A Least-Squares Approach to Sample-Based Prior Elicitation

Abstract

An expert who supplies examples of a quantity often also signals how plausible each one is; when is that signal worth using? We study eliciting a Bayesian prior from an expert who provides example points together with their approximate likelihoods. We propose fitting the prior by least squares--minimizing the squared discrepancy between a parametric density and the elicited likelihoods--which defines an M-estimator that remains well posed even for families whose moments do not exist. We establish consistency and asymptotic normality, and prove--under explicit regularity conditions, comprising a well-separation and a uniform-concentration requirement that we verify for the families considered--a non-asymptotic O(1/sqrt(n)) Berry--Esseen bound on its sampling distribution, uniform and nonuniform, by extending a result of Pinelis for maximum-likelihood estimators to the M-estimation setting. We then relax the assumptions that most limit the method in practice. Experts need not report on the density's own scale: an unknown reporting scale can be profiled out in closed form and estimated jointly, at no asymptotic cost for location families. The theory extends to multivariate parameters, where a directional Berry--Esseen bound follows from the multivariate delta method applied to a smooth implicit proxy for the estimator. An additive error floor in the noise model removes a degeneracy in the optimal design, making optimal designs interior. Simulations for normal and beta families confirm the predicted n^-1 error rate and the accuracy of the normal approximation at moderate sample sizes. Finally, we compare the estimator with the sample-only maximum-likelihood baseline, derive an explicit threshold on the expert's reporting noise below which the elicited likelihoods provably reduce estimation error, and calibrate that threshold against eleven datasets of human frequency judgments.