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Inferring the value of a property of a large stochastic system is a difficulttask when the number of samples is insufficient to reliably estimate theprobability distribution. The Bayesian estimator of the property of interestrequires the knowledge of the prior distribution, and in many situations, it isnot clear which prior should be used. Several estimators have been developed sofar, in which the proposed prior was individually tailored for each property ofinterest; such is the case, for example, for the entropy, the amount of mutualinformation, or the correlation between pairs of variables. In this paper wepropose a general framework to select priors, valid for arbitrary properties.We first demonstrate that only certain aspects of the prior distributionactually affect the inference process. We then expand the sought prior as alinear combination of a one-dimensional family of indexed priors, each of whichis obtained through a maximum entropy approach with constrained mean value ofthe property under study. In many cases of interest, only one or very fewcomponents of the expansion turn out to contribute to the Bayesian estimator,so it is often valid to only keep a single component. The relevant component isselected by the data, so no handcrafted priors are required. We test theperformance of this approximation with a few paradigmatic examples, and showthat it performs well in comparison to the ad-hoc methods previously proposedin the literature. Our method highlights the connection between Bayesianinference and equilibrium statistical mechanics, since the most relevantcomponent of the expansion can be argued to be the one with the righttemperature.

Inferring a property of a large system from a small number of samples