THE BAYES DECONVOLUTION PROBLEM By

An unknown prior density g(θ) has yielded realizations Θ1,Θ2, . . . ,ΘN . They are unobservable, but each Θi produces an observable value Xi according to a known probability mechanism, for instance Xi ∼ Poisson(Θi). We wish to estimate g(θ) from the observed sample X1, X2, . . . , XN . Traditional asymptotic calculations are discouraging, indicating very slow nonparametric rates of convergence. Here we show that parametric exponential family modeling of g(θ) can give useful estimates in moderate-sized samples. A variety of real and artificial examples illustrates the methodology. Covariate information can be incorporated into the deconvolution process, leading to a more detailed theory of Generalized Linear Mixed Models.