Transformers can do Bayesian model selection, but only with meaningful tokens

Prior work showed transformers can do Bayesian filtering within a fixed hypothesis class. But can they identify the correct hypothesis class itself? In a new preprint titled "Bayesian Wind Tunnels for Model Selection," Siddhartha Dalal, Vishal Misra, and Abhay Parekh create controlled environments where ground-truth posteriors over hypothesis classes are known exactly. Using fixed-point-free involutions—a purely relational function where f(f(x))=x—a tiny 2.8M-parameter transformer achieves 0.01-bit entropy agreement with the Bayesian optimum across multiple seeds. The model also handles non-nested comparisons (involutions vs. 3-cycles) with class-posterior MAE under 0.001, proving genuine model selection beyond simplicity bias.

The key finding is a sharp perceptual access boundary: when the discriminative statistic requires arithmetic (modular addition or multiplication), model selection succeeds with integer tokens but fails completely with opaque symbols that change meaning each episode. This failure holds even after 112x scaling (2.8M to 316M parameters), but a stationarity control—where opaque tokens keep fixed relabelings—succeeds (0.009-bit MAE), showing stable semantics enable circuit compilation. Probing frontier LLMs reveals qualitative Bayesian behavior but a large calibration gap (~55x). The work has major implications for understanding when transformers truly generalize vs. exploit token meaning.