  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.
