The probability of an outcome was taken to be something closely related to the frequency with which that outcome could be expected to occur in a large number of identical repeated trials of some specified kind. A fundamental theorem of probability theory, Bayes' Theorem, relates the probability of a hypothesis on the basis of evidence to the conditional probability of the evidence, given the truth of the hypothesis and the initial probability that the hypothesis is true. "Nonpathological" means, essentially, that the initial probability distribution will give probability zero to any region of microconditions that have zero probability in the familiar, standard measure. Any non-pathological initial probability distribution for a mixing system will, as time goes into past infinity, also evolve into a probability distribution that is coarsely like the equilibrium distribution. The probability distribution is rationalized by showing that it is the uniquely stationary one that gives probability zero to regions given probability zero in that very natural measure.
The probability of an outcome was taken to be something closely related to the frequency with which that outcome could be expected to occur in a large number of identical repeated trials of some specified kind. A fundamental theorem of probability theory, Bayes' Theorem, relates the probability of a hypothesis on the basis of evidence to the conditional probability of the evidence, given the truth of the hypothesis and the initial probability that the hypothesis is true. "Nonpathological" means, essentially, that the initial probability distribution will give probability zero to any region of microconditions that have zero probability in the familiar, standard measure. Any non-pathological initial probability distribution for a mixing system will, as time goes into past infinity, also evolve into a probability distribution that is coarsely like the equilibrium distribution. The probability distribution is rationalized by showing that it is the uniquely stationary one that gives probability zero to regions given probability zero in that very natural measure.
Definitions The concept of probability is of considerable importance in optics, as in any situation in which the outcome of a given trial or measurement is uncertain. Under these conditions it is desirable to be able to associate a measure with the likelihood of the outcome or the event in question; such a measure is called the probability of the event. Several different definitions of probability have been adopted at various times in the past. The classical definition is based on an exhaustive enumeration of the possible outcomes of an experiment or trial. If the trial has N distinguishable, mutually exclusive outcomes, which are equally likely to occur, and if n out of these N possible outcomes have an attribute or characteristic that we call ‘success’, then the probability of success in any one trial is given by the ratio n/N . For example, if we roll a die, and if each of the six digits is equally likely to be on top when the die comes to rest, there are N = 6 distinguishable outcomes. If we identify success with an even number, for example, then since there are three different ways in which success can be achieved, it follows that the probability of success when the die is rolled is given by 3/6 = 1/2. Unfortunately, an exhaustive enumeration of all possibilities is not always feasible. Another common definition of probability is based on the notion of relative frequency of success.
Every vertex of the magic cube of physics discussed so far introduced a major element in the development of physics and the whole of science. Mechanics introduced the basic concept of equations of motion, or time-evolution of a physical system given certain initial condition and the associated idea of determinism, even with its generalization in the form of chaos, the dependence of the outcomes of the temporal evolution on the initial conditions. These concepts taken for many particles, or elements of a physical system, led to the probabilistic description of many particle systems, in the realm of statistical physics and thermodynamics, whereby the averages over time of various observable quantities were related to the ensemble averages, that involve classical, so-called Kolmogorov, probability measures, at least for certain “equilibrium” situations. The second law of thermodynamics introduced the concept of entropy and its probabilistic increase, the first truly emergent, probabilistic law in physics (temperature and pressure, being other emergent quantities, apart from entropy, the measure of disorder of a statistical system). In the context of Newtonian gravity, one gets, at least in retrospect the geometric description of gravity in non-relativistic, Galilean spacetime. Galilean relativity, based on the asymmetric mixing of space and time, with the absolute, universal time, true for all observers, gets generalized in special theory of relativity to observer dependent space and observer dependent time, viewed as “slices” of a new concept of spacetime, with the new concept of causal structure, that distinguish timelike, null and spacelike events. This in turn is made into a full dynamical geometric structure of spacetime with a relativistic description of gravity stemming from curvature of spacetime, and a dynamical causal structure, tied to gravity. In our next story, the story of quantum theory, we will encounter fundamental, intrinsic probabilities, that can interfere, unlike the classical probabilities of statistical thermodynamics, and which are consistent with equations of motion of individual elements, and not, as in statistical thermodynamics, with temporal averages over the time evolution of many elements. This is the big new lesson of quantum theory, that transcends the lessons of classical spacetime physics.
We formalise in Lean4/Mathlib the core repeated-trial theorem underlying volume-based deterministic accounts of quantum statistics. The setting is a finite-measure ontic state space (Σ, µL) equipped with a measurable preparation region Ω0 ⊆ Σ, a deterministic measurable flow Φ : Σ → Σ, and a fixed measurable ontic outcome region. The key modelling assumption is explicit: repeated trials are represented by a repeated-preparation model in which each trial begins from an initial microstate distributed according to the conditional preparation measure on Ω0. For a fixed outcome region, the empirical frequencies of the associated indicator observables converge almost surely to the corresponding normalised ontic weight. The formalisation isolates the exact logical dependencies of the result, separates what is proved inside Lean from what is imported from Mathlib, and distinguishes the physical assumptions of the repeated-preparation model from the law-of-large-numbers machinery used in the convergence step. In the current repository, the theorem is exported as LF1_main_theorem_ae. It is stated for a repeated-trial model T : S.TrialModel Ω and a fixed outcome region O : S.OutcomeRegion, with pairwise independence of the associated indicator random variables as the only non-trivial repeated-trial hypothesis supplied explicitly at the final stage. Integrability and identical-distribution properties are discharged internally from the trial-model structure. Within the Constraint-Surface Dynamics programme, this paper formalises the statistical backbone of volume typicality in the notation of TN0. It does not, by itself, provide a machine-verified derivation of Born-form weights, a projective measure bridge, or the mixed-state and post-measurement layers. Its role is narrower and more precise: to show that, once deterministic ontic events and a repeated-preparation scheme have been specified, empirical frequencies converge almost surely to the corresponding real-valued normalised ontic weights.
Foundational issues in statistical mechanics and the more general question of how probability is to be understood in the context of physical theories are both areas that have been neglected by philosophers of physics. This book fills an important gap in the literature by providing a most systematic study of how to interpret probabilistic assertions in the context of statistical mechanics. The book explores both subjectivist and objectivist accounts of probability, and takes full measure of work in the foundations of probability theory, in statistical mechanics, and in mathematical theory. It will be of particular interest to philosophers of science, physicists and mathematicians interested in foundational issues, and also to historians of science.
We propose a simple abstract formalisation of the act of observation, in which the system and the observer are assumed to be in a pure state and their interaction deterministically changes the states such that the outcome can be read from the state of the observer after the interaction. If the observer consistently realizes the outcome which maximizes the likelihood ratio that the outcome pertains to the system under study (and not to his own state), he will be called Bayes-optimal. We calculate the probability if for each trial of the experiment the observer is in a new state picked randomly from his set of states, and the system under investigation is taken from an ensemble of identical pure states. For classical statistical mixtures, the relative frequency resulting from the maximum likelihood principle is an unbiased estimator of the components of the mixture. For repeated Bayes-optimal observation in case the state space is complex Hilbert space, the relative frequency converges to the Born rule. Hence, the principle of Bayes-optimal observation can be regarded as an underlying mechanism for the Born rule. We show the outcome assignment of the Bayes-optimal observer is invariant under unitary transformations and contextual, but the probability that results from repeated application is non-contextual. The proposal gives a concise interpretation for the meaning of the occurrence of a single outcome in a quantum experiment as the unique outcome that, relative to the state of the system, is least dependent on the state of the observe at the instant of measurement.