Density estimation on small datasets

TL;DR

This paper introduces a field-theoretic Bayesian method for accurately estimating smooth probability distributions from small datasets in one dimension, effectively quantifying uncertainty without tuning parameters.

Contribution

It presents an exact nonparametric Bayesian approach that handles strong non-Gaussian constraints non-perturbatively, improving density estimation with limited data.

Findings

Provides an exact Bayesian posterior for 1D density estimation.

Effectively reduces uncertainty by accounting for non-Gaussian constraints.

Includes software implementation of the method.

Abstract

How might a smooth probability distribution be estimated, with accurately quantified uncertainty, from a limited amount of sampled data? Here we describe a field-theoretic approach that addresses this problem remarkably well in one dimension, providing an exact nonparametric Bayesian posterior without relying on tunable parameters or large-data approximations. Strong non-Gaussian constraints, which require a non-perturbative treatment, are found to play a major role in reducing distribution uncertainty. A software implementation of this method is provided.