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We consider nonparametric estimation of a mixed discrete‐continuous distribution under anisotropic smoothness conditions and a possibly increasing number of support points for the discrete part of the distribution. For these settings, we derive lower bounds on the estimation rates. Next, we consider a nonparametric mixture of normals model that uses continuous latent variables for the discrete part of the observations. We show that the posterior in this model contracts at rates that are equal to the derived lower bounds up to a log factor. Thus, Bayesian mixture of normals models can be used for (up to a log factor) optimal adaptive estimation of mixed discrete‐continuous distributions. The proposed model demonstrates excellent performance in simulations mimicking the first stage in the estimation of structural discrete choice models. 

This article develops a Markov chain Monte Carlo (MCMC) method for a class of models that encompasses finite and countable mixtures of densities and mixtures of experts with a variable number of mixture components. The method is shown to maximize the expected probability of acceptance for cross-dimensional moves and to minimize the asymptotic variance of sample average estimators under certain restrictions. The method can be represented as a retrospective sampling algorithm with an optimal choice of auxiliary priors and as a reversible jump algorithm with optimal proposal distributions. The method is primarily motivated by and applied to a Bayesian nonparametric model for conditional densities based on mixtures of a variable number of experts. The mixture of experts model outperforms standard parametric and nonparametric alternatives in out of sample performance comparisons in an application to Engel curve estimation. The proposed MCMC algorithm makes estimation of this model practical.