  We consider a non-parametric Bayesian model for conditional densities. The
model is a finite mixture of normal distributions with covariate dependent
multinomial logit mixing probabilities. A prior for the number of mixture
components is specified on positive integers. The marginal distribution of
covariates is not modeled. We study asymptotic frequentist behavior of the
posterior in this model. Specifically, we show that when the true conditional
density has a certain smoothness level, then the posterior contraction rate
around the truth is equal up to a log factor to the frequentist minimax rate of
estimation. An extension to the case when the covariate space is unbounded is
also established. As our result holds without a priori knowledge of the
smoothness level of the true density, the established posterior contraction
rates are adaptive. Moreover, we show that the rate is not affected by
inclusion of irrelevant covariates in the model. In Monte Carlo simulations, a
version of the model compares favorably to a cross-validated kernel conditional
density estimator.
