  Bayesian models of group learning are studied in Economics since the 1970s.
and more recently in computational linguistics. The models from Economics
postulate that agents maximize utility in their communication and actions. The
Economics models do not explain the ``probability matching" phenomena that are
observed in many experimental studies. To address these observations, Bayesian
models that do not formally fit into the economic utility maximization
framework were introduced. In these models individuals sample from their
posteriors in communication. In this work we study the asymptotic behavior of
such models on connected networks with repeated communication. Perhaps
surprisingly, despite the fact that individual agents are not utility
maximizers in the classical sense, we establish that the individuals ultimately
agree and furthermore show that the limiting posterior is Bayes optimal.
  We explore the interpretation of our results in terms of Large Language
Models (LLMs). In the positive direction our results can be interpreted as
stating that interaction between different LLMs can lead to optimal learning.
However, we provide an example showing how misspecification may lead LLM agents
to be overconfident in their estimates.
