Inferring the eccentricity distribution

TL;DR

This paper introduces a hierarchical Bayesian method to accurately infer the true eccentricity distribution of binary stars and exoplanets, correcting biases inherent in traditional estimators and applicable to various astronomical measurements.

Contribution

The paper develops a hierarchical probabilistic approach that corrects bias in eccentricity estimation and can be applied broadly to any finite-precision astronomical data.

Findings

Reduces bias in eccentricity distribution estimates

Applicable to various astronomical measurements

Provides a flexible hierarchical Bayesian framework

Abstract

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.