Inferring whether intelligence resides in observable behavior is a foundational challenge in AI. This paper introduces a Bayesian intelligence theory for agents such as language models. Each prompt is treated as a possibly imperfect internal experiment; the agent updates a full‑support prior via Bayes' rule and faithfully reports the posterior over all possible answers. Repeating the same prompt under a fixed state draws fresh, independent outcomes from the same unobserved experiment. We prove that the agent's behavior can be explained in this way if and only if its reports are not fully contradictory—i.e., there always exists at least one state that remains possible across all reports for all prompts. Report frequencies and the magnitudes of positive probabilities impose no further constraints. We further propose and characterize a behavioral intelligence order: if the behaviors of two agents can be interpreted as one having access to a more informative experiment, there must exist a coupling of their report distributions such that the more informative agent's report excludes every answer excluded by the other. Finally, we show the difficulty of aggregating coarse reports from intelligent agents: unless an agent reports a belief about the complete state of the world, the optimal aggregation can assign arbitrary weight to states that have not been excluded. These results provide a theoretical basis for when an agent's behavior can be deemed intelligent and highlight the challenge of rejecting Bayesian rationality.
Review: The work offers a rigorous Bayesian lens to link observable behavior with hidden experiments, giving a fresh framework for assessing the rationality of language models while warning of the aggregation pitfalls inherent in incomplete information.