Probabilistic Grammars for Equation Discovery

TL;DR

This paper introduces probabilistic context-free grammars for equation discovery, enabling more flexible, efficient, and Bayesian-friendly symbolic regression by encoding soft constraints and priors over equations.

Contribution

It proposes the use of probabilistic grammars in symbolic regression, enhancing flexibility and efficiency over deterministic methods, and establishing a foundation for Bayesian approaches.

Findings

Probabilistic grammars improve search efficiency in equation discovery.

Soft constraints via probabilities better encode parsimony principles.

Framework supports Bayesian methods for symbolic regression.

Abstract

Equation discovery, also known as symbolic regression, is a type of automated modeling that discovers scientific laws, expressed in the form of equations, from observed data and expert knowledge. Deterministic grammars, such as context-free grammars, have been used to limit the search spaces in equation discovery by providing hard constraints that specify which equations to consider and which not. In this paper, we propose the use of probabilistic context-free grammars in equation discovery. Such grammars encode soft constraints, specifying a prior probability distribution on the space of possible equations. We show that probabilistic grammars can be used to elegantly and flexibly formulate the parsimony principle, that favors simpler equations, through probabilities attached to the rules in the grammars. We demonstrate that the use of probabilistic, rather than deterministic grammars, in the context of a Monte-Carlo algorithm for grammar-based equation discovery, leads to more efficient equation discovery. Finally, by specifying prior probability distributions over equation spaces, the foundations are laid for Bayesian approaches to equation discovery.