A wide variety of priors have been proposed for nonparametric Bayesian estimation of conditional distributions, and there is a clear need for theorems providing conditions on the prior for large support, as well as weak and strong posterior consistency. Estimation of an uncountable collection of conditional distributions across different regions of the predictor space is a challenging problem, which differs in some important ways from density and mean regression estimation problems. We first introduce new notions of weak and strong neighborhoods that are applicable to conditional distributions. Focusing on a broad class of priors formulated as predictor-dependent mixtures of Gaussian kernels, we provide sufficient conditions under which weak and strong posterior consistency hold. This theory is illustrated by showing that the conditions are satisfied for a class of generalized stick-breaking process mixtures in which the stick-breaking lengths are constructed through mapping continuous stochastic processes to the unit interval using a monotone differentiable link function. Probit stick-breaking processes provide a computationally convenient special case. We also provide a set of sufficient conditions to ensure strong and weak posterior consistency using fixed-$\pi$ dependent Dirichlet process mixtures of Gaussians.
Keywords: Asymptotics; Bayesian nonparametrics; Density regression; Large support; Probit stick-breaking process; Dependent Dirichlet process
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