Bayesian Wind Tunnels for Model Selection

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[Submitted on 1 Jul 2026]

Title:Bayesian Wind Tunnels for Model Selection

View a PDF of the paper titled Bayesian Wind Tunnels for Model Selection, by Siddhartha R Dalal and Vishal Misra and Abhay Parekh

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Abstract:Prior work has shown that transformers can perform exact Bayesian filtering within a fixed

hypothesis class. Can they also perform Bayesian model selection -- identifying the correct

hypothesis class from data? We introduce model-selection Bayesian wind tunnels: controlled

environments where ground-truth posteriors over hypothesis classes are available in closed

form. Using fixed-point-free involutions -- whose defining property f(f(x))=x is purely

relational -- a 2.8M-parameter transformer achieves 0.01-bit entropy agreement with the

Bayesian optimum (3 seeds), with both integer tokens and opaque symbols whose meanings

change every episode. This extends to non-nested comparisons: involutions vs. 3-cycles

(where neither class is a subset of the other) achieve class-posterior MAE under 0.001,

demonstrating genuine model selection beyond simplicity/subset bias. We then identify a

sharp perceptual access condition: when the discriminative statistic requires arithmetic --

modular addition (rotations) or multiplication (f(x)=cx mod p) -- model selection succeeds

with integer tokens but fails completely with opaque symbols, and this boundary persists

under 112x scaling (2.8M to 316M parameters). A stationarity control confirms the operative

factor: opaque tokens with a fixed relabeling succeed (0.009-bit MAE), showing that stable

semantics, not integer identity, enable circuit compilation. Header subtask diagnostics

localize the failure to the composition of header inversion with arithmetic rather than

header parsing itself. Probing frontier LLMs on the same tasks shows qualitative Bayesian

behavior but a large calibration gap (~55x), measured through lossy probes and therefore

directional rather than exact.

Comments: 27 pages

Subjects:

Machine Learning (cs.LG); Machine Learning (stat.ML)

ACM classes: I.2.6; G.3

Cite as: arXiv:2607.19379 [cs.LG]

(or arXiv:2607.19379v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2607.19379

arXiv-issued DOI via DataCite (pending registration)

Submission history

From: Vishal Misra [view email] [v1] Wed, 1 Jul 2026 04:05:08 UTC (124 KB)

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