish to compute the estimator, exp(x), where x is some sample point that is contaminated with noise. Suppose the variance of the noise is sigma, and we know the noise is gaussian. Then the estimator exp(x-sigma^2/2) will be unbiased.

Free Energy Links

Radford Neal has written a paper on computing free energy differences. The paper presents the Linked Importance Sampling method, but describes other methods as well. The paper contains references to both the physics literature and the statistics literature, and uses physics language.

If you're really interested in free energy differences, check out Arjun Acharya's thesis, Free Energy differences: Representations, estimators, and sampling strategies (also on arxiv). The review chapter covers several methods, but most of the thesis deals with the Phase Mapping technique, which increases the overlap between phases. It does not eliminate the problem entirely, however, and many of the other methods can be used to deal with the remaining issues.

[Edit 2/27/06: updated links from comments, and revised the entry]