Partition-based conditional density estimation

¹ IPANEMA USR 3461 CNRS/MCC, BP 48 Saint Aubin, F-91192 Gif-sur-Yvette, France
² SELECT/Inria Saclay IdF, Laboratoire de Mathématiques Faculté des Sciences d’Orsay, Université Paris-Sud 11, F-91405 Orsay Cedex, France
Erwan.LePennec@inria.fr

Received: 9 December 2011
Revised: 9 July 2012

Abstract

We propose a general partition-based strategy to estimate conditional density with candidate densities that are piecewise constant with respect to the covariate. Capitalizing on a general penalized maximum likelihood model selection result, we prove, on two specific examples, that the penalty of each model can be chosen roughly proportional to its dimension. We first study a classical strategy in which the densities are chosen piecewise conditional according to the variable. We then consider Gaussian mixture models with mixing proportion that vary according to the covariate but with common mixture components. This model proves to be interesting for an unsupervised segmentation application that was our original motivation for this work.