Axioms Of Probability Research Articles

A probability space of continuous-time discrete value stochastic process with Markov property

Getting acquainted with the theory of stochastic processes we can read the following statement: "In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it". The classical results for limited stochastic and intensity matrices goes back to Kolmogorov at least late 40-s. But for some infinity matrices the sum of probabilities of all trajectories is less than 1. Some years ago I constructed physical models of simulation of any stochastic processes having a stochastic or an intensity matrices and I programmed it. But for computers I had to do some limitations - set of states at present time had to be limited, at next time - not necessarily. If during simulation a realisation accepted a state out of the set of limited states - the simulation was interrupted. I saw that I used non-quadratic, half-infinity stochastic and intensity matrices and that the set of trajectories was bigger than for quadratic ones. My programs worked good also for stochastic processes described in literature as without probability space. I asked myself: did the probability space for these experiments not exist or were only set of events incompleted? This paper shows that the second hipothesis is true.

A geometric principle of indifference

That one's degrees of belief at any one time obey the axioms of probability theory is widely regarded as a necessary condition for static rationality. Many theorists hold that it is also a sufficient condition, but according to critics this yields too subjective an account of static rationality. However, there are currently no good proposals as to how to obtain a tenable stronger probabilistic theory of static rationality. In particular, the idea that one might achieve the desired strengthening by adding some symmetry principle to the probability axioms has appeared hard to maintain. Starting from an idea of Carnap and drawing on relatively recent work in cognitive science, this paper argues that conceptual spaces provide the tools to devise an objective probabilistic account of static rationality. Specifically, we propose a principle that derives prior degrees of belief from the geometrical structure of concepts.

The probability of simple versus complex causal models in causal analyses.

Complex causal models, accompanied by causal analyses based on large correlation matrices, are more common in the social sciences than are simple causal models accompanied by a single correlation coefficient. The increased complexity of the former, relative to the latter, seems to carry with it an augmented scientific respectability or credibility. In contrast, the axioms of probability suggest an argument in the opposite direction.

The paradigm of complex probability and Chebyshev’s inequality

ABSTRACTThe system of axioms for probability theory laid in 1933 by Andrey Nikolaevich Kolmogorov can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Therefore, we create the complex probability set , which is the sum of the real set with its corresponding real probability, and the imaginary set with its corresponding imaginary probability. Hence, all stochastic experiments are performed now in the complex set instead of the real set . The objective is then to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Consequently, the corresponding probability in the whole set is always equal to one and the outcome of the random experiments that follow any probability distribution in is now predicted totally in . Subsequently, it follows that, chance and luck in is replaced by total determinism in . Consequently, by subtracting the chaotic factor from the degree of our knowledge of the stochastic system, we evaluate the probability of any random phenomenon in . This novel complex probability paradigm will be applied to the established theorem of Pafnuty Chebyshev's inequality and to extend the concepts of expectation and variance to the complex probability set .

The paradigm of complex probability and analytic nonlinear prognostic for vehicle suspension systems

ABSTRACTAdding three novel axioms to Kolmogorov's five established probability axioms which were laid in 1933 extends classical probability theory to encompass the imaginary set of numbers. Therefore, all random experiments will be executed in the complex set which is the sum of the real set with its real probability component, and the corresponding imaginary set with its imaginary probability component. The purpose of extending Kolmogorov's axioms is to add supplementary imaginary dimensions to any random event that occurs in the ‘real’ laboratory and hence to be able to evaluate the associated complex probability in the complex plane which is always equal to one. My purpose in this work is to link the complex probability paradigm to the vehicle suspension system analytic prognostic in the nonlinear damage accumulation case. Hence, by calculating the parameters of the new prognostic model, we will be able to determine the magnitude of the chaotic factor, the degree of our knowledge, the complex probability, the system failure and survival probabilities, and the remaining useful lifetime probability, after that N load cycles have been applied to the suspension, and which are all functions of the system degradation subject to random effects.

The Event: An Underexamined Risk Concept.

Some of the terms used in risk assessment and management are poorly and even contradictorily defined. One such term is "event," which arguably describes the most basic of all risk-related concepts. The author cites two contemporary textbook interpretations of "event" that he contends are incorrect and misleading. He then examines the concept of an event in A. N. Kolmogorov's probability axioms and in several more-current textbooks. Those concepts are found to be too narrow for risk assessments and inconsistent with the actual usage of "event" by risk analysts. The author goes on to define and advocate linguistic definitions of events (as opposed to mathematical definitions)-definitions constructed from natural language. He argues that they should be recognized for what they are: the de facto primary method of defining events.

A cumulative failure probability model for cleavage fracture in ferritic steels

The fundamental physical assumption of plastic deformation as a prerequisite to the cleavage fracture event demands to rule out the contribution of elastic stress components to the occurrence of fracture from both the pure elastic zone and the elastoplastic zone. It also commands the threshold stress for cleavage fracture to be no less than the yielding stress. Based on this understanding, a model of the cumulative failure probability for cleavage fracture in ferritic steels is developed. Further, a self-consistent statistical model of cleavage fracture toughness is also derived. The Beremin model does not conform to the normality axiom of probability, as it defies the premise of plastic yielding before cleavage fracture by counting in the elastic stress components inside the plastic zone to calculate the cumulative probability. Examples are given in several homogeneous uniaxial and multiaxial loading conditions to highlight the differences between the Beremin model and the presented model in probability calculation, model calibration, and interpretation of the temperature and constraint dependence of microscopic cleavage fracture stress.

A non-probabilist principle of higher-order reasoning

The author uses a series of examples to illustrate two versions of a new, nonprobabilist principle of epistemic rationality, the special and general versions of the metacognitive, expected relative frequency (MERF) principle. These are used to explain the rationality of revisions to an agent’s degrees of confidence in propositions based on evidence of the reliability or unreliability of the cognitive processes responsible for them—especially reductions in confidence assignments to propositions antecedently regarded as certain—including certainty-reductions to instances of the law of excluded middle or the law of noncontradiction in logic or certainty-reductions to the certainties of probabilist epistemology. The author proposes special and general versions of the MERF principle and uses them to explain the examples, including the reasoning that would lead to thoroughgoing fallibilism—that is, to a state of being certain of nothing (not even the MERF principle itself). The author responds to the main defenses of probabilism: Dutch Book arguments, Joyce’s potential accuracy defense, and the potential calibration defenses of Shimony and van Fraassen by showing that, even though they do not satisfy the probability axioms, degrees of belief that satisfy the MERF principle minimize expected inaccuracy in Joyce’s sense; they can be externally calibrated in Shimony and van Fraassen’s sense; and they can serve as a basis for rational betting, unlike probabilist degrees of belief, which, in many cases, human beings have no rational way of ascertaining. The author also uses the MERF principle to subsume the various epistemic akrasia principles in the literature. Finally, the author responds to Titelbaum’s argument that epistemic akrasia principles require that we be certain of some epistemological beliefs, if we are rational.

Investigation of properties of time-dependent bell inequalities in Wigner’s form for nonstationary and open quantum systems

A radically new class of Bell inequalities in Wigner’s form was obtained on the basis of Kolmorov’s axiomatization of probability theory and the hypothesis of locality. These inequalities take explicitly into account the dependence on time (time-dependent Bell inequalities in Wigner’s form). By using these inequalities, one can propose a means for experimentally testing Bohr’ complementarity principle in the relativistic region. The inequalities in question open broad possibilities for studying correlations of nonrelativistic and relativistic quantum systems in external fields. The violation of the time-dependent inequalities in quantum mechanics was studied by considering the behavior of a pair of anticorrelated spins in a constant external magnetic field and oscillations of neutral pseudoscalar mesons. The decay of a pseudoscalar particle to a fermion–antifermion pair is considered within quantum field theory. In order to test experimentally the inequalities proposed in the present study, it is not necessary to perform dedicated noninvasive measurements required in the Leggett–Garg approach, for example.

The Propensity Interpretation of Probability: A Re-evaluation

Single-case and long-run propensity theories are among the main objective interpretations of probability. There have been various objections to these theories, e.g. that it is difficult to explain why propensities should satisfy the probability axioms and, worse, that propensities are at odds with these axioms, that the explication of propensities is circular and accordingly not informative, and that single-case propensities are metaphysical and accordingly non-scientific. We consider various propensity theories of probability and their prospects in light of these objections. We argue that while propensity theories face challenges, these challenges do not undermine their validity as prospective interpretations of probability in science.

Hamilton–Jacobi meet Möbius

Adaptation of the Hamilton–Jacobi formalism to quantum mechanics leads to a cocycle condition, which is invariant under D-dimensional Mobius transformations with Euclidean or Minkowski metrics. In this paper we aim to provide a pedagogical presentation of the proof of the Möbius symmetry underlying the cocycle condition. The Möbius symmetry implies energy quantization and undefinability of quantum trajectories, without assigning any prior interpretation to the wave function. As such, the Hamilton–Jacobi formalism, augmented with the global Möbius symmetry, provides an alternative starting point, to the axiomatic probability interpretation of the wave function, for the formulation of quantum mechanics and the quantum spacetime. The Möbius symmetry can only be implemented consistently if spatial space is compact, and correspondingly if there exist a finite ultraviolet length scale. Evidence for nontrivial space topology may exist in the cosmic microwave background radiation.

A note on cancellation axioms for comparative probability

We prove that the generalized cancellation axiom for incomplete comparative probability relations introduced by Ríos Insua (Theory Decis 33:83–100, 1992) and Alon and Lehrer (J Econ Theory 151:476–492, 2014) is stronger than the standard cancellation axiom for complete comparative probability relations introduced by Scott (J Math Psychol 1:233–247, 1964), relative to their other axioms for comparative probability in both the finite and infinite cases. This result has been suggested but not proved in the previous literature.

Measuring the overall incoherence of credence functions

Many philosophers hold that the probability axioms constitute norms of rationality governing degrees of belief. This view, known as subjective Bayesianism, has been widely criticized for being too idealized. It is claimed that the norms on degrees of belief postulated by subjective Bayesianism cannot be followed by human agents, and hence have no normative force for beings like us. This problem is especially pressing since the standard framework of subjective Bayesianism only allows us to distinguish between two kinds of credence functions—coherent ones that obey the probability axioms perfectly, and incoherent ones that don’t. An attractive response to this problem is to extend the framework of subjective Bayesianism in such a way that we can measure differences between incoherent credence functions. This lets us explain how the Bayesian ideals can be approximated by humans. I argue that we should look for a measure that captures what I call the ‘overall degree of incoherence’ of a credence function. I then examine various incoherence measures that have been proposed in the literature, and evaluate whether they are suitable for measuring overall incoherence. The competitors are a qualitative measure that relies on finding coherent subsets of incoherent credence functions, a class of quantitative measures that measure incoherence in terms of normalized Dutch book loss, and a class of distance measures that determine the distance to the closest coherent credence function. I argue that one particular Dutch book measure and a corresponding distance measure are particularly well suited for capturing the overall degree of incoherence of a credence function.

Reflection and Self-Trust

The Reflection principle expresses a kind of epistemic deference to one’s future self. There is a plausible intuitive argument to the effect that, if one believes one will reason well and gain information over time, then one ought to satisfy Reflection. There are also associated formal arguments that show that, if one’s beliefs about one’s current and future selves meet certain criteria, then one is committed by the axioms of probability to satisfy Reflection. The formal arguments, however, rely on an assumption that has no apparent relevance to the intuitive argument: the assumption that one has perfect access to one’s beliefs. This paper explains why it is advantageous to replace the perfect access assumption with a self-trust assumption. The self-trust assumption is superior from a formal point of view, since it is formally weaker than the perfect access assumption, and from an intuitive point of view, since it leads to an improved formulation of the intuitive argument.

What probability probably isn't

Joyce and others have claimed that degrees of belief are estimates of truth-values and that the probability axioms are conditions of admissibility for these estimates with respect to a scoring rule penalizing inaccuracy. In this article, I argue that the claim that the rules of probability are truth-directed in this way depends on an assumption that is both implausible and lacks any supporting evidence, strongly suggesting that the probability axioms have nothing intrinsically to do with truth-directedness.

Dutch Books, Coherence, and Logical Consistency

AbstractIn this paper I present a new way of understanding Dutch Book Arguments: the idea is that an agent is shown to be incoherent iff (s)he would accept as fair a set of bets that would result in a loss under any interpretation of the claims involved. This draws on a standard definition of logical inconsistency. On this new understanding, the Dutch Book Arguments for the probability axioms go through, but the Dutch Book Argument for Reflection fails. The question of whether we have a Dutch Book Argument for Conditionalization is left open.

Safeguards of a Disunified Mind

The papers focuses on pragmatic arguments for various rationality constraints on a decision maker’s state of mind: on her beliefs or preferences. An argument of this kind typically targets constraint violations. It purports to show that a violator of a given constraint can be confronted with a decision problem in which she will act to her guaranteed disadvantage. Dramatically put, she can be exploited by a clever bookie who doesn’t know more than the agent herself. Examples of pragmatic arguments of this kind are synchronic Dutch Books, for the standard probability axioms, diachronic Dutch Books, for the more controversial principles of reflection and conditionalization, and Money Pumps, for the acyclicity requirement on preferences. The paper suggests that the proposed exploitation set-ups share a common feature. If the violator of a given constraint is logically and mathematically competent, and if she prefers to be better off rather than worse off, she can be exploited only if she is disunified in her decision-making, i.e. only if she makes decisions on various issues she faces separately rather than jointly. Unification in decision making is relatively unproblematic in synchronic contexts, but it may be costly and inconvenient diachronically. On this view, therefore, pragmatic arguments should be seen as delivering conditional recommendations: If you want to afford disunification, then you’d better satisfy these constraints. They identify safeguards of a disunified mind. Isaac Levi’s position on these matters is diametrically different. According to Levi, only synchronic pragmatic arguments are valid (indeed, categorically so). The diachronic ones, he argues, lack any validity at all. This line of reasoning is questioned in the paper.

The Stability Theory of Belief

This essay develops a joint theory of rational (all-or-nothing) belief and degrees of belief. The theory is based on three assumptions: the logical closure of rational belief; the axioms of probability for rational degrees of belief; and the so-called Lockean thesis, in which the concepts of rational belief and rational degree of belief figure simultaneously. In spite of what is commonly believed, this essay will show that this combination of principles is satisfiable (and indeed nontrivially so) and that the principles are jointly satisfied if and only if rational belief is equivalent to the assignment of a stably high rational degree of belief. Although the logical closure of belief and the Lockean thesis are attractive postulates in themselves, initially this may seem like a formal “curiosity”; however, as will be argued in the rest of the essay, a very reasonable theory of rational belief can be built around these principles that is not ad hoc and that has various philosophical features that are plausible independently. In particular, this essay shows that the theory allows for a solution to the Lottery Paradox, and it has nice applications to formal epistemology. The price that is to be paid for this theory is a strong dependency of belief on the context, where a context involves both the agent's degree of belief function and the partitioning or individuation of the underlying possibilities. But as this essay argues, that price seems to be affordable. This essay develops a joint theory of rational (all-or-nothing) belief and degrees of belief. The theory is based on three assumptions: the logical closure of rational belief; the axioms of probability for rational degrees of belief; and the so-called Lockean thesis, in which the concepts of rational belief and rational degree of belief figure simultaneously. In spite of what is commonly believed, I will show that this combination of principles is satisfiable (and indeed nontrivially so) and that the principles are jointly satisfied if and only if rational belief is equivalent to the assignment of a stably high rational degree of belief. Although the logical closure of belief and the Lockean thesis are attractive postulates in themselves, initially this may seem like a formal “curiosity”; however, as I am going to argue in the rest of the essay, a very reasonable theory of rational belief can be built around these principles that is not ad hoc but that has various philosophical features that are plausible independently.

A personal history of Bayesian statistics

The history of Bayesian statistics is traced, from a personal perspective, through various strands and via its re‐genesis during the 1960s to the current day. Emphasis is placed on broad‐sense Bayesian methodology that can be used to meaningfully analyze observed datasets. Over 750 people in science, medicine, and socioeconomics, who have influenced the evolution of the Bayesian approach into the powerful paradigm that it is today, are highlighted. The frequentist/Bayesian controversy is addressed, together with the ways in which many Bayesians combine the two ideologies as a Bayes/non‐Bayes compromise, e.g., when drawing inferences about unknown parameters or when investigating the choice of sampling model in relation to its real‐life background. A number of fundamental issues are discussed and critically examined, and some elementary explanations for nontechnical readers and some personal reminiscences are included. Some of the Bayesian contributions of the 21st century are subjected to more detailed critique, so that readers may learn more about the quality and relevance of the ongoing research. A recent resolution of Lindley's paradox by Baskurt and Evans is reported. The axioms of subjective probability are reassessed, some state‐of‐the‐art alternatives to Leonard Savage's axioms of utility are discussed, and Deborah Mayo and Michael Evan's refutation of Allan Birnbaum's 1962 justification of the likelihood principle in terms of the sufficiency and conditionality principles is addressed.WIREs Comput Stat2014, 6:80–115. doi: 10.1002/wics.1293This article is categorized under:Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory

The main two arguments for probabilism are flawed

Probabilism, the view that agents have numerical degrees of beliefs that conform to the axioms of probability, has been defended by the vast majority of its proponents by way of either of two arguments, the Dutch Book Argument and the Representation Theorems Argument. In this paper I argue that both arguments are flawed. The Dutch Book Argument is based on an unwarranted, ad hoc premise that cannot be dispensed with. The Representation Theorems Argument hinges on an invalid implication.