Cyclic thermodynamics of Bayesian updating: class-relative costs, Gaussian modes, and the limits of universality

Abstract

Bayesian updating determines an ideal posterior after an observation, but it does not determine the physical resources required to produce, expose, and reset a posterior representation. We formulate Bayesian inference as a cyclic task only after specifying an admissible physical class: the charged joint system, available reservoirs and controls, output interface, accuracy criterion, cycle schedule, and closure convention. This specification is necessary because posterior sampling, posterior parameter storage, and recording the observation are distinct physical tasks. For finite-state registers and linear-Gaussian harmonic samplers, we identify an exact conditional free energy relation for a prescribed posterior register representation. When the register is conditioned on the evidence, its average nonequilibrium free energy relative to a prior-equilibrium reference equals \(k_{\mathrm B}T\) times the expected posterior–prior relative entropy. For an exact posterior sampler, however, the unconditional register marginal can already coincide with the prior; the corresponding resource is therefore carried by evidence–register correlations and by conditional access to the output, not by marginal register free energy alone. The associated controller, evidence carrier, and residual correlations must be included whenever they are retained within the charged cycle. For Gaussian updates, the prior-whitened signal-to-noise spectrum decomposes the conditional resource into independent statistical modes. We also analyze an overdamped harmonic realization with a finite preparation interval. Its finite-time contribution is treated as a class-specific, endpoint-consistent control cost and is checked against the exact moment dynamics for the stated control family. These constructions provide solvable benchmarks for well-defined posterior output tasks. They do not imply an architecture-independent energy cost per Bayesian bit or per unit of information gain.