Functional regression approximate Bayesian computation for Gaussian process density estimation

A novel Bayesian nonparametric method is proposed for hierarchical modelling on a set of related density functions, where grouped data in the form of samples from each density function are available. Borrowing strength across the groups is a major challenge in this context. To address this problem, a hierarchically structured prior, defined over a set of univariate density functions using convenient transformations of Gaussian processes, is introduced. Inference is performed through approximate Bayesian computation (ABC) via a novel functional regression adjustment. The performance of the proposed method is illustrated via simulation studies and an analysis of rural high school exam performance in Brazil.