Sparse Density Trees and Lists: An Interpretable Alternative to High-Dimensional Histograms

TL;DR

This paper introduces sparse, interpretable density estimation models using trees and lists for high-dimensional categorical data, improving visualization, sparsity, and accuracy over traditional histograms.

Contribution

The authors develop novel Bayesian sparse density models based on trees and lists, offering flexible control over model complexity and improved high-dimensional density estimation.

Findings

Models outperform traditional histograms in high dimensions.

The methods provide interpretable visualizations of density.

Application to crime analysis demonstrates practical utility.

Abstract

We present sparse tree-based and list-based density estimation methods for binary/categorical data. Our density estimation models are higher dimensional analogies to variable bin width histograms. In each leaf of the tree (or list), the density is constant, similar to the flat density within the bin of a histogram. Histograms, however, cannot easily be visualized in more than two dimensions, whereas our models can. The accuracy of histograms fades as dimensions increase, whereas our models have priors that help with generalization. Our models are sparse, unlike high-dimensional fixed-bin histograms. We present three generative modeling methods, where the first one allows the user to specify the preferred number of leaves in the tree within a Bayesian prior. The second method allows the user to specify the preferred number of branches within the prior. The third method returns density lists (rather than trees) and allows the user to specify the preferred number of rules and the length of rules within the prior. The new approaches often yield a better balance between sparsity and accuracy of density estimates than other methods for this task. We present an application to crime analysis, where we estimate how unusual each type of modus operandi is for a house break-in.