Random irregular histograms

TL;DR

This paper introduces a Bayesian method for constructing irregular histograms that automatically determines the number and location of bins, achieving optimal convergence rates and competitive performance in density estimation.

Contribution

It presents a fully Bayesian approach to irregular histogram construction with automatic bin selection and proven consistency and optimal convergence rates.

Findings

Histogram estimate is consistent under mild conditions.

Method attains near-minimax convergence rate for Hölder densities.

Performs comparably to existing methods in simulations.

Abstract

We propose a new method of histogram construction, providing a fully Bayesian approach to irregular histograms. Our procedure applies Bayesian model selection to a piecewise constant model of the underlying distribution, resulting in a method that selects both the number of bins as well as their location based on the data in a fully automatic fashion. We show that the histogram estimate is consistent with respect to the Hellinger metric under mild regularity conditions, and that it attains a convergence rate equal to the minimax rate (up to a logarithmic factor) for H\"{o}lder continuous densities. Simulation studies indicate that the new method performs comparably to other histogram procedures, both for minimizing the estimation error and for identifying modes. A software implementation is included as supplementary material.