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A unified interpretation of probability

This paper proposes a modified, partition-based frequency interpretation of probability that retains von Mises' core requirement of repeatable events with stabilized frequencies while resolving classic philosophical problems, unifying Bayesian approaches, and offering new perspectives on quantum puzzles.

Original authors: Louis Vervoort

Published 2026-10-01

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Louis Vervoort

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

For over a century, scientists and philosophers have debated what probability actually is. Is it a measure of how often something happens in the real world, like the frequency of rain in a specific season? Or is it a measure of how strongly a person believes an event will occur, a reflection of their own knowledge or ignorance? This question sits at the intersection of mathematics, physics, and philosophy. While the mathematical rules for calculating probability are well established, applying those rules to the messy reality of the natural world has proven difficult. The core of the problem is determining what counts as a valid experiment. If you roll a die, the outcome seems random, but if you knew every detail of the throw—the force, the angle, the air resistance—you could predict the result with certainty. This tension between the apparent randomness of nature and the underlying laws of physics has led to many competing theories, none of which have fully satisfied everyone.

A researcher named Louis Vervoort, writing from the Higher School of Economics in Moscow, proposes a way to resolve this long-standing debate. His work suggests that the confusion stems from looking at probability as a property of a single object or a single event, rather than as a property of a whole system interacting with its environment. He revisits an older idea called the frequency interpretation, which argues that probability is simply the rate at which an event repeats over time. While this idea has faced heavy criticism for being too rigid or relying on impossible infinite series, Vervoort argues that the critics missed the point. He proposes a modified version that keeps the core of the frequency idea but adds a crucial detail: to define a probability, you must first define the specific conditions under which the experiment happens.

The author's central insight is that you cannot talk about the probability of a die landing on a six without describing the entire setup. This setup includes the die itself, the hand or machine that throws it, and the surface it lands on. Vervoort calls this complete arrangement a "system." If you change the hand, the surface, or the air around the die, you change the system, and you change the probability. In his view, probability only exists for systems that can be repeated under these specific, controlled conditions. When you repeat the experiment enough times, the results settle into a stable pattern. For example, if you throw a fair die thousands of times under the same conditions, the number of sixes will stabilize around one-sixth of the total throws. This stabilization is the key. If the results jump around erratically without settling, then the system is not truly probabilistic in the scientific sense, or the conditions were not controlled well enough.

This approach solves several famous puzzles that have confused mathematicians for generations. One such puzzle, known as Bertrand's paradox, asks for the probability that a randomly drawn line inside a circle is shorter than the side of a triangle inscribed in that circle. Depending on how you choose to draw the line, you get three different correct answers. Vervoort explains that this is not a failure of math, but a failure to define the experiment. The question is ambiguous because it does not specify the method of drawing the line. Once you specify the exact physical process—how the line is initiated and how the result is measured—the ambiguity disappears, and there is only one correct probability. The same logic applies to other confusing scenarios, such as the Monty Hall problem, where the answer depends entirely on the specific rules of the game and the information available to the player.

The paper also addresses the role of human knowledge. Some theories suggest that probability is subjective, changing based on what a person knows. For instance, if you throw a die and a friend with a high-speed camera sees the result before you do, their probability for that throw is 100 percent, while yours remains one-sixth. Vervoort argues that this does not mean probability is subjective. Instead, it means you and your friend are describing two different systems. Your system is a standard die throw where the outcome is unknown. Your friend's system includes the camera and the knowledge of the outcome, making it a different, deterministic system. The probability value itself remains an objective feature of the specific system being observed, not a reflection of the observer's mind.

Furthermore, the author suggests that this framework helps clarify some of the most baffling aspects of quantum physics, the branch of science that deals with the smallest particles. In quantum mechanics, the act of measuring a particle seems to change its behavior. Critics often argue that this implies reality is fundamentally dependent on the observer. Vervoort contends that this is simply a case of the measuring device being part of the system. Just as the table and the hand are part of the die-throwing system, the detector is part of the quantum system. The probability of a particle appearing in a certain place depends on the entire experimental setup, including the detector. When physicists realize that the "observer" is just a component of the physical conditions, many of the philosophical mysteries surrounding quantum mechanics become less about magic and more about the precise definition of the experiment.

The author concludes that probability is an objective feature of the physical world, but it is a feature of systems-in-context, not of isolated objects. It is a theory about how things behave when repeated under specific conditions. By focusing on the conditions—the initiating forces, the test objects, and the probing environments—scientists can unify different ways of thinking about chance. This view allows them to treat probability as a physical theory, similar to gravity or electromagnetism, where the numbers describe real, measurable patterns in nature. While the idea of "infinite" repetitions is a useful mathematical tool, the author argues that in the real world, we only need enough repetitions to see the pattern stabilize within a reasonable margin of error. This practical adjustment makes the theory robust enough to apply to everything from rolling dice to the behavior of gas molecules and the decay of atoms, offering a single, coherent way to understand the role of chance in our universe.

Technical Summary: A Unified Interpretation of Probability

Problem Statement
The paper addresses the persistent philosophical and foundational "problem of interpretation" regarding the concept of probability. While Kolmogorov's axiomatic calculus provides a rigorous mathematical framework, it does not define what a "probabilistic event" is or how to apply the theory to real-world physical systems. The literature is fragmented among competing interpretations: the classical (Laplacian) view, the frequency view (von Mises), the subjective/Bayesian view, and the propensity view. This pluralism leads to paradoxes (e.g., Bertrand's paradox, the Monty Hall problem) and confusion regarding the nature of single-case probabilities and the objectivity of quantum mechanical predictions. The author argues that the frequency interpretation, specifically the version proposed by Richard von Mises, has been unfairly discredited and contains the essential elements for a unified, objective, and physically grounded interpretation of scientific probability.

Methodology
The author employs a unification strategy guided by the "MCT-thesis" (a heuristic from the broader book context), aiming to synthesize existing interpretations into a single coherent model. The methodology involves:

Re-evaluating von Mises: Critically analyzing von Mises' frequency interpretation, retaining the core concept of "frequency stabilisation" while discarding the complex and controversial notion of "infinite collectives" and "invariance under place selection."
Partitioning Probabilistic Systems: Introducing a structural analysis of probabilistic events by partitioning them into three subsystems: the test object (the system being measured), the initiating system (the mechanism starting the event), and the probing system (the environment or detector observing the outcome).
Defining $\rho$ -Systems: Proposing that probability is not a property of isolated objects but of " $\rho$ -systems" (systems-in-context) that exhibit frequency stabilisation under well-defined, repeatable conditions.
Introducing Precision ( $\delta$ ): Reformulating the definition of probability to include an experimental precision parameter, moving away from the strict counterfactual requirement of an infinite limit ( $n \to \infty$ ) toward a pragmatic definition based on convergence within a precision interval $\delta$ .
Case Analysis: Applying this modified framework to resolve classic paradoxes and unify subjective interpretations (specifically Jaynes' Bayesianism) and quantum mechanical puzzles.

Key Contributions and Definitions
The paper proposes two central definitions to ground the interpretation in empirical reality:

DEF1 (Frequency Stabilisation): A system $S_C$ $S_{C}$ possesses frequency stabilisation (characterized by precision $\delta$ $δ$ ) if:
1. It is possible to repeat $n$ "identical" experiments on a test object $T$ under identical conditions $C$ (partitioned into initiating and probing subsystems).
2. The relative frequencies of outcomes $R_j$ converge toward constant numbers as $n$ grows, such that the ratio lies within the interval $[P - \delta, P + \delta]$ .
DEF2 (Probability): For a system exhibiting frequency stabilisation (per DEF1), the probability $P(R_j)$ is defined as the relative frequency of the outcome in the series for sufficiently large $n$ .

The author introduces the concept of $\rho$ -systems (or $\rho$ -random systems) to distinguish genuine probabilistic systems (where frequency stabilisation occurs) from "quasi-probabilities" (where stabilisation fails due to uncontrolled variables or changing conditions).

Results and Solutions to Specific Problems
The unified model yields several specific results (R1–R4):

R1: Single-Case Probabilities and Propensity: The model dissolves the problem of single-case probabilities. It argues that attributing a quantitative probability to a single event (e.g., a specific atom decaying) is only meaningful if that event is part of a repeatable $\rho$ -system. The "propensity" interpretation is shown to be a trimmed-down variant of the frequency interpretation, where propensity is simply the objective tendency of a specific experimental setup (conditions) to generate specific frequencies.
R2: Bertrand's Paradox: The paradox is resolved by recognizing that "drawing a chord randomly" is an experimentally ambiguous phrase. Different methods of "randomization" (initiating conditions) constitute different $\rho$ -systems, each with its own valid probability. The paradox arises from failing to specify the initiating and probing conditions.
R3: Unification with Subjective/Bayesian Interpretations: The paper argues that subjective interpretations, particularly Jaynes' Bayesianism, can be unified with frequentism. The "prior information" or "knowledge state" in Bayesianism corresponds to the specific experimental conditions ( $C$ ) in the frequency model. Conditional probabilities $P(H|D)$ are reinterpreted as objective frequencies of outcomes in automated experiments where the "data" $D$ is a condition of the setup. Thus, subjective probability is reframed as an objective measure relative to specific conditions.
R4: Quantum Puzzles: Many quantum mechanical puzzles, including the EPR debate and contextuality (Kochen-Specker theorem), are attributed to an imprecise interpretation of probability. The author argues that quantum properties depend intrinsically on the "entire experimental setup" (the $\rho$ -system). The "observer effect" is not a mystical collapse but a consequence of the fact that probability is defined relative to the probing subsystem. Commutation relations and contextuality are interpreted as constraints on which properties can be measured simultaneously within a single $\rho$ -system.

Significance and Claims
The paper claims that probability theory, beyond its mathematical formalism, is fundamentally a physical theory describing an objective category of nature: "objective chance." Its significance lies in:

Unification: It provides a single framework that encompasses classical chance games, natural stochastic phenomena, and quantum mechanics, subsuming the classical and subjective interpretations as special cases or operational shortcuts.
Objectivity: It grounds probability in the physical reality of repeatable systems and frequency stabilisation, countering the view that probability is merely a measure of human ignorance or belief.
Resolution of Paradoxes: By explicitly partitioning systems into test objects, initiators, and probes, the model clarifies the conditions under which probability is defined, thereby dissolving classic paradoxes that arise from ambiguous experimental definitions.
Epistemic vs. Ontic: The paper clarifies the distinction between the value of a probability (which is objective) and the ascription of determinism or indeterminism to a system (which depends on the observer's knowledge of the conditions). It suggests that what appears as "indeterminism" may simply be a lack of knowledge regarding the full $\rho$ -system conditions.

The author concludes that while the operationalist elements of the definition (reliance on protocols and conditions) are unavoidable, they do not undermine the ontological status of probability as a real, objective feature of the world, applicable even to systems where no human observer is present.

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