Coherent Previsions Research Articles

Systems of Precision: Coherent Probabilities on Pre-Dynkin Systems and Coherent Previsions on Linear Subspaces.

In the literature on imprecise probability, little attention is paid to the fact that imprecise probabilities are precise on a set of events. We call these sets systems of precision. We show that, under mild assumptions, the system of precision of a lower and upper probability form a so-called (pre-)Dynkin system. Interestingly, there are several settings, ranging from machine learning on partial data over frequential probability theory to quantum probability theory and decision making under uncertainty, in which, a priori, the probabilities are only desired to be precise on a specific underlying set system. Here, (pre-)Dynkin systems have been adopted as systems of precision, too. We show that, under extendability conditions, those pre-Dynkin systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum probability are equivalent to coherence from the imprecise probability literature. On this basis, we spell out a lattice duality which relates systems of precision to credal sets of probabilities. We conclude the presentation with a generalization of the framework to expectation-type counterparts of imprecise probabilities. The analogue of pre-Dynkin systems turns out to be (sets of) linear subspaces in the space of bounded, real-valued functions. We introduce partial expectations, natural generalizations of probabilities defined on pre-Dynkin systems. Again, coherence and extendability are equivalent. A related but more general lattice duality preserves the relation between systems of precision and credal sets of probabilities.

The consumer’s demand functions defined to study contingent consumption plans

Given two probability distributions expressing returns on two single risky assets of a portfolio, we innovatively define two consumer’s demand functions connected with two contingent consumption plans. This thing is possible whenever we coherently summarize every probability distribution being chosen by the consumer. Since prevision choices are consumption choices being made by the consumer inside of a metric space, we show that prevision choices can be studied by means of the standard economic model of consumer behavior. Such a model implies that we consider all coherent previsions of a joint distribution. They are decomposed inside of a metric space. Such a space coincides with the consumer’s consumption space. In this paper, we do not consider a joint distribution only. It follows that we innovatively define a stand-alone and double risky asset. Different summary measures of it characterizing consumption choices being made by the consumer can then be studied inside of a linear space over {mathbb {R}}. We show that it is possible to obtain different summary measures of probability distributions by using two different quadratic metrics. In this paper, our results are based on a particular approach to the origin of the variability of probability distributions. We realize that it is not standardized, but it always depends on the state of information and knowledge of the consumer.

Coherent lower and upper conditional previsions defined by Hausdorff inner and outer measures to represent the role of conscious and unconscious thought in human decision making

The model of coherent lower and upper conditional previsions, based on Hausdorff inner and outer measures, is proposed to represent the preference orderings and the equivalences, respectively assigned by the conscious and unconscious thought in human decision making under uncertainty. Complexity of partial information is represented by the Hausdorff dimension of the conditioning event. When the events, that describe the decision problem, are measurable is represented to the s-dimensional Hausdorff outer measure, where s is the Hausdorff dimension of the conditioning event, an optimal decision can be reached. The model is applied and discussed in Linda’s Problem and the conjunction fallacy is resolved.

Disintegration property of coherent upper conditional previsions with respect to Hausdorff outer measures for unbounded random variables

In a metric space , coherent upper conditional previsions are defined by Hausdorff outer measures on the linear space of all absolutely Choquet integrable random variables. They are proven to satisfy the disintegration property on every non-null partition if Ω is a set with positive and finite Hausdorff outer measure in its Hausdorff dimension. Coherent upper conditional probabilities are obtained as restrictions to the indicators function and it is proven they can be extended as finitely additive probabilities in the sense of Dubins. Examples are given in the discrete metric space and in the Euclidean metric space.

A Quadratic and Linear Metric Characterizing the Sampling Design with Fixed Sample Size Considered From a Geometric Viewpoint

The first-order and second-order inclusion probabilities are chosen by the statistician. They are subjective probabilities. We innovatively define univariate and bivariate random quantities whose logically possible values are samples of a given size in order to obtain the first-order and second-order inclusion probabilities by means of their coherent previsions. We consider linear maps connected with univariate random quantities as well as bilinear maps connected with bivariate random quantities. The covariance of two univariate random quantities that are the components of a bivariate random quantity has been expressed by means of two bilinear maps. We show that a univariate random quantity denoted by S is complementary to the univariate Horvitz-Thompson estimator. We identify a quadratic and linear metric with regard to two univariate random quantities representing deviations that we innovatively define. We use the α-criterion of concordance introduced byGini in order to identify it. It is a statistical criterion that we innovatively apply to probability.

An Essential Analogy Between Coherent Previsions of Random Gains and Well-Behaved Preferences

We deal with a unified approach to an integrated and simplified formulation of the decision-making theory in its two subjective components, probability and utility. We show a choice model based on an application of fundamental microeconomic principles to the two-dimensional convex set of all coherent previsions of two random gains. Such a model is well-founded because we find out an analogy between properties of well-behaved preferences and the ones of coherent previsions of random gains. Coherence properties of the notion of price or prevision of a random gain are based on economic criteria of the decision-making theory. In particular, additivity property of price tells us that our decision-maker is not risk-averse but he is risk-neutral. Therefore, the certain gain equivalent to a random gain coincides with a coherent price of this random gain.

Space of alternatives as a foundation of a mathematical model concerning decision-making under conditions of uncertainty

We show a mathematical model based on “a priori” possible data and coherent subjective probabilities. A set of possible alternatives is viewed as a set of all possible samples whose size is equal to 1 selected from a finite population. Such a finite population coincides with those coherent previsions of a univariate random quantity representing all possible alternatives considered “a priori”. We consider a discrete probability distribution of all possible samples. We approximately get the standardized normal distribution from this probability distribution. Within this context an event is not a measurable set so we do not consider random variables viewed as measurable functions into a probability space characterized by a σ-algebra. Anyway, a parameter space is always provided with a metric structure that we introduce after studying the range of possibility. This metric structure is useful in order to obtain different quantitative measures that allow us of considering meaningful relationships between random quantities. When we study multivariate random quantities we introduce antisymmetric tensors satisfying simplification and compression reasons with respect to these random quantities into this metric structure.

Probabilities Obtained by Means of Hyperhomographies into a Quadruple Random Quantity

I realized that it is possible to construct an original and well-organized theory of multiple random quantities by accepting the principles of the theory of concordance into the domain of subjective probability. A very important point relevant to such a construction is consequently treated in this paper by showing that a coherent prevision of a bivariate random quantity coincides with the notion of -product of two vectors while a coherent prevision of a quadruple random quantity coincides with the notion of -product of two affine tensors. Metric properties of the notion of -product mathematically characterize both the notion of coherent prevision of a generic bivariate random quantity and the notion of coherent prevision of a generic quadruple random quantity. Coherent previsions of bivariate and quadruple random quantities can be used in order to obtain fundamental metric expressions of bivariate and quadruple random quantities.

INCREASING OF SECURITY DURING INFORMATION TRANSFERING AT PSYCICAL LEVEL

The need and the urgency of the task of ensuring the safe transfer of information relevant to the security of the country is obvious in view of its particular importance. In the study of data protection for this kind of data, it was revealed that the most optimal and effective is the development of a hardware and software complex that identifies potential attacks aimed at data transmission systems at the physical level, and also implements an attack prediction technique that is more accurate and takes into consideration the random component of this process. The theory of the coherent lower previsions theory of Peter Walley laid the basis of the technique for predicting attacks. The implemented equipment is designed for testing communication channels at the physical level. A special feature of this device is automatic monitoring of both the output parameters of the device itself and the parameters of the communication line for compliance with regulatory and technical requirements. After a formalized statement of the problem and information about the initial data, a measurement of the accuracy of the forecasting model is made by means of the Fisher criterion. The results of the implementation showed that the number of prevented attacks as a result of the implementation of the developed software complex has increased.

Uses and computation of imprecise probabilities from statistical data and expert arguments

Imprecise probabilities and the theory of coherent previsions offer a rigorous and powerful framework for modelling subjective uncertainty and solving problems of statistical inference, decision making or risk analysis. The paper introduces formulas for computing imprecise probabilities when statistical data and expert arguments are available to a subject. We then show how to use these imprecise probabilities (dialectical probabilities) for comparing the likelihood of events, conditioning on events, comparing decisions, computing optimal decisions or assessing financial risk. We apply the method to stock trading and show in this experiment the added value both of imprecision and of expert arguments derived from Technical Analysis.

Bayes' Theorem Bounds for Convex Lower Previsions

In this paper we consider some bounds for lower previsions that are either coherent or, more generally, centered convex. We focus on bounds concerning the classical product and Bayes’ rules, discussing first weak product rules and some of their implications for coherent lower previsions. We then generalise a well-known lower bound, which is a (weak) version for events and coherent lower probabilities of Bayes’ theorem, to the case of random variables and (centered) convex previsions. We obtain a family of bounds and show that one of them is undominated in all cases. Some applications are outlined, and it is shown that 2-monotonicity, which ensures that the bound is sharp in the case of events, plays a much more limited role in this general framework.

The fundamental theorems of prevision and asset pricing

We explore two connections between the concepts of coherence, as defined by de Finetti, and arbitrage-free asset pricing in financial markets. We contrast these concepts when random quantities may be unbounded. And we discuss some of the consequences for arbitrage theory when coherent previsions are merely finitely (but not countably) additive.

Uncertainty modelling and conditioning with convex imprecise previsions

Two classes of imprecise previsions, which we termed convex and centered convex previsions, are studied in this paper in a framework close to Walley’s and Williams’ theory of imprecise previsions. We show that convex previsions are related with a concept of convex natural extension, which is useful in correcting a large class of inconsistent imprecise probability assessments, characterised by a condition of avoiding unbounded sure loss. Convexity further provides a conceptual framework for some uncertainty models and devices, like unnormalised supremum preserving functions. Centered convex previsions are intermediate between coherent previsions and previsions avoiding sure loss, and their not requiring positive homogeneity is a relevant feature for potential applications. We discuss in particular their usage in (financial) risk measurement. In a final part we introduce convex imprecise previsions in a conditional environment and investigate their basic properties, showing how several of the preceding notions may be extended and the way the generalised Bayes rule applies.

Variety of judgements admitted in imprecise statistical reasoning

Intending the present paper as a contribution towards precautionary decision-making, the authors give a brief overview and summary of their experience in the modelling of different judgements admitted within the framework of the theory of coherent imprecise previsions and of possible use in reliability analysis. The paper specifically focuses on direct, comparative, structural and unreliable judgements.

Coherent lower previsions and Choquet integrals

Walley (Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, London, 1991) has given a helpful general framework of uncertain reasoning which is grounded on a behavioural foundation, and extends de Finetti's work. For some important applications like the expected utility theory it might be a tool, alternatively to the techniques of Choquet integration. Therefore, this paper tries to clarify the relation between Walley's central concept of coherent lower prevision and the Choquet integration theory. In one direction we deal with the problem to represent coherent lower previsions by Choquet integrals, and obtain general results using Greco's representation theorem. Another aspect of investigation is to compare Walley's concept to define integrals of coherent fuzzy measures, the so-called natural extensions, with Choquet integrals. We succeed in exploring completely the relationship.

Interval-valued finite Markov chains

The requirement that precise state and transition probabilities be available is often not realistic because of cost, technical difficulties or the uniqueness of the situation under study. Expert judgements, generic data, heterogeneous and partial information on the occurrences of events may be sources of the probability assessments. All this source information cannot produce precise probabilities of interest without having to introduce drastic assumptions often of quite an arbitrary nature. in this paper the theory of interval-valued coherent previsions is employed to generalise discrete Markov chains to interval-valued probabilities. A general procedure of interval-valued probability elicitation is analysed as well. In addition, examples are provided.

Integration and Conditioning in Numerical Possibility Theory

The paper discusses integration and some aspects of conditioning in numerical possibility theory, where possibility measures have the behavioural interpretation of upper probabilities, that is, systems of upper betting rates. In such a context, integration can be used to extend upper probabilities to upper previsions. It is argued that the role of the fuzzy integral in this context is limited, as it can only be used to define a coherent upper prevision if the associated upper probability is 0–1-valued, in which case it moreover coincides with the Choquet integral. These results are valid for arbitrary coherent upper probabilities, and therefore also relevant for possibility theory. It follows from the discussion that in a numerical context, the Choquet integral is better suited than the fuzzy integral for producing coherent upper previsions starting from possibility measures. At the same time, alternative expressions for the Choquet integral associated with a possibility measure are derived. Finally, it is shown that a possibility measure is fully conglomerable and satisfies Walley's regularity axiom for conditioning, ensuring that it can be coherently extended to a conditional possibility measure using both the methods of natural and regular extension.

Statistical Reasoning with Imprecise Probabilities.

Part 1 Reasoning and behaviour: interpretations of probability beliefs and behaviour inference and decision reasoning and rationality assessment strategies survey of related work. Part 2 Coherent previsions: possibility probability currency upper and lower previsions avoiding sure loss coherence basic properties of coherent previsions coherent probabilities linear previsions and additive probabilities examples of coherent previsions interpretations of prevision and probability objections to behavioural theories of probability. Part 3 Extensions, envelopes and decisions: natural extension extension from a field lower envelopes of linear previsions linear extension invariant linear previsions compactness and extreme points of M(P) desirability and preference equivalent models for beliefs decision making. Part 4 Assessment and elicitation: a general elicitation procedure finitely-generated models and simplex representations steps in assessment process classificatory probability comparative probability other types of assessment. Part 5 The importance of imprecision: uncertainty, indeterminacy and imprecision sources from imprecision information from Bernoulli trials prior-data conflict Bayesian noninformative prioirs indecision axioms of precision practical reasons for precision Bayesian sensitivity analysis second-order probabilities fuzzy sets maximum entropy the Dempster-Shafer theory of belief functions. Part 6 Conditional previsions: updated and contingent previsions separate coherence coherence with unconditional previsions the generalized Bayes rule coherence axioms examples of conditional previsions extension of conditional and marginal previsions conglomerability countable additivity conditioning on events of probability zero updating beliefs. Part 7 Coherent statistical models: general concepts of coherence sampling models coherence of sampling model and posterior previsions inferences from improper priors confidence intervals and relevant subsets proper prior previsions standard Bayesian inference inferences from imprecise priors joint prior previsions. Part 8 Statistical reasoning: a general theroy of natural extension extension to prior previsions extension to predictive previsions extension to posterior previsions posteriors for imprecise sampling models the likelihood principle. Part 9 Structural judgements: independent events independent experiments constructing joint previsions from independent marginals permutability exchangeability Robust Bernoulli models structural judgements. Appendices: verifying coherence N-coherence win and place betting on horses topological structure and L and P separating hyperplane theorems desirability upper and lower variances operational measuremnet procedures the World Cup football experiment regular extension W-coherence.

Coherent qualitative probability

This note introduces the concepts of coherent, positive coherent, and strongly coherent qualitative probability. The first interesting result is that such a qualitative probability can be extended from a given domain (not necessarily an algebra) to an arbitrary larger one. The most important result of this paper consists in proving that coherent qualitative probabilities can be represented by the Finetti's coherent previsions.

A remark on failures of conglomerability of prevision

Given a coherent conditional prevision we obtain, by means of the Yosida-Hewitt Decomposition Theorem, an upper bound of its failures of conglomerability with respect to a random number and a denumerable partition. In this way we get an extension of Theorem 2.3 of [4] to coherent previsions.