The problem of nonparametric estimation of the conditional density of a
response, given a vector of explanatory variables, is classical and of
prominent importance in many prediction problems since the conditional density
provides a more comprehensive description of the association between the
response and the predictor than, for instance, does the regression function.
The problem has applications across different fields like economy, actuarial
sciences and medicine. We investigate empirical Bayes estimation of conditional
densities establishing that an automatic data-driven selection of the prior
hyper-parameters in infinite mixtures of Gaussian kernels, with
predictor-dependent mixing weights, can lead to estimators whose performance is
on par with that of frequentist estimators in being minimax-optimal (up to
logarithmic factors) rate adaptive over classes of locally H\"older smooth
conditional densities and in performing an adaptive dimension reduction if the
response is independent of (some of) the explanatory variables which,
containing no information about the response, are irrelevant to the purpose of
estimating its conditional density