Regularized Maximum Likelihood Estimation and Feature Selection in Mixtures-of-Experts Models

Mixture of Experts (MoE) are successful models for modeling heterogeneous data in many statistical learning problems including regression, clustering and classification. Generally fitted by maximum likelihood estimation via the well-known EM algorithm, their application to high-dimensional problems is still therefore challenging. We consider the problem of fitting and feature selection in MoE models, and propose a regularized maximum likelihood estimation approach that encourages sparse solutions for heterogeneous regression data models with potentially high-dimensional predictors. Unlike state-of-the art regularized MLE for MoE, the proposed modelings do not require an approximate of the penalty function. We develop two hybrid EM algorithms: an Expectation-Majorization-Maximization (EM/MM) algorithm, and an EM algorithm with coordinate ascent algorithm. The proposed algorithms allow to automatically obtaining sparse solutions without thresholding, and avoid matrix inversion by allowing univariate parameter updates. An experimental study shows the good performance of the algorithms in terms of recovering the actual sparse solutions, parameter estimation, and clustering of heterogeneous regression data

Effect fusion using model-based clustering

In social and economic studies many of the collected variables are measured on a nominal scale, often with a large number of categories. The definition of categories is usually not unambiguous and different classification schemes using either a finer or a coarser grid are possible. Categorisation has an impact when such a variable is included as covariate in a regression model: a too fine grid will result in imprecise estimates of the corresponding effects, whereas with a too coarse grid important effects will be missed, resulting in biased effect estimates and poor predictive performance. To achieve automatic grouping of levels with essentially the same effect, we adopt a Bayesian approach and specify the prior on the level effects as a location mixture of spiky normal components. Fusion of level effects is induced by a prior on the mixture weights which encourages empty components. Model-based clustering of the effects during MCMC sampling allows to simultaneously detect categories which have essentially the same effect size and identify variables with no effect at all. The properties of this approach are investigated in simulation studies. Finally, the method is applied to analyse effects of high-dimensional categorical predictors on income in Austria