Statistical model discovery is a challenging search over a vast space of models subject to domain-specific constraints. Efficiently searching over this space requires expertise in modeling and the problem domain. Motivated by the domain knowledge and programming capabilities of large language models (LMs), we introduce a method for language model driven automated statistical model discovery. We cast our automated procedure within the principled framework of Box’s Loop: the LM iterates between proposing statistical models represented as probabilistic programs, acting as a modeler, and critiquing those models, acting as a domain expert. By leveraging LMs, we do not have to define a domain-specific language of models or design a handcrafted search procedure, which are key restrictions of previous systems. We evaluate our method in three settings in probabilistic modeling: searching within a restricted space of models, searching over an open-ended space, and improving expert models under natural language constraints (e.g., this model should be interpretable to an ecologist). Our method identifies models on par with human expert designed models and extends classic models in interpretable ways. Our results highlight the promise of LM-driven model discovery.