# Bayesian Symbolic Regression via Posterior Sampling

arXiv: https://arxiv.org/abs/2512.10849 · HTML: https://mlsanity.com/paper/2512.10849/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Builds on (most cited first, 8 of 8)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [Bayesian Symbolic Regression](https://mlsanity.com/paper/1910.08892/) (2019)
- [A Bayesian machine scientist to aid in the solution of challenging scientific problems](https://mlsanity.com/paper/2004.12157/) (2020)
- [Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery](https://mlsanity.com/paper/2206.10540/) (2022)
- [Automated Learning of Interpretable Models with Quantified Uncertainty](https://mlsanity.com/paper/2205.01626/) (2022)
- [Improving Genetic Programming for Symbolic Regression with Equality Graphs](https://mlsanity.com/paper/2501.17848/) (2025)

## Built on by (most cited first, 3 of 3)

- [Symbolic Density Estimation for Discrete Distributions](https://mlsanity.com/paper/2605.21813/) (2026)
- [Are you sure? A Comprehensive and Comprehensible Survey of Uncertainty Quantification in Symbolic Regression](https://mlsanity.com/paper/2606.06567/) (2026)
- [Bayesian Symbolic Regression with Entropic Reinforcement Learning](https://mlsanity.com/paper/2608.09617/) (2026)

## Abstract (arXiv)

Symbolic regression is a powerful tool for discovering governing equations directly from data, but its sensitivity to noise hinders its broader application. This paper introduces a Sequential Monte Carlo (SMC) framework for Bayesian symbolic regression that approximates the posterior distribution over symbolic expressions, enhancing robustness and enabling uncertainty quantification for symbolic regression in the presence of noise. Differing from traditional genetic programming approaches, the SMC-based algorithm combines probabilistic selection, adaptive tempering, and the use of normalized marginal likelihood to efficiently explore the search space of symbolic expressions, yielding parsimonious expressions with improved generalization. When compared to standard genetic programming baselines, the proposed method better deals with challenging, noisy benchmark datasets. The reduced tendency to overfit and enhanced ability to discover accurate and interpretable equations paves the way for more robust symbolic regression in scientific discovery and engineering design applications.