Abstract:We equip the space of beliefs with a cost geometry (what it costs to pass from one belief to another): optimal transport in Wasserstein space, reweighted conformally by Fisher information (the price of the precision at stake), distinct from the Fisher-Rao metric. In the setting we consider, a finite machine maintains a digital twin of a system; observing the territory through finite, noisy sensors, we model its coherent output as a belief: a probability density over states, the Bayes posterior. Certainty (the perfect twin) is denied twice, by observation and by physics, both read off the Fisher information. On the conformal class, essentially location-scale, three results emerge, all invariants of one change of cost unit. A wall: a well-posed inference rejects certainty to infinite distance as soon as the cost dominates the Fisher information (necessity conjectured beyond power laws). An honesty: an honest (eikonal) cost, each nat the same length everywhere, selects the geometries proportional to the Fisher information. A rigidity: these geometries are hyperbolic, and the Stam bound crowns the Gaussian, the most hyperbolic location-scale belief. Changing the unit dilates the geometry yet preserves the wall, the curvature ordering, and the extremality of the Gaussian: an absolute cost says nothing, only relative cost carries meaning, the value -1/4 being one of its images. The cost of reaching a given precision then has a geometric floor diverging at certainty. Thermodynamics fixes the cost unit and motivates this framework; the results are geometric, in nats.
From: Laurent Caraffa Dr [view email]
[v1]
Fri, 19 Jun 2026 16:41:28 UTC (158 KB)
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