Bayesian Model Discovery with Language Models: Closing the Loop between Hypotheses and Experiments

Abstract

Scientific model discovery requires proposing candidate models of an unknown system and designing experiments that discriminate among them. However, LLM-based approaches typically rely on in-context learning to predict outcomes and select experiments, lacking explicit model evaluation to systematically guide hypothesis search, while Bayesian approaches enable principled model comparison but are confined to a predefined candidate set which results in limited search spaces. To this end, we introduce BMD, which closes the loop between LLM-driven hypothesis generation and Bayesian experimental design. Specifically, an LLM proposes model families, while a numerical engine fits their parameters, weighs them by model evidence, and averages their predictions. Then, BMD chooses the next experimental conditions by maximizing the joint expected information gain over families and their parameters, while feeding the updated beliefs and residual diagnostics back to the LLM to guide its next proposals. Across four scientific discovery benchmarks, BMD achieves lowest final prediction error than in-context learning and LLM-based scientific agents while recovering exact functional forms more often, with ablations supporting numerical feedback and the joint objective.