  This paper shows that large nonparametric classes of conditional multivariate
densities can be approximated in the Kullback--Leibler distance by different
specifications of finite mixtures of normal regressions in which normal means
and variances and mixing probabilities can depend on variables in the
conditioning set (covariates). These models are a special case of models known
as "mixtures of experts" in statistics and computer science literature.
Flexible specifications include models in which only mixing probabilities,
modeled by multinomial logit, depend on the covariates and, in the univariate
case, models in which only means of the mixed normals depend flexibly on the
covariates. Modeling the variance of the mixed normals by flexible functions of
the covariates can weaken restrictions on the class of the approximable
densities. Obtained results can be generalized to mixtures of general location
scale densities. Rates of convergence and easy to interpret bounds are also
obtained for different model specifications. These approximation results can be
useful for proving consistency of Bayesian and maximum likelihood density
estimators based on these models. The results also have interesting
implications for applied researchers.
