[Submitted on 17 Jun 2026]
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Abstract:To leading order under local asymptotic normality (LAN), the confidence density a worker emits over a chunk of size $n$ is a Gibbs--Boltzmann measure $\exp\{-\beta E(\theta)\}$ whose inverse temperature is the sample size, $\beta=n$. Three consequences are exact in the Gaussian/linear case and first-order otherwise: disjoint chunks carry independent Boltzmann factors, so the MapReduce \emph{reduce}, read literally, is a partition function $Z=\int\prod_k h_k\,d\theta$ whose mode is precision-weighted (inverse-variance) pooling; frequentist consistency is the zero-temperature limit $T=1/n\to0$
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