# Automated Learning of Interpretable Models with Quantified Uncertainty

arXiv: https://arxiv.org/abs/2205.01626 · HTML: https://mlsanity.com/paper/2205.01626/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 43 references that are in the map.
- First posted on arXiv 12 Apr 2022.

## Builds on (most cited first, 11 of 11)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Techniques for Interpretable Machine Learning](https://mlsanity.com/paper/1808.00033/) (2018)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [Explainable Machine Learning in Deployment](https://mlsanity.com/paper/1909.06342/) (2019)
- [PMLB: A Large Benchmark Suite for Machine Learning Evaluation and Comparison](https://mlsanity.com/paper/1703.00512/) (2017)
- [SymbolicGPT: A Generative Transformer Model for Symbolic Regression](https://mlsanity.com/paper/2106.14131/) (2021)
- [Bayesian Symbolic Regression](https://mlsanity.com/paper/1910.08892/) (2019)
- [Where are we now? A large benchmark study of recent symbolic regression methods](https://mlsanity.com/paper/1804.09331/) (2018)
- [Stop Explaining Black Box Machine Learning Models for High Stakes Decisions and Use Interpretable Models Instead](https://mlsanity.com/paper/1811.10154/) (2018)
- [Bayesian System ID: Optimal management of parameter, model, and measurement uncertainty](https://mlsanity.com/paper/2003.02359/) (2020)

## Built on by (most cited first, 8 of 8)

- [Priors for symbolic regression](https://mlsanity.com/paper/2304.06333/) (2023)
- [Interpretable Symbolic Regression for Data Science: Analysis of the 2022 Competition](https://mlsanity.com/paper/2304.01117/) (2023)
- [Discovering stochastic partial differential equations from limited data using variational Bayes inference](https://mlsanity.com/paper/2306.15873/) (2023)
- [Complexity-Aware Deep Symbolic Regression with Robust Risk-Seeking Policy Gradients](https://mlsanity.com/paper/2406.06751/) (2024)
- [Prediction Intervals and Confidence Regions for Symbolic Regression Models based on Likelihood Profiles](https://mlsanity.com/paper/2209.06454/) (2022)
- [Bayesian Symbolic Regression via Posterior Sampling](https://mlsanity.com/paper/2512.10849/) (2025)
- [Can Synthetic Data Improve Symbolic Regression Extrapolation Performance?](https://mlsanity.com/paper/2511.22794/) (2025)
- [Are you sure? A Comprehensive and Comprehensible Survey of Uncertainty Quantification in Symbolic Regression](https://mlsanity.com/paper/2606.06567/) (2026)

## Abstract (arXiv)

Interpretability and uncertainty quantification in machine learning can provide justification for decisions, promote scientific discovery and lead to a better understanding of model behavior. Symbolic regression provides inherently interpretable machine learning, but relatively little work has focused on the use of symbolic regression on noisy data and the accompanying necessity to quantify uncertainty. A new Bayesian framework for genetic-programming-based symbolic regression (GPSR) is introduced that uses model evidence (i.e., marginal likelihood) to formulate replacement probability during the selection phase of evolution. Model parameter uncertainty is automatically quantified, enabling probabilistic predictions with each equation produced by the GPSR algorithm. Model evidence is also quantified in this process, and its use is shown to increase interpretability, improve robustness to noise, and reduce overfitting when compared to a conventional GPSR implementation on both numerical and physical experiments.