# Bayesian Symbolic Regression

arXiv: https://arxiv.org/abs/1910.08892 · HTML: https://mlsanity.com/paper/1910.08892/

## Key facts

- Cited by 78 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Oct 2019.

## Built on by (most cited first, 30 of 78)

- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Interpretable Scientific Discovery with Symbolic Regression: A Review](https://mlsanity.com/paper/2211.10873/) (2022)
- [Symbolic Regression via Neural-Guided Genetic Programming Population Seeding](https://mlsanity.com/paper/2111.00053/) (2021)
- [LLM and Simulation as Bilevel Optimizers: A New Paradigm to Advance Physical Scientific Discovery](https://mlsanity.com/paper/2405.09783/) (2024)
- [Alpha-GPT: Human-AI Interactive Alpha Mining for Quantitative Investment](https://mlsanity.com/paper/2308.00016/) (2023)
- [Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl](https://mlsanity.com/paper/2305.01582/) (2023)
- [Rethinking Symbolic Regression Datasets and Benchmarks for Scientific Discovery](https://mlsanity.com/paper/2206.10540/) (2022)
- [Deep Generative Symbolic Regression](https://mlsanity.com/paper/2401.00282/) (2023)
- [Predicting Ordinary Differential Equations with Transformers](https://mlsanity.com/paper/2307.12617/) (2023)
- [Taylor Genetic Programming for Symbolic Regression](https://mlsanity.com/paper/2205.09751/) (2022)
- [Symbolic regression via MDLformer-guided search: from minimizing prediction error to minimizing description length](https://mlsanity.com/paper/2411.03753/) (2024)
- [Call for Action: towards the next generation of symbolic regression benchmark](https://mlsanity.com/paper/2505.03977/) (2025)
- [A Neural-Guided Dynamic Symbolic Network for Exploring Mathematical Expressions from Data](https://mlsanity.com/paper/2309.13705/) (2023)
- [From Kepler to Newton: Explainable AI for Science](https://mlsanity.com/paper/2111.12210/) (2021)
- [GSR: A Generalized Symbolic Regression Approach](https://mlsanity.com/paper/2205.15569/) (2022)
- [A Neural Symbolic Model for Space Physics](https://mlsanity.com/paper/2503.07994/) (2025)
- [Informed Equation Learning](https://mlsanity.com/paper/2105.06331/) (2021)
- [GENSR: Symbolic Regression Based in Equation Generative Space](https://mlsanity.com/paper/2602.20557/) (2026)
- [AI Descartes: Combining Data and Theory for Derivable Scientific Discovery](https://mlsanity.com/paper/2109.01634/) (2021)
- [Automated Learning of Interpretable Models with Quantified Uncertainty](https://mlsanity.com/paper/2205.01626/) (2022)
- [Inferring the Structure of Ordinary Differential Equations](https://mlsanity.com/paper/2107.07345/) (2021)
- [Symbolic Expression Transformer: A Computer Vision Approach for Symbolic Regression](https://mlsanity.com/paper/2205.11798/) (2022)
- [GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equations](https://mlsanity.com/paper/2312.10913/) (2023)
- [In Context Learning and Reasoning for Symbolic Regression with Large Language Models](https://mlsanity.com/paper/2410.17448/) (2024)
- [The Need for Verification in AI-Driven Scientific Discovery](https://mlsanity.com/paper/2509.01398/) (2025)
- [Knowledge Integration for Physics-informed Symbolic Regression Using Pre-trained Large Language Models](https://mlsanity.com/paper/2509.03036/) (2025)
- [Quant 4.0: Engineering Quantitative Investment with Automated, Explainable and Knowledge-driven Artificial Intelligence](https://mlsanity.com/paper/2301.04020/) (2022)
- [MESSY Estimation: Maximum-Entropy based Stochastic and Symbolic densitY Estimation](https://mlsanity.com/paper/2306.04120/) (2023)
- [GFN-SR: Symbolic Regression with Generative Flow Networks](https://mlsanity.com/paper/2312.00396/) (2023)
- [Improving Monte Carlo Tree Search for Symbolic Regression](https://mlsanity.com/paper/2509.15929/) (2025)

## Abstract (arXiv)

Interpretability is crucial for machine learning in many scenarios such as quantitative finance, banking, healthcare, etc. Symbolic regression (SR) is a classic interpretable machine learning method by bridging X and Y using mathematical expressions composed of some basic functions. However, the search space of all possible expressions grows exponentially with the length of the expression, making it infeasible for enumeration. Genetic programming (GP) has been traditionally and commonly used in SR to search for the optimal solution, but it suffers from several limitations, e.g. the difficulty in incorporating prior knowledge; overly-complicated output expression and reduced interpretability etc. To address these issues, we propose a new method to fit SR under a Bayesian framework. Firstly, Bayesian model can naturally incorporate prior knowledge (e.g., preference of basis functions, operators and raw features) to improve the efficiency of fitting SR. Secondly, to improve interpretability of expressions in SR, we aim to capture concise but informative signals. To this end, we assume the expected signal has an additive structure, i.e., a linear combination of several concise expressions, whose complexity is controlled by a well-designed prior distribution. In our setup, each expression is characterized by a symbolic tree, and the proposed SR model could be solved by sampling symbolic trees from the posterior distribution using an efficient Markov chain Monte Carlo (MCMC) algorithm. Finally, compared with GP, the proposed BSR(Bayesian Symbolic Regression) method saves computer memory with no need to keep an updated 'genome pool'. Numerical experiments show that, compared with GP, the solutions of BSR are closer to the ground truth and the expressions are more concise. Meanwhile we find the solution of BSR is robust to hyper-parameter specifications such as the number of trees.