Axioms Of Probability Research Articles

On the Irrationality of Being in Two Minds.

This article presents a general framework that allows irrational decision making to be theoretically investigated and simulated. Rationality in human decision making under uncertainty is normatively prescribed by the axioms of probability theory in order to maximize utility. However, substantial literature from psychology and cognitive science shows that human decisions regularly deviate from these axioms. Bistable probabilities are proposed as a principled and straight forward means for modeling (ir)rational decision making, which occurs when a decision maker is in “two minds”. We show that bistable probabilities can be formalized by positive-operator-valued projections in quantum mechanics. We found that (1) irrational decision making necessarily involves a wider spectrum of causal relationships than rational decision making, (2) the accessible information turns out to be greater in irrational decision making when compared to rational decision making, and (3) irrational decision making is quantum-like because it violates the Bell–Wigner polytope.

E.T. Jaynes’s Solution to the Problem of Countable Additivity

Philosophers cannot agree on whether the rule of Countable Additivity should be an axiom of probability. Edwin T. Jaynes attacks the problem in a way which is original to him and passed over in the current debate about the principle: he says the debate only arises because of an erroneous use of mathematical infinity. I argue that this solution fails, but I construct a different argument which, I argue, salvages the spirit of the more general point Jaynes makes. I argue that in Jaynes’s objective Bayesianism we might have good reasons to adopt Countable Additivity, and some of the major problems this adoption is known to entail need not worry us. In particular, I propose to adopt this new angle on Countable Additivity in Jon Williamson’s version of objective Bayesianism.

Some properties of <i>η</i>-convex stochastic processes

<abstract> <p>The stochastic processes is a significant branch of probability theory, treating probabilistic models that develop in time. It is a part of mathematics, beginning with the axioms of probability and containing a rich and captivating arrangement of results following from those axioms. In probability, a convex function applied to the expected value of an random variable is always bounded above by the expected value of the convex function of the random variable. The definition of <italic>η</italic>-convex stochastic process is introduced in this paper. Moreover some basic properties of <italic>η</italic>-convex stochastic process are derived. We also derived Jensen, Hermite-Hadamard and Ostrowski type inequalities for <italic>η</italic>-convex stochastic process.</p> </abstract>

The logic of qualitative probability

In this paper we present a theory of qualitative probability. The usual approach of earlier work was to specify a binary operator ⪯ on formulas with ϕ⪯ψ having the intended interpretation that the event expressed by ϕ is no more probable than that expressed by ψ. We generalise these approaches by extending the domain of the operator ⪯ from the set of events to the set of finite sequences of events. If Φ and Ψ are finite sequences of events, Φ⪯Ψ has the intended interpretation that the combined probabilities of the elements of Φ are no greater than those of Ψ. A sound and complete axiomatisation for this operator over finite outcome sets is given. We argue that our approach is more perspicuous and intuitive than previous accounts. As well, we show that the approach is sufficiently expressive to capture the results of axiomatic probability theory and to encode rational linear inequalities. We also prove that our approach generalises the two major accounts for finite outcome sets.

Popper's Axiomatic Probability System and the Value-Assignment Problem

In the standard theory of probability, developed by Kolmogorov, the concept of conditional probability is defined with what is known as the ratio formula: the probability of A given B is the ratio between the probability of A and B and the probability of B, i.e. P(A|B)= (P(AB))/(P(B)). Clearly, this ratio is not defined when the probability of the condition, P(B), is 0. According to Popper, this problem, which is known as the zero-denominator problem, shows a serious conceptual shortcoming of the standard Kolmogorovian theory of probability. In order to overcome this shortcoming, Popper developed an alternative axiomatic theory of probability where conditional probability is taken as primitive. It should be noted that this axiomatic probability theory is different than and independent from Popper’s philosophy of probability which is based on the propensity approach. Popper claims that his axiomatic theory is a better fit for the use of probability in the philosophy of science and statistics. Based on this claim, it is often stated that Popper’s theory is conceptually superior to Kolmogorov’s theory. The ultimate aim of this paper is to evaluate this claim by analyzing Popper’s axiomatic theory within the context of the zero-denominator problem.

The joint aggregation of beliefs and degrees of belief

The article proceeds upon the assumption that the beliefs and degrees of belief of rational agents satisfy a number of constraints, including: (1) consistency and deductive closure for belief sets, (2) conformity to the axioms of probability for degrees of belief, and (3) the Lockean Thesis concerning the relationship between belief and degree of belief. Assuming that the beliefs and degrees of belief of both individuals and collectives satisfy the preceding three constraints, I discuss what further constraints may be imposed on the aggregation of beliefs and degrees of belief. Some possibility and impossibility results are presented. The possibility results suggest that the three proposed rationality constraints are compatible with reasonable aggregation procedures for belief and degree of belief.

Hugo Dionizy Steinhaus – droga do współczesnej teorii prawdopodobieństwa

Hugo Steinhaus (1887–1972) ukończył studia matematyczne i filozoficzne na Uniwersytecie Lwowskim. W latach 1905–1911 przebywał w Getyndze, pracując nad doktoratem pod opieką Davida Hilberta. W 1920 roku został profesorem Uniwersytetu Jana Kazimierza we Lwowie. Skupione wokół niego i Stefana Banacha grono wybitnych matematyków tworzyło silny ośrodek matematyczny specjalizujący się w analizie funkcjonalnej. Po II wojnie światowej osiedlił się we Wrocławiu, gdzie współtworzył matematyczne środowisko naukowe, a następnie wrocławską szkołę zastosowań matematyki. Jest autorem i współautorem ponad 250 prac naukowych i publikacji popularyzujących matematykę. W 1923 roku H. Steinhaus opublikował w czasopiśmie „Fundamenta Mathematicae” wyniki zawierające aksjomatyczny opis pewnej miary prawdopodobieństwa określonej na podzbiorach przestrzeni nieskończonych ciągów zero‑jedynkowych. Celem niniejszego artykułu jest próba ukazania istotnej roli, jaką wyniki te odegrały w procesie aksjomatyzacji prawdopodobieństwa, zakończonej w 1933 roku publikacją Andrieja Kołmogorowa.

The Quantum Mind: Alternative Ways of Reasoning with Uncertainty

Human reasoning about and with uncertainty is often at odds with the principles of classical probability. Order effects, conjunction biases, and sure-thing inclinations suggest that an entirely different set of probability axioms could be developed and indeed may be needed to describe such habits. Recent work in diverse fields, including cognitive science, economics, and information theory, explores alternative approaches to decision theory. This work considers more expansive theories of reasoning with uncertainty while continuing to recognize the value of classical probability. In this paper, we discuss one such alternative approach, called quantum probability, and explore its applications within decision theory. Quantum probability is designed to formalize uncertainty as an ontological feature of the state of affairs, offering a mathematical model for entanglement, de/coherence, and interference, which are all concepts with unique onto-epistemological relevance for social theorists working in new and trans-materialisms. In this paper, we suggest that this work be considered part of the quantum turn in the social sciences and humanities. Our aim is to explore different models and formalizations of decision theory that attend to the situatedness of judgment. We suggest that the alternative models of reasoning explored in this article might be better suited to queries about entangled mathematical concepts and, thus, be helpful in rethinking both curriculum and learning theory.

Recursive Neural Networks for Density Estimation Over Generalized Random Graphs.

Structured data in the form of labeled graphs (with variable order and topology) may be thought of as the outcomes of a random graph (RG) generating process characterized by an underlying probabilistic law. This paper formalizes the notions of generalized RG (GRG) and probability density function (pdf) for GRGs. Thence, a "universal" learning machine (combining the encoding module of a recursive neural network and a radial basis functions' network) is introduced for estimating the unknown pdf from an unsupervised sample of GRGs. A maximum likelihood training algorithm is presented and constrained so as to ensure that the resulting model satisfies the axioms of probability. Techniques for preventing the model from degenerate solutions are proposed, as well as variants of the algorithm suitable to the tasks of graphs classification and graphs clustering. The major properties of the machine are discussed. The approach is validated empirically through experimental investigations in the estimation of pdfs for synthetic and real-life GRGs, in the classification of images from the Caltech Benchmark data set and molecules from the Mutagenesis data set, and in clustering of images from the LabelMe data set.

Multivariate normal maximum likelihood with both ordinal and continuous variables, and data missing at random.

A novel method for the maximum likelihood estimation of structural equation models (SEM) with both ordinal and continuous indicators is introduced using a flexible multivariate probit model for the ordinal indicators. A full information approach ensures unbiased estimates for data missing at random. Exceeding the capability of prior methods, up to 13 ordinal variables can be included before integration time increases beyond 1 s per row. The method relies on the axiom of conditional probability to split apart the distribution of continuous and ordinal variables. Due to the symmetry of the axiom, two similar methods are available. A simulation study provides evidence that the two similar approaches offer equal accuracy. A further simulation is used to develop a heuristic to automatically select the most computationally efficient approach. Joint ordinal continuous SEM is implemented in OpenMx, free and open-source software.

The paradigm of complex probability and Ludwig Boltzmann's entropy

ABSTRACTThe Andrey Nikolaevich Kolmogorov's classical system of probability axioms can be extended to encompass the imaginary set of numbers and this by adding to his original five axioms an additional three axioms. Hence, any experiment can thus be executed in what is now the complex probability set 𝒞 which is the sum of the real set ℛ with its corresponding real probability and the imaginary set ℳ with its corresponding imaginary probability. The objective here is to evaluate the complex probabilities by considering supplementary new imaginary dimensions to the event occurring in the ‘real’ laboratory. Whatever the probability distribution of the input random variable in ℛ is, the corresponding probability in the whole set 𝒞 is always one, so the outcome of the random experiment in 𝒞 can be predicted totally and perfectly. The result indicates that chance and luck in ℛ is replaced now by total determinism in 𝒞 . This is the consequence of the fact that the probability in 𝒞 is got by subtracting the chaotic factor from the degree of our knowledge of the stochastic system. This novel complex probability paradigm will be applied to Ludwig Boltzmann's classical concept of entropy in thermodynamics and in statistical mechanics.

Soft-Constrained Neural Networks for Nonparametric Density Estimation

The paper introduces a robust connectionist technique for the empirical nonparametric estimation of multivariate probability density functions (pdf) from unlabeled data samples (still an open issue in pattern recognition and machine learning). To this end, a soft-constrained unsupervised algorithm for training a multilayer perceptron (MLP) is proposed. A variant of the Metropolis–Hastings algorithm (exploiting the very probabilistic nature of the present MLP) is used to guarantee a model that satisfies numerically Kolmogorov’s second axiom of probability. The approach overcomes the major limitations of the established statistical and connectionist pdf estimators. Graphical and quantitative experimental results show that the proposed technique can offer estimates that improve significantly over parametric and nonparametric approaches, regardless of (1) the complexity of the underlying pdf, (2) the dimensionality of the feature space, and (3) the amount of data available for training.

Impossibility Results for Rational Belief

AbstractThere are two ways of representing rational belief: qualitatively as yes‐or‐no belief, and quantitatively as degrees of belief. Standard rationality conditions are: (i) consistency and logical closure, for qualitative belief, (ii) satisfaction of the probability axioms, for quantitative belief, and (iii) a relationship between qualitative and quantitative beliefs in accordance with the Lockean thesis. In this paper, it is shown that these conditions are inconsistent with each of three further rationality conditions: fallibilism, open‐mindedness, and invariance under independent conceptual expansions. Restrictions of the Lockean thesis that have been suggested in the literature cannot remove the inconsistency. In the outlook we discuss two alternative ways of dealing with this problem: restricting conjunctive closure or going for a dual system account.

How generalizable is good judgment? A multi-task, multi-benchmark study

AbstractGood judgment is often gauged against two gold standards – coherence and correspondence. Judgments are coherent if they demonstrate consistency with the axioms of probability theory or propositional logic. Judgments are correspondent if they agree with ground truth. When gold standards are unavailable, silver standards such as consistency and discrimination can be used to evaluate judgment quality. Individuals are consistent if they assign similar judgments to comparable stimuli, and they discriminate if they assign different judgments to dissimilar stimuli. We ask whether “superforecasters”, individuals with noteworthy correspondence skills (see Mellers et al., 2014) show superior performance on laboratory tasks assessing other standards of good judgment. Results showed that superforecasters either tied or out-performed less correspondent forecasters and undergraduates with no forecasting experience on tests of consistency, discrimination, and coherence. While multifaceted, good judgment may be a more unified than concept than previously thought.

Financial and risk modelling with semicontinuous covariances

We are encountering deep financial crisis. One of the basic problems we face is how to predict financial developments. Current modelling is clearly insufficient since continuity of autocorrelation oversmoothes the empirical financial data. We prove that geometric probability to chose positive definite continuous covariance by financial data is zero. We also have the empirical evidence for this fact from real stock data. In order to overcome this gap we introduce the novel abc class of semicontinuous covariance functions, which integrates both continuous and discontinuous covariances. The abc class of covariances is much more flexible, powerful, and realistic than its continuous counterpart. It gives answer to the question why financial markets face negative interest rates, cyclic downbreaks, and other difficulties nowadays. As it is clearly illustrated in this paper, the experimenter (e.g. the financial institution) can choose only up to some extent whether abc class member will be positive definite everywhere. To the best knowledge of the authors this fits well to the own Kolmogorov’s opinions on axiomatic theory of probability. The Kolmogorov’s axiomatic theory of probability was developed as one possible alternative in order to utilize the flexibility of Lebesgue integral in probability theory, but in many experiments e.g. in physics and finance, one should go beyond this axiomatics. That means one should accept negative probabilities (and consequently negative definite covariance functions). In this paper we clarify that the modeler’s choice of positive/negative definite functions should be considered within broader Information Theory setup. The introduced abc class is purely topologically defined, with soft regularity conditions on covariances, which are still applicable for increasing and infill domain asymptotics for regression problems, kriging, and finance. These conditions are related to semicontinuous maps of Ornstein Uhlenbeck (OU) processes. We provide several novel results for optimal design of random fields with abc covariances, random walks in finance, and probabilities of ruins related to shocks (e.g. by earthquakes). In particular, we construct a random walk model with semicontinuous covariance. The novel abc class provides a proper environment for unifying several contradictions on IS/ML model in the economic literature. All mentioned novel economical applications are hardly visible from currently used “continuous covariance” framework.

On Noncontextual, Non-Kolmogorovian Hidden Variable Theories

One implication of Bell's theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig (2015) to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., a finite collection of probability zero events whose disjunction exhausts the space of experimental possibilities.

Strategy Constrained by Cognitive Limits, and the Rationality of Belief-Revision Policies

Strategy is formally defined as a complete plan of action for every contingency in a game. Ideal agents can evaluate every contingency. But real people cannot do so, and require a belief-revision policy to guide their choices in unforeseen contingencies. The objects of belief-revision policies are beliefs, not strategies and acts. Thus, the rationality of belief-revision policies is subject to Bayesian epistemology. The components of any belief-revision policy are credences constrained by the probability axioms, by conditionalization, and by the principles of indifference and of regularity. The principle of indifference states that an agent updates his credences proportionally to the evidence, and no more. The principle of regularity states that an agent assigns contingent propositions a positive (but uncertain) credence. The result is rational constraints on real people’s credences that account for their uncertainty. Nonetheless, there is the open problem of non-evidential components that affect people’s credence distributions, despite the rational constraint on those credences. One non-evidential component is people’s temperaments, which affect people’s evaluation of evidence. The result is there might not be a proper recommendation of a strategy profile for a game (in terms of a solution concept), despite agents’ beliefs and corresponding acts being rational.

What Is ‘Real’ in Probabilism?

ABSTRACTThis paper defends two related claims about belief: first, the claim that, unlike numerical degrees of belief, comparative beliefs are primitive and psychologically real; and, second, the claim that the fundamental norm of Probabilism is not that numerical degrees of belief should satisfy the probability axioms, but rather that comparative beliefs should satisfy certain constraints.

Pragmatic Considerations on Comparative Probability

While pragmatic arguments for numerical probability axioms have received much attention, justifications for axioms of qualitative probability have been less discussed. We offer an argument for the requirement that an agent’s qualitative (comparative) judgments be probabilistically representable, inspired by, but importantly different from, the Money Pump argument for transitivity of preference and Dutch book arguments for quantitative coherence. The argument is supported by a theorem, to the effect that a subject is systematically susceptible to dominance given her preferred acts, if and only if the subject’s comparative judgments preclude representation by a standard probability measure (or set of such measures).

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.