Bayesian nonparametric methods for conditional distributions

In the first paper, we propose a flexible class of priors for density estimation avoiding discrete mixtures, based on random nonlinear functions of a uniform latent variable with an additive residual. Although discrete mixture modeling has formed the backbone of the literature on Bayesian density estimation incorporating covariates, the use of discrete mixtures leads to some well known disadvantages. We propose an alternative class of priors based on random nonlinear functions of a uniform latent variable with an additive residual. The induced prior for the density is shown to have desirable properties including ease of centering on an initial guess for the density, posterior consistency and straightforward computation via Gibbs sampling.;In the second paper, we propose a Bayesian variable selection method involving non-parametric residuals, noting that the majority of literature has focused on the parametric counterpart. We generalize methods and asymptotic theory established for mixtures of g-priors to linear regression models with unknown residuals characterized by DP location mixture. We propose a mixture of semiparametric g-priors allowing for straightforward posterior computation via a stochastic search variable selection algorithm. In addition, Bayes factor and variable selection consistency is shown to result under a class of proper priors on g allowing the number of candidate predictors p to increase much faster than sample size n while making sparsity assumption on the true model size.;Our third paper is motivated by the fact that although there are standard algorithms for estimating minimum length credible intervals for scalars, there are no such methods for estimating minimum volume credible sets for vectors and functions. We propose a minimum volume covering ellipsoids (MVCE) approach for vector valued parameters, guaranteed to construct credible regions with probability ≥ 1 - alpha, while yielding highest posterior density regions under asymptotic normality. For one-dimensional random curves, our proposed approach starts with a MVCE region evaluated at finitely many knots, and then interpolates between the knots linearly or relying on Lipschitz continuity. For multivariate random surfaces, our approach uses Delaunay triangulations to approximate the credible region. Frequentist coverage properties and computational efficiency compared with frequentist alternatives are assessed through simulation studies.