Axioms Of Probability Research Articles

Countable Additivity and Subjective Probability

While there are several arguments on either side, it is far from clear as to whether or not countable additivity is an acceptable axiom of subjective probability. I focus here on de Finetti's central argument against countable additivity and provide a new Dutch book proof of the principle, To argue that if we accept the Dutch book foundations of subjective probability, countable additivity is an unavoidable constraint.

An expert system for coherent assessment of probabilities in multigraph models

In this paper, we present a method that allows a coherent assessment of probabilistic Bayesian networks over sets of nodes or variables that share a common subset. We motivate and illustrate the models with a diseases-symptoms knowledge base defined by multiple Bayesian networks. Since the parameter values for the joint probability distribution of diseases and symptoms must satisfy some compatibility conditions, the probability assessment becomes complicated and can be monitored by an expert system, in order to maintain coherence in the knowledge base. The bases of this expert system are presented. Once the human expert selects one parameter in the model to be assessed, the expert system calculates and provides the human expert with a range of feasible values of the parameter. Any value of the parameter within this range is coherent with the probability axioms. Then, the human expert can specify either a single value or an interval for that parameter in the feasible range. Accordingly, the expert system then updates the feasible ranges for all other parameters. The process can be repeated until all the parameters have been specified. At the end, the expert system gives a final set of values or intervals for all parameters. We apply the proposed system to the dependent-symptoms and the independent-symptoms probabilistic models. We give two numerical examples to illustrate the method. © 1999 John Wiley & Sons, Inc. Networks 33: 193–206, 1999

A Nonpragmatic Vindication of Probabilism

The pragmatic character of the Dutch book argument makes it unsuitable as an “epistemic” justification for the fundamental probabilist dogma that rational partial beliefs must conform to the axioms of probability. To secure an appropriately epistemic justification for this conclusion, one must explain what it means for a system of partial beliefs to accurately represent the state of the world, and then show that partial beliefs that violate the laws of probability are invariably less accurate than they could be otherwise. The first task can be accomplished once we realize that the accuracy of systems of partial beliefs can be measured on a gradational scale that satisfies a small set of formal constraints, each of which has a sound epistemic motivation. When accuracy is measured in this way it can be shown that any system of degrees of belief that violates the axioms of probability can be replaced by an alternative system that obeys the axioms and yet is more accurate in every possible world. Since epistemically rational agents must strive to hold accurate beliefs, this establishes conformity with the axioms of probability as a norm of epistemic rationality whatever its prudential merits or defects might be.

Responses to Maher, and to Kelly, Schulte, and Juhl

1. Continuing Dutch Flow. My idea, in Section 2 of Bayes and Beyond, was not to give a new proof of a Dutch Book theorem but rather to present a new reading of an existing theorem-proved, e.g., as in Howson and Urbach-framed in such terms that challenges to the premises based on considerations of preference or utility could be bypassed as irrelevant. The aim was to forge a reading such that systematic substitution of flow terms for the usual betting terms would result in unproblematic premises and Dutch Book theorems more directly expressing Howson and Urbach's insight, namely that an incoherent belief system-one violating the probability axioms-can be revealed to be defective in the sense of containing a logical inconsistency. Thus, the defectiveness is clearly epistemic, and there is no need to bring in considerations of prudential or decision-theoretic rationality. (Of course, one may well bring in such considerations if one wants to answer such questions as When should we 'reject' an inconsistent belief system? But clearly that is a distinct question at a different level from the one explicitly addressed by Dutch Flow arguments, viz. For what epistemic reason should degrees of belief conform to the probability axioms?) The inconsistency involved in violating a probability axiom takes the canonical form of judgments that a set of belief-tests-say a finite set, in the simplest case-consists of only or fairtests when indeed they are provably not all neutral, where 'neutral' was defined to mean having zero expected flow, i.e., xPr(h) = yPr(-,h) as explained in Section 2. This notion of neutral belief-test is to be substituted for

Not all (possibly) "random" sequences are created equal.

The need to assess the randomness of a single sequence, especially a finite sequence, is ubiquitous, yet is unaddressed by axiomatic probability theory. Here, we assess randomness via approximate entropy (ApEn), a computable measure of sequential irregularity, applicable to single sequences of both (even very short) finite and infinite length. We indicate the novelty and facility of the multidimensional viewpoint taken by ApEn, in contrast to classical measures. Furthermore and notably, for finite length, finite state sequences, one can identify maximally irregular sequences, and then apply ApEn to quantify the extent to which given sequences differ from maximal irregularity, via a set of deficit (def(m)) functions. The utility of these def(m) functions which we show allows one to considerably refine the notions of probabilistic independence and normality, is featured in several studies, including (i) digits of e, pi, radical2, and radical3, both in base 2 and in base 10, and (ii) sequences given by fractional parts of multiples of irrationals. We prove companion analytic results, which also feature in a discussion of the role and validity of the almost sure properties from axiomatic probability theory insofar as they apply to specified sequences and sets of sequences (in the physical world). We conclude by relating the present results and perspective to both previous and subsequent studies.

Dutch books, dutch strategies and what they show about rationality

Bayesians typically appeal to the so-called Dutch Book argument to justify their claim that a rational agent’s degrees of belief must obey the axioms of probability. The argument has also been extended in support of two other Bayesian principles: the rule of conditionalization, which states that when the evidence E is learned the new probability (degree of belief ) for the proposition A should be equal to the old probability of A given E, and the principle of reflection which states that an agent’s current subjective probability for A, conditional on the supposition that she will later attach a probability to A equal to r, must equal r. 2 Considerable attention has been focused on the structure of the basic argument to show that degrees of belief must satisfy the probability axioms and on whether it established this claim. Recently, a number of authors have suggested that what the basic argument really shows is that degrees of belief which fail to satisfy the axioms suffer from a form of inconsistency and because of this are defective. In his paper, “Clever Bookies and Coherent Beliefs”, David Christensen not only endorses this view, but further argues that on this reading of the basic Dutch Book argument, it follows that the argument cannot adequately be extended to justify belief change by conditionalization nor to justify the principle of reflection. 3 In this paper, I argue that Christensen’s criticisms of the Dutch Book arguments for conditionalization and reflection are unsatisfactory. Nevertheless, I agree that the arguments do not show that an agent who violates these principles must be irrational. In the final section of the paper, I suggest that the Dutch Book arguments for conditionalization and reflection do lend support for these principles as ideals of certain sorts, although they do not show that they are requirements of rationality.

Hidden Assumptions in the Dutch Book Argument

Probabilistic theories of rationality claim that degrees of belief have to satisfy the probability axioms in order to be rational. A standard argument to support this claim is the Dutch Book argument. This paper tries to show that, in spite of its popularity, the Dutch Book argument does not provide a foundation for normative theories of rationality. After a presentation of the argument and some of its criticisms a problem is pointed out: the Dutch Book argument applies only to situations with a specific formal structure. Several attempts to justify the argument for more general situations are considered and rejected. The only way to remedy the shortcoming, it is argued, seems to be the acceptance of a far-reaching and highly implausible empirical hypothesis.

Uncertainty about probability: a reconciliation with the subjectivist viewpoint

The use of probability distributions to represent uncertainty about probabilities (rather than events) has long been a subject of controversy among theorists. Many have concluded that it is inherently meaningless to be uncertain about a probability, because this appears to violate the subjectivists' assumption that individuals can develop unique and precise probability judgments. Others have found the concept of uncertainty about probability to be both intuitively appealing and potentially useful. This paper presents a resolution of this question, indicating that at least one type of uncertainty about probabilities (that arising from uncertainty about the underlying events on which those probabilities are conditioned) is consistent with the subjective theory of probability. Another type of uncertainty (namely, that arising from cognitive imprecision) appears not to be consistent with the axioms of subjective probability as they are currently formulated, Distinguishing between these two sources of uncertainty is an important step in resolving the persistent theoretical controversy over uncertainty about probability. We also show that uncertainty about probability will be potentially relevant to decision-making whenever the expected value of the relevant performance measure is nonlinear in the probability in question.

Finite Linear Qualitative Probability

Many years ago, Bruno de Finetti asked whether certain axioms for a comparative probability ordering relation ≻ on the subsets of ann-element set that are necessary for the existence of an order-preserving probability measure are also sufficient. This was answered in the negative forn⩾5 by Kraft, Pratt, and Seidenberg [Annals of Mathematical Statistics30(1959), 408–419]. The present paper extends their analysis of comparative probabilities that satisfy de Finetti's axioms but lack order-preserving measures when it is assumed also that ≻ is a linear order. We refer to a linear order on the subsets of {1, 2, …, n} that satisfies de Finetti's axioms as an LQP (linear qualitative probability), and say that it is an NLQP (nonrepresentable LQP) if it has no order-preserving probability measure. The paper characterizes all NLQPs forn=5, shows that everyn⩾5 has an NLQP that violates the simplest extension of de Finetti's additivity axiom yet has an order-preserving measure on the family of subsets of every (n−1)-element set, and proves that asn→∞ there is no upper bound on the number of ≻ comparisons needed to verify the nonexistence of an order-preserving measure. Special examples for smallnillustrate other facets of NLQPs, including the necessity of considering multiplicities of an ≻ comparison in testing whether an LQP has an order-preserving measure.

Non-Archimedean probability: Frequency and axiomatics theories

We propose a new theory of probability based on the general principle of the statistical stabilization of relative frequencies. According to this principle it is possible to consider the statistical stabilization not only with respect to the standard real topology on the field of rational numbersQ but also with respect to an arbitrary topology onQ. The case ofp-adic (and more general non-Archimedean) topologies is the most important. Our frequency theory of Probability is a fruitful extension of the frequency theory of R. von Mises[18]. It's well known that the axiomatic theory of Kolmogorov uses the frequency theory as one of the foundations. And a new general frequency theory can be considered as the base for the general axiomatic theory of probability (Kolmogorov's theory is a particular case of this theory which corresponds to the real topology of the statistical stabilization onQ). The situation in the theory of probability becomes similar to that in modern geometry. The Kolmogorov axiomatics (as the Euclidean) is only one of the possibilities, and we have generated a great number of different non-Kolmogorov theories of probability. The applications top-adic quantum mechanics and field theory are considered.

A source of Bayesian priors

Establishing reasonable, prior distributions remains a significant obstacle for the construction of probabilistic expert systems. Human assessment of chance is often relied upon for this purpose, but this has the drawback of being inconsistent with axioms of probability. This article advances a method for extracting a coherent distribution of probability from human judgment. The method is based on a psychological model of probabilistic reasoning, followed by a correction phase using linear programming.

A Source of Bayesian Priors

Establishing reasonable, prior distributions remains a significant obstacle for the construction of probabilistic expert systems. Human assessment of chance is often relied upon for this purpose, but this has the drawback of being inconsistent with axioms of probability. This article advances a method for extracting a coherent distribution of probability from human judgment. The method is based on a psychological model of probabilistic reasoning, followed by a correction phase using linear programming.

The foundations of probability and quantum mechanics

Taking as starting point two familiar interpretations of probability, we develop these in a perhaps unfamiliar way to arrive ultimately at an improbable claim concerning the proper axiomatization of probability theory: the domain of definition of a point-valued probability distribution is an orthomodular partially ordered set. Similar claims have been made in the light of quantum mechanics but here the motivation is intrinsically probabilistic. This being so the main task is to investigate what light, if any, this sheds on quantum mechanics. In particular it is important to know under what conditions these point-valued distributions can be thought of as derived from distribution-pairs of upper and lower probabilities on boolean algebras. Generalising known results this investigation unsurprisingly proves unrewarding. In the light of this failure the next topic investigated is how these generalized probability distributions are to be interpreted.

A Limiting Frequency Approach to Probability Based on the Weak Law of Large Numbers

Von Mises defined a “physical” probability as a strict limit of the relative frequency of occurrence of an event in repeated trials. As a result of a number of criticisms of von Mises's approach, the more favored approach became the “propensity” interpretation. It is argued here that this interpretation is not compelling and that the only problem in von Mises's approach is the assumption that the relative frequency converges in a strict sense. This problem is then remedied by deducing the axioms of probability theory from the assumption that the relative frequency converges only in the sense of the weak law of large numbers.

Subjective expected utility theory revisited: A reductio ad absurdum paradox

A paradox is posed and analyzed in which people reverse their preferences for information on probabilities versus prizes once the range of the unknown probabilities is sufficiently narrowed. This reversal is shown to be incompatible with both objective expected utility (EU) as well as subjective versions in which the same probability transformation applies to all random variables. Experimental data are presented showing that the reversals occur with small, medium and large payoffs. The present paradox is compared with those of Allais and Ellsberg, and found to differ in substantive ways. It raises further questions about the normative status of expected utility theory, especially its treatment of probability and value. The paradox specifically calls into question EU's substitution and compound probability axioms.

Dutch Book Arguments and Consistency

I consider Dutch Book arguments for three principles of classical Bayesianism: (i) agents' belief-probabilities are consistent only if they obey the probability axioms. (ii) beliefs are updated by Bayesian conditionalisation. (iii) that the so-called Principal Principle connects statistical and belief probabilities. I argue that while there is a sound Dutch Book argument for (i), the standard ones for (ii) based on the Lewis-Teller strategy are unsound, for reasons pointed out by Christensen. I consider a type of Dutch Book argument for (iii), where the statistical probability is a von Mises one.

Probability and Random Processes for Electrical Engineering

Probability and Random Processes for Electrical Engineering

Reappraisal of the use of conditional probability in early expert systems

A qualitative method of reasoning under uncertainty using Bayesian probability was originally proposed by Duda et al. However, this has been found to be 80% reliable at best in practice, and better techniques are being looked for by those who seek greater accuracy. The paper concludes that Bayesian approaches should not be abandoned. A Bayesian approach can be used as a sound quantitative technique, provided that the axioms of probability are properly observed and the nature of Bayes' theorem itself is understood. The paper documents some lessons that can be learned from the previous use of Bayes' theorem, and assists in the understanding of the way that it works in practice. The limitations of assuming conditional independence are emphasized, and the consequent equal ratios that must exist between various pairs of joint probabilities are highlighted. This leads to validity checks that can be made by an expert-system designer using local calculations and Bayesian update methods. It is also demonstrated that, for even modest numbers of antecedents, the assumption that these are independent overwhelms the information given by the expert, who, when considering a particular piece of evidence, has control of little more than the 'all other antecedents true' and the 'all other antecedents false' scenarios. The same conclusions may be pertinent to more recent work, but this is not explored in depth.

Output probability from a general combinational network

In this paper, we present the method which calculates the probability of the output of a general combinational network being 1, when the probabilities is given for each input being 1. Also we present the method which derives the output probability expression in terms of a given set of input probabilities. The method is based on a basic probability axiom and Binary Decision Diagrams. A Boolean function for a combinational network is transformed into a Binary Decision Diagram and the output probability expression is derived from it. Therefore a derivation of the output probability for a combinational network can be straightforward. Some examples are showed that the method using Binary Decision Diagrams is simple and efficient. We can see a variation of the output probability for a combinational network according to the given set of input probabilities.

Interval probability theory for evidential support

An interval theory of probability is presented for use as a measure of evidential support in knowledge-based systems. An interval number is used to capture, in a relatively simple manner, features of fuzziness and incompleteness. The vertex method is used for the interval analysis. A new parameter (also an interval number), p, called the degree of' dependence is introduced. The relationship of this interval probability with the theories of Dempster-Shafer, fuzzy sets, and Baldwin's support logic are discussed. The advantage of the theory is that it is based on a development of the axioms of probability, but allows that evidential support for a conjecture be separated from evidential support for the negation of the conjecture.