**P**ierre Jacob and Alexandre Thiéry just arXived a highly pertinent paper on the most debated issue of non-negative unbiased estimators (of positive quantities). If you remember that earlier post of mine, I mentioned the issue in connection with the Russian roulette estimator(s) of Mark Girolami et al. And, as Pierre and Alexandre point out in the paper, there is also a clear and direct connection with the Bernoulli factory problem. And with our Vanilla Rao-Blackwellisation technique (sadly overlooked, once more!).

**T**he first thing I learned from the paper is how to turn a converging sequence into an unbiased estimator. If *(E _{n})* is this converging sequence, with limit μ, then

is unbiased..! Amazing. Even though the choice of the distribution of N matters towards getting a finite variance estimator, this transform is simply amazing. (Of course, once one looks at it, one realises it is the “old” trick of turning a series into a sequence and vice-versa. Still…!) And then you can reuse it into getting an unbiased estimator for almost any transform of μ.

**T**he second novel thing in the paper is the characterisation of impossible cases for non-negative unbiased estimators. For instance, if the original sequence has an unbounded support, there cannot be such an estimator. If the support is an half-line, the transform must be ~~monotonous~~ monotonic. If the support is a bounded interval (a,b), then the transform must be bounded from below by a polynomial bound

(where the extra-parameters obviously relate to the transform). (In this later case, the authors also show how to derive a Bernoulli estimator from the original unbiased estimator.)