Interpolating Conditional Density Trees

TL;DR

This paper introduces and compares tree-based algorithms for flexible, accurate conditional density estimation over continuous variables, enabling efficient learning of complex joint distributions in high-dimensional data.

Contribution

It presents novel Bayesian network structure-learning algorithms that utilize nonuniform density trees, improving accuracy and efficiency in modeling complex continuous data.

Findings

Tree-based algorithms outperform discretization methods in density estimation accuracy.

Nonuniform leaf densities lead to better modeling of complex nonlinear relationships.

Models can be learned efficiently over dozens of variables from thousands of data points.

Abstract

Joint distributions over many variables are frequently modeled by decomposing them into products of simpler, lower-dimensional conditional distributions, such as in sparsely connected Bayesian networks. However, automatically learning such models can be very computationally expensive when there are many datapoints and many continuous variables with complex nonlinear relationships, particularly when no good ways of decomposing the joint distribution are known a priori. In such situations, previous research has generally focused on the use of discretization techniques in which each continuous variable has a single discretization that is used throughout the entire network. \ In this paper, we present and compare a wide variety of tree-based algorithms for learning and evaluating conditional density estimates over continuous variables. These trees can be thought of as discretizations that vary according to the particular interactions being modeled; however, the density within a given leaf of the tree need not be assumed constant, and we show that such nonuniform leaf densities lead to more accurate density estimation. We have developed Bayesian network structure-learning algorithms that employ these tree-based conditional density representations, and we show that they can be used to practically learn complex joint probability models over dozens of continuous variables from thousands of datapoints. We focus on finding models that are simultaneously accurate, fast to learn, and fast to evaluate once they are learned.