Kolmogorov Probability Theory Research Articles

A Supraclassical Probabilistic Entailment Relation

The paper presents an original supraclassical nontrivial plausible entailment relation $\vapprox$ that employs Kolmogorov's probability theory. Its crucial feature is the primitiveness of a conditional probability, which one calculates with the help of the method of truth tables for classical propositional logic. I study the properties of the entailment relation in question. In particular, I show that while being supraclassical, i.e., all classical entailments and valid formulas are $\vapprox$-valid, but not vice versa, it is not trivial and enjoys the same form of inconsistency as classical entailment ⊧ does. I specify the place of the proposed probability entailment relation in certain classifications of nonclassical entailment relations. In particular, I use Douven's analysis of some probabilistic entailment relations that contains dozens of properties that are crucial for any probabilistic entailment relation, as well as Hlobil's choosing your nonmonotonic logic: shopper’s guide, due to the fact that $\vapprox$ is not monotonic, and Cobreros, Egré, Ripley, van Rooij's entailment relations for tolerant reasoning. At last, I perform a comparative analysis of classical, the proposed, and some other entailment relations closely related to the latter: those introduced by Bocharov, Markin, Voishvillo, Degtyarev, Ivlev, where the last two entailment relations are based on the so-called principle of reverse deduction, which is an intuitively acceptable way to connect classical and probabilistic entailment relations.

Maximum Probability Theorem: A Framework for Probabilistic Machine Learning

We present a theoretical framework of probabilistic learning derived from the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">maximum probability (MP) theorem</i> shown in this article. In this probabilistic framework, a model is defined as an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">event</i> in the probability space, and a model or the associated <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">event</i> —either the true underlying model or the parameterized model—has a quantified probability measure. This quantification of a model's probability measure is derived by the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">MP theorem</i> , in which we have shown that an event's probability measure has an upper bound given its conditional distribution on an arbitrary random variable. Through this alternative framework, the notion of model parameters is encompassed in the definition of the model or the associated <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">event</i> . Therefore, this framework deviates from the conventional approach of assuming a prior on the model parameters. Instead, the regularizing effects of assuming prior over parameters are imposed through maximizing probabilities of models or, according to information theory, minimizing the information content of a model. The probability of a model in our framework is invariant to reparameterization and is solely dependent on the model's likelihood function. In addition, rather than maximizing the posterior in a conventional Bayesian setting, the objective function in our alternative framework is defined as the probability of set operations (e.g., intersection) on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">event</i> of the true underlying model and the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">event</i> of the model at hand. Our theoretical framework adds clarity to probabilistic learning through solidifying the definition of probabilistic models, quantifying their probabilities, and providing a visual understanding of objective functions. <p xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>Impact Statement</i>—The choice of prior distribution over the parameters of probabilistic machine learning models determines the regularization of learning algorithms in the Bayesian perspective. The complexity in choice of prior over the parameters and the form of regularization is relative to the complexity of the models being used. Thereby, finding priors for parameters of complex models is often not tractable. We address this problem by uncovering the maximum probability (MP) theorem as a direct consequence of Kolmogorov's probability theory. Through the lens of the MP theorem, the process of regularizing models is understood and automated. The regularization process is defined as the maximization of the probability of the model. The probability of the model is understood by the MP theorem and is determined by the behavior of the model. The effects of maximizing the probability of the model can be backpropagated in a gradient-based optimization process. Consequently, the MP framework provides a form of black-box regularization and eliminates the need for case-by-case analysis of models to determine priors.

Postulating the Theory of Experience and Chance

The aim of the paper is the axiomatic justification of the theory of experience and chance, one of the dual halves of which is the Kolmogorov probability theory. The author’s main idea was the natural inclusion of Kolmogorov’s axiomatics of probability theory in a number of general concepts of the theory of experience and chance. The main result of this work is the axiom of co~event, intended for the sake of constructing a theory formed by dual theories of believabilities and probabilities, each of which itself is postulated by its own Kolmogorov system of axioms. Of course, other systems of postulating the theory of experience and chance can be imagined, however, in this work a preference is given to a system of postulates that is able to describe in the most simple manner the results of what I call an experienced-random experiment

Infinitesimal Probabilities Based on Grossone

In finite probability theory, the only probability zero event is the impossible one, but in standard Kolmogorov probability theory, probability zero events occur all the time. Prominent logicians, probability experts and philosophers of probability, including Carnap, Kemeny, Shimony, Savage, De Finetti, Jeffrey, have successfully argued that a sound probability should be regular, that is, only the impossible event should have zero probability. This intuition is shared by physicists too. Totality is another desideratum which means that every event should be assigned a probability. Regularity and totality are achievable in rigorous mathematical terms even for infinite events via hyper-reals valued probabilities. While the mathematics of these theories is not objectionable, some philosophical arguments purport to show that infinitesimal probabilities are inherently problematic. In this paper, we present a simpler and natural construction—based on Sergeyev’s calculus with Grossone (in a formalism inspired by Lolli) enriched with infinitesimals—of a regular, total, finitely additive, uniformly distributed probability on infinite sets of positive integers. These probability spaces—which are inspired by and parallels the construction of classical probability—will be briefly studied. In this framework, De Finetti fair lottery has the natural solution and Williamson’s objections against infinitesimal probabilities are mathematically refuted.

On MV-Algebraic Versions of the Strong Law of Large Numbers.

Many-valued (MV; the many-valued logics considered by Łukasiewicz)-algebras are algebraic systems that generalize Boolean algebras. The MV-algebraic probability theory involves the notions of the state and observable, which abstract the probability measure and the random variable, both considered in the Kolmogorov probability theory. Within the MV-algebraic probability theory, many important theorems (such as various versions of the central limit theorem or the individual ergodic theorem) have been recently studied and proven. In particular, the counterpart of the Kolmogorov strong law of large numbers (SLLN) for sequences of independent observables has been considered. In this paper, we prove generalized MV-algebraic versions of the SLLN, i.e., counterparts of the Marcinkiewicz–Zygmund and Brunk–Prokhorov SLLN for independent observables, as well as the Korchevsky SLLN, where the independence of observables is not assumed. To this end, we apply the classical probability theory and some measure-theoretic methods. We also analyze examples of applications of the proven theorems. Our results open new directions of development of the MV-algebraic probability theory. They can also be applied to the problem of entropy estimation.

Role of Information and its Processing in Statistical Analysis

This paper discusses how real-life statistical analysis/inference deviates from ideal environments. More specifically, there often exist models that have equal statistical power as the actual data-generating model, given only limited information and information processing/computation capacity. This means that misspecification actually has two problems: first with misspecification around the model we wish to find, and that an actual data-generating model may never be discovered. Thus the role information - this includes data - plays on statistical inference needs to be considered more heavily than often done. A game defining pseudo-equivalent models is presented in this light. This limited information nature effectively casts a statistical analyst as a decider in decision theory facing an identical problem: trying best to form credence/belief of some events, even if it may end up not being close to objective probability. The sleeping beauty problem is used as a study case to highlight some properties of real-life statistical inference. Bayesian inference of prior updates can lead to wrong credence analysis when prior is assigned to variables/events that are not (statistical identification-wise) identifiable. A controversial idea that Bayesianism can go around identification problems in frequentist analysis is brought to more doubts. This necessitates re-defining how Kolmogorov probability theory is applied in real-life statistical inference, and what concepts need to be fundamental.

Editorial: Quantum Structures in Cognitive and Social Science.

Editorial: Quantum Structures in Cognitive and Social Science.

Möbius operators and non-additive quantum probabilities in the Birkhoff–von Neumann lattice

The properties of quantum probabilities are linked to the geometry of quantum mechanics, described by the Birkhoff-von Neumann lattice. Quantum probabilities violate the additivity property of Kolmogorov probabilities, and they are interpreted as Dempster-Shafer probabilities. Deviations from the additivity property are quantified with the Mobius (or non-additivity) operators which are defined through Mobius transforms, and which are shown to be intimately related to commutators. The lack of distributivity in the Birkhoff-von Neumann lattice Lambda , causes deviations from the law of the total probability (which is central in Kolmogorov's probability theory). Projectors which quantify the lack of distributivity in Lambda , and also deviations from the law of the total probability, are introduced. All these operators, are observables and they can be measured experimentally. Constraints for the Mobius operators, which are based on the properties of the Birkhoff-von Neumann lattice (which in the case of finite quantum systems is a modular lattice), are derived.Application of this formalism in the context of coherent states, generalizes coherence to multi-dimensional structures.

Time-dependent Bell inequalities in a Wigner form

We derive a new class of time-dependent Bell inequalities in Wigner form under the assumption of locality in the framework of Kolmogorov's probability theory. We consider violation of the obtained inequalities for three cases: spin correlations in an external magnetic field; oscillations of neutral pseudoscalar mesons; and decays of a pseudoscalar into a fermion-antifermion pair.

Quantum probabilities as Dempster-Shafer probabilities in the lattice of subspaces

The orthocomplemented modular lattice of subspaces ${\cal L}[H(d)]$L[H(d)], of a quantum system with d-dimensional Hilbert space H(d), is considered. A generalized additivity relation which holds for Kolmogorov probabilities is violated by quantum probabilities in the full lattice ${\cal L}[H(d)]$L[H(d)] (it is only valid within the Boolean subalgebras of ${\cal L}[H(d)]$L[H(d)]). This suggests the use of more general (than Kolmogorov) probability theories, and here the Dempster-Shafer probability theory is adopted. An operator ${\mathfrak {D}}(H_1, H_2)$D(H1,H2), which quantifies deviations from Kolmogorov probability theory is introduced, and it is shown to be intimately related to the commutator of the projectors ${\mathfrak {P}}(H_1), {\mathfrak {P}}(H_2)$P(H1),P(H2), to the subspaces H1, H2. As an application, it is shown that the proof of the inequalities of Clauser, Horne, Shimony, and Holt for a system of two spin 1/2 particles is valid for Kolmogorov probabilities, but it is not valid for Dempster-Shafer probabilities. The violation of these inequalities in experiments supports the interpretation of quantum probabilities as Dempster-Shafer probabilities.

Realization of the probability laws in the quantum central limit theorems by a quantum walk

Since a limit distribution of a discrete-time quantum walk on the line was derived in 2002, a lot of limit theorems for quantum walks with a localized initial state have been reported. On the other hand, in quantum probability theory, there are four notions of independence (free, monotone, commuting, and boolean independence) and quantum central limit theorems associated to each independence have been investigated. The relation between quantum walks and quantum probability theory is still unknown. As random walks are fundamental models in the Kolmogorov probability theory, can the quantum walks play an important role in quantum probability theory? To discuss this problem, we focus on a discrete-time 2-state quantum walk with a non-localized initial state and present a limit theorem. By using our limit theorem, we generate probability laws in the quantum central limit theorems from the quantum walk.

The Bell phenomenon in classical frameworks

The experimentally tested occurrence of probabilities that do not find a representation within the usual Kolmogorov probability theory is mathematically formalized through the notion of the Bell phenomenon for probability measures and for observables. Reference is made to a physically natural definition of observables that includes both the classical and the quantum usual versions. It is shown that even a classical framework can host the Bell phenomenon, provided fuzzy observables are called into play.

What Price Substitutivity? A Note on Probability Theory

Teddy Seidenfeld recently claimed that Kolmogorov's probability theory transgresses the Substitutivity Law. Underscoring the seriousness of Seidenfeld's charge, the author shows that (Popper's version of) the law, to wit:If (∀D)(Pr(B,D) = Pr(C,D)), then Pr(A,B) = Pr(A,C),follows from justfive constraints on Pr of the most elementary and most basic sort.

Alternative axiomatizations of elementary probability theory.

1 Outline In this paper I offer some new axiomatizations of elementary (finite) probability theory. The new axiomatizations have the following distinctive features. First, both conditional and unconditional probabilities are simultaneously axiomatized. Second, the axiomatizations are extensions of Kolmogorov probability theory, and have the usual definitions linking conditional and unconditional probabilities as theorems. Third, while both probabilities are simultaneously axiomatized, the resultant axioms are about as simple as the usual axiomatizations of conditional probability alone. Fourth, the axioms adopted have strong and direct justifications independently of the interpretation of the probability function (axiomatization L however is only a permissible extension of intuitions). The reader who wishes to review the axiomatizations before their rationale can at this point go to section 5 (the main axiomatization is system KK). 2 What is unsatisfactor y about usual axiomatizations? I offer alternative axiomatizations because I am dissatisfied with the customary ones. But this dissatisfaction has a special nature. In improving existing axiomatizations, one might reduce the number of axioms, or simplify them, or adopt axioms that make proofs easier and more elegant. Although in my own axiomatizations I am concerned with these factors, none of these traditional formal motivations is the source of my dissatisfaction. Instead, I am basically concerned about how the relationship between conditional and unconditional probabilities is developed (for some related misgivings, see de Finetti [2], pp. 81-83). There are two probability functions about which we have intuitions. The basic formal difference is that unconditional probability is a one-place function, and conditional probability is a twoplace function. As a result, there are two probabilistic concepts that can be axiomatized, and the two can be axiomatized independently. What makes a function a one-place probability function, and what makes a function a two-place probability function, are separate questions, and the correct answer for each involves a separate characterizing theory.