Differentiability and overlap concentration in optimal Bayesian inference

doi:10.1093/imaiai/iaag002

Differentiability and overlap concentration in optimal Bayesian inference

2026-01-01

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DOI：10.1093/imaiai/iaag002

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Abstract

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In this short note, we consider models of optimal Bayesian inference of finite-rank tensor products. We add to the model a linear channel parametrized by $h$. We show that at every interior differentiable point $h$ of the free energy (associated with the model), the overlap concentrates at the gradient of the free energy and the minimum mean-square error converges to a related limit. In other words, the model is replica-symmetric at every differentiable point. At any signal-to-noise ratio, such points $h$ form a full-measure set (hence $h=0$ belongs to the closure of these points). For a sufficiently low signal-to-noise ratio, we show that every interior point is a differentiable point.

Keywords:

statistical inference

Hamilton-Jacobi equation

replica symmetry

Differentiability and overlap concentration in optimal Bayesian inference

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