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Bayesian Hierarchical Mixtures of Experts

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.