  The Hierarchical Mixture of Experts (HME) is a well-known tree-based model
for regression and classification, based on soft probabilistic splits. In its
original formulation it was trained by maximum likelihood, and is therefore
prone to over-fitting. Furthermore the maximum likelihood framework offers no
natural metric for optimizing the complexity and structure of the tree.
Previous attempts to provide a Bayesian treatment of the HME model have relied
either on ad-hoc local Gaussian approximations or have dealt with related
models representing the joint distribution of both input and output variables.
In this paper we describe a fully Bayesian treatment of the HME model based on
variational inference. By combining local and global variational methods we
obtain a rigourous lower bound on the marginal probability of the data under
the model. This bound is optimized during the training phase, and its resulting
value can be used for model order selection. We present results using this
approach for a data set describing robot arm kinematics.
