Is probability a "normal" physical quantity?

P. T. Landsberg

Yu.I. Alimov, Yu. A. Kravtsov

The informal aspects, arising in the interpretation of physical experiments, of the theory of probability and mathematical statistics are discussed. The conditions that verifying experiments must satisfy are presented and the role of heuristic (extralogical) assertions is analyzed using the example of mathematical expectation. The principal hypotheses implicit in experiments are enumerated: the principle of reproducibility (the past will be repeated in the future); the principle of reasonable sufficiency; and, the statistical principle (better to predict something rather than nothing). Considerable attention is devoted to Fisher and multisample confidence intervals. It is noted that Fisher confidence intervals are inconsistent. The arguments for introducing contrivances into practical calculations of probabilities are enumerated: incompleteness of any system of hypotheses; subjective estimates of probabilities; adjoining of statistical ensembles; nonstationariness and instability; rare phenomena; and, the use of classical probabilities and the law of large numbers. It is concluded that the relative frequency of appearance (empirical probability) is a normal physical quantity in the sense that it admits physical measurement. Its abnormality is manifested in the fact that it is burdened, more than other physical quantities, with conventions and hypotheses which must be specially checked (verified).

María Carla Galavotti

Francesco R. Ruggeri

For the case of physical quantities with units, such as mass, time, length etc, a change in the value of the quantity makes sense by itself in addition to a relative change. In other words, one does not need to know the mass of an object to understand what a change of 1 kg is. Probability, on the other hand, is a strictly relative quantity and so needs to be normalized. If it is not normalized, one needs to know relative probabilities. To say that an unnormalized probability changes by a value of 10 does not mean anything. Given that equations often solve for an unnormalized probability function, we argue that the notion of the relative probability should appear in equations. Even in the case of ln(P(y)), called information in information theory, for which there is no derivative, taking a derivative, say d/dy, immediately creates a relative probability derivative. Thus the form ln(P) is in keeping with the notion of a relative derivative. Here we examine the idea of a relative derivative and its association with information ln(P), Fisher information, quantum mechanics and the Fokker-Planck equation as well as classical statistical balance in the presence of a potential.

Charles Bruce Cook

This work derives a normalized residual, D, for sustained probability structure using classical statistical and electromagnetic considerations prior to the introduction of quantum mechanics. The construction begins by isolating two independently measurable quantities: Normalized covariance extent (v), representing the declared multidimensional spread of a system. Persistence (p), representing the survival of normalized relational structure across nonzero separation. Classical controls are used to establish independent null boundaries. Independent Brownian increments provide a finite-extent, zero-persistence stochastic control, while electromagnetic coherence theory supplies a physical persistence measure through the standard mutual-coherence formalism. These controls demonstrate that extent and persistence are physically distinct quantities. The paper proves that any sufficiently regular residual respecting both null boundaries factorizes as W(v,p)=vpH(v,p) Within an explicitly defined economy class that excludes unsupported parameters, thresholds, coefficient fields, and additional variables, saturation selects the normalized residual D=vp Importantly, the residual is constructed and fixed before any appeal to quantum mechanics. Quantum systems therefore enter only as later tests of the framework rather than as sources of the definition. The manuscript further develops operational definitions, falsification criteria, transferability requirements, and a programme for investigating whether physical probability possesses measurable internal structure that can be mapped across domains. The work is intended as a foundational probability-domain study rather than a claim of a completed theory of quantum gravity. Its primary objective is the isolation and characterization of a pre-quantum residual built from independently measurable extent and persistence, together with a framework for testing whether that residual appears across broader areas of physics.

Romano Scozzafava