  Standard maximum-likelihood estimators for binary-star and exoplanet
eccentricities are biased high, in the sense that the estimated eccentricity
tends to be larger than the true eccentricity. As with most non-trivial
observables, a simple histogram of estimated eccentricities is not a good
estimate of the true eccentricity distribution. Here we develop and test a
hierarchical probabilistic method for performing the relevant meta-analysis,
that is, inferring the true eccentricity distribution, taking as input the
likelihood functions for the individual-star eccentricities, or samplings of
the posterior probability distributions for the eccentricities (under a given,
uninformative prior). The method is a simple implementation of a hierarchical
Bayesian model; it can also be seen as a kind of heteroscedastic deconvolution.
It can be applied to any quantity measured with finite precision--other orbital
parameters, or indeed any astronomical measurements of any kind, including
magnitudes, parallaxes, or photometric redshifts--so long as the measurements
have been communicated as a likelihood function or a posterior sampling.
