Symbolic Density Estimation for Discrete Distributions

TL;DR

This paper presents symbolic density estimation (SDE), an unsupervised method that automatically derives closed-form discrete probability distributions using structured search and domain priors.

Contribution

It introduces SDE, a novel framework combining evolutionary search and validity-aware inference to recover interpretable discrete distributions automatically.

Findings

Successfully recovers all benchmark distribution families with accurate parameters.

Extends to complex distributions like zero inflation and finite mixtures.

Improves goodness-of-fit with interpretable mixture models in real data.

Abstract

Discrete probability laws underpin statistical modeling, yet the catalog of interpretable distributions has expanded only gradually through centuries of case-by-case mathematical derivations. We introduce symbolic density estimation (SDE), an unsupervised framework that automatically recovers closed-form probability mass functions by composing elementary analytic operations within a structured search space. Our method integrates domain-specific structural priors with evolutionary search and a validity-aware inference stage, and it extends to richer distribution families such as zero inflation and finite mixtures. To support systematic evaluation and future research, we contribute a benchmark dataset spanning a broad collection of commonly used discrete distributions. The proposed algorithm recovers all benchmark families with accurate parameter estimates. A real data application shows that it identifies concise and interpretable mixture models that improve goodness-of-fit over standard models.