  We consider nonparametric estimation of a mixed discrete-continuous
distribution under anisotropic smoothness conditions and possibly increasing
number of support points for the discrete part of the distribution. For these
settings, we derive lower bounds on the estimation rates in the total variation
distance. 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 optimal adaptive estimation of mixed discrete-continuous
distributions.
