On non-negative unbiased estimators

We study the existence of algorithms generating almost surely non-negative unbiased estimators. We show that given a non-constant real-valued function $f$ and a sequence of unbiased estimators of $\lambda \in \mathbb{R}$, there is no algorithm yielding almost surely non-negative unbiased estimators of $f(\lambda) \in \mathbb{R}^+$. The study is motivated by pseudo-marginal Monte Carlo algorithms that rely on such non-negative unbiased estimators. These methods allow "exact inference" in intractable models, in the sense that integrals with respect to a target distribution can be estimated without any systematic error, even though the associated probability density function cannot be evaluated pointwise. We discuss the consequences of our results on the applicability of pseudo-marginal algorithms and thus on the possibility of exact inference in intractable models. We illustrate our study with particular choices of functions $f$ corresponding to known challenges in statistics, such as exact simulation of diffusions, inference in large datasets and doubly intractable distributions.