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We investigate a class of hierarchical mixtures-of-experts (HME) models whereexponential family regression models with generalized linear mean functions ofthe form psi(ga+fx^Tfgb) are mixed. Here psi(...) is the inverse link function.Suppose the true response y follows an exponential family regression model withmean function belonging to a class of smooth functions of the form psi(h(fx))where h(...)in W_2^infty (a Sobolev class over [0,1]^{s}). It is shown that theHME probability density functions can approximate the true density, at a rateof O(m^{-2/s}) in L_p norm, and at a rate of O(m^{-4/s}) in Kullback-Leiblerdivergence. These rates can be achieved within the family of HME structureswith no more than s-layers, where s is the dimension of the predictor fx. It isalso shown that likelihood-based inference based on HME is consistent inrecovering the truth, in the sense that as the sample size n and the number ofexperts m both increase, the mean square error of the predicted mean responsegoes to zero. Conditions for such results to hold are stated and discussed.

Hierarchical Mixtures-of-Experts for Exponential Family Regression  Models with Generalized Linear Mean Functions: A Survey of Approximation and  Consistency Results