Subjective Probability Theory Research Articles

Logic, rationality and knowledge in Ramsey's thought: reassessing ‘human logic’

This paper reconsiders Frank Ramsey's essay on subjective probability (1926) as a consistent way to articulate logic, rationality and knowledge. The first part of the essay builds an axiomatic theory of subjective probability based on ‘formal logic’, defining rationality as choice-consistency. The second part seems to open up different horizons: the evaluation of degrees of belief by ‘human logic’. Because of the interest Keynes (1931) had taken in ‘human logic’, it was considered to be a possible alternative to the formal logic underlying the neoclassical theory of individual behaviour. The analysis of Ramsey's method in the entire paper, the relation between logic and rationality it constructs and the conception of uncertainty it reveals, lead me to note on the contrary that Ramsey's human logic was a complementary logic rather than an alternative to formal logic. Defining a standard to evaluate beliefs formation according to a frequentist criterion, it completes a normative representation of rationality which supports an original theory of knowledge that appears more in line with further developments of neoclassical methodology than with Keynesian economics.

A scoping inquiry into the potential contribution of Subjective Probability Theory, Dempster–Shafer Theory and Possibility Theory in accommodating degrees of belief in traveller behaviour research

There is a small but growing interest in traveller behaviour research on investigating ways to identify and quantify degrees of belief (as subjective probabilities or other propositions) associated with behavioural responses, especially in the context of popular travel choice methods such as stated choice experiments, as a way of adding to our understanding of decision making in real-world contexts that are associated with inevitable risk and uncertainty. This paper reviews three major theories that are not well known in the transportation literature that have been developed in psychology and decision sciences to accommodate belief, namely Subjective Probability Theory, Dempster–Shafer Theory and Possibility Theory. We focus on how degrees of belief are measured in these theories. The key elements of each theoretical approach are compared, including their mathematical properties and evidence patterns. Despite their being few applications to date in transportation, the review promotes the relevance of accounting for degrees of belief in travel choice analysis.

The Law Doesn't Say Much About Base Rates

In his recent contribution “Neglect the Base Rate: It’s the law!” (http://ssrn.com/abstract=2192423), Prof. Engel makes the point that the law forbids making group-to-individual inferences based on the relative frequency of a property of interest in a reference class (the “base rate”), unless the members of the reference class were “acting in concert” or the base rate information serves as exculpating, rather than inculpating, evidence in criminal cases. While making his argument, Prof. Engel also criticizes a “Bayesian” approach to evidence evaluation, i.e. an approach that posits that the partial beliefs of judges should conform to the axioms of probability theory, suggesting that it leads to conclusions which are incompatible with substantive law. Prof. Engel develops his arguments based on a series of examples. In this response, I purport to show that, i) subjective probability theory does not lead to solutions of the problems presented by Prof. Engel that are contrary to substantive law; and, ii) the law does not generally prohibit the use of base rate information where it admits other relevant evidence. In fact, the law does not say much about the use of base rate information at all, because it is hardly ever directly relevant outside the world of hypothetical cases of legal scholars. On the other hand, base rate information can be epistemically useful, and a blanket prohibition cannot be read from the law.

Trust‐oriented QoS‐aware composite service selection based on genetic algorithms

SUMMARYService selection in service‐oriented computing has emerged to be an increasingly important research area. From the client's point of view, in addition to the QoS of a service or a service composition, the trust level becomes an important part. As the complexity of invocation in service composition has been greatly increased, a comprehensive mechanism, which could evaluate both the subjective aspect as trust expression and the objective aspect as QoS, is needed. In this paper, we provide a formal service composition architecture for service selection. In addition, we propose a trust evaluation method for the service composition plan based on the subjective probability theory, based on them, our trust‐oriented genetic algorithm (TOGA) is proposed to find a near‐optimal service composition plan with QoS constraints. Experimental results have illustrated that our proposed approach can discover the near‐optimal solution efficiently. Copyright © 2013 John Wiley & Sons, Ltd.

Bayesian decision analysis: principles and practice

Bayesian decision analysis supports principled decision making in complex domains. This textbook takes the reader from a formal analysis of simple decision problems to a careful analysis of the sometimes very complex and data rich structures confronted by practitioners. The book contains basic material on subjective probability theory and multi-attribute utility theory, event and decision trees, Bayesian networks, influence diagrams and causal Bayesian networks. The author demonstrates when and how the theory can be successfully applied to a given decision problem, how data can be sampled and expert judgements elicited to support this analysis, and when and how an effective Bayesian decision analysis can be implemented. Evolving from a third-year undergraduate course taught by the author over many years, all of the material in this book will be accessible to a student who has completed introductory courses in probability and mathematical statistics--Provided by publisher.

A Novel Ranking Method Based on Subjective Probability Theory for Evolutionary Multiobjective Optimization

Most of the engineering problems are modeled as evolutionary multiobjective optimization problems, but they always ask for only one best solution, not a set of Pareto optimal solutions. The decision maker′s subjective information plays an important role in choosing the best solution from several Pareto optimal solutions. Generally, the decision‐making processing is implemented after Pareto optimality. In this paper, we attempted to incorporate the decider′s subjective sense with Pareto optimality for chromosomes ranking. A new ranking method based on subjective probability theory was thus proposed in order to explore and comprehend the true nature of the chromosomes on the Pareto optimal front. The properties of the ranking rule were proven, and its transitivity was presented as well. Simulation results compared the performance of the proposed ranking approach with the Pareto‐based ranking method for two multiobjective optimization cases, which demonstrated the effectiveness of the new ranking approach.

Simple axioms for countably additive subjective probability

This paper refines Savage’s theory of subjective probability for the case of countably additive beliefs. First, I replace his continuity axioms P6 and P7 with a simple modification of Arrow’s (1970) Monotone Continuity. Second, I relax Savage’s primitives: in my framework, the class of events need not be a σ-algebra, and acts need not have finite or bounded range. By varying the domains of acts and events, I obtain a unique extension of preference that parallels Caratheodory’s unique extension of probability measures. Aside from subjective expected utility, I characterize exponential time discounting in a setting with continuous time and an arbitrary consumption set.

Subjective Trust Inference in Composite Services

In Service-Oriented Computing (SOC) environments, the trustworthiness of each service is critical for a service client when selecting one from a large pool of services. The trust value of a service is usually in the range of [0,1] and is evaluated from the ratings given by service clients, which represent the subjective belief of these service clients on the satisfaction of delivered services. So a trust value can be taken as the subjective probability, with which one party believes that another party can perform an action in a certain situation. Hence, subjective probability theory should be adopted in trust evaluation. In addition, in SOC environments, a service usually invokes other services offered by different service providers forming a composite service. Thus, the global trust of a composite service should be evaluated based on complex invocation structures. In this paper, firstly, based on Bayesian inference, we propose a novel method to evaluate the subjective trustworthiness of a service component from a series of ratings given by service clients. Secondly, we interpret the trust dependency caused by service invocations as conditional probability, which is evaluated based on the subjective trust values of service components. Furthermore, we propose a joint subjective probability method to evaluate the subjective global trust of a composite service on the basis of trust dependency. Finally, we introduce the results of our conducted experiments to illustrate the properties of our proposed subjective global trust inference method.

Probability and Uncertainty in Economic Modeling

The article considers the paradigm of subjective probability and expected utility theory with respect to their applications in the theory of decision-making. Advantages and shortcomings of Savage’s axiomatic in the subjective probability theory are analyzed, the models of beliefs formation are considered. The authors propose a new approach to the analysis of decision-making — a multiple priors model, where an agent attributes to each event not a single probability, but a range of probabilities.

Ranking functions and rankings on languages

The Spohnian paradigm of ranking functions is in many respects like an order-of-magnitude reverse of subjective probability theory. Unlike probabilities, however, ranking functions are only indirectly—via a pointwise ranking function on the underlying set of possibilities W—defined on a field of propositions A over W. This research note shows under which conditions ranking functions on a field of propositions A over W and rankings on a language L are induced by pointwise ranking functions on W and the set of models for L , Mod L , respectively.

Any reasonable cost function can be used for a posteriori probability approximation

In this paper, we provide a straightforward proof of an important, but nevertheless little known, result obtained by Lindley in the framework of subjective probability theory. This result, once interpreted in the machine learning/pattern recognition context, puts new light on the probabilistic interpretation of the output of a trained classifier. A learning machine, or more generally a model, is usually trained by minimizing a criterion-the expectation of the cost function-measuring the discrepancy between the model output and the desired output. In this letter, we first show that, for the binary classification case, training the model with any "reasonable cost function" can lead to Bayesian a posteriori probability estimation. Indeed, after having trained the model by minimizing the criterion, there always exists a computable transformation that maps the output of the model to the Bayesian a posteriori probability of the class membership given the input. Then, necessary conditions allowing the computation of the transformation mapping the outputs of the model to the a posteriori probabilities are derived for the multioutput case. Finally, these theoretical results are illustrated through some simulation examples involving various cost functions.

The impartial observer theorem of social ethics

I would like to dedicate this paper to the memory of Louis-André Gérard-Varet. His premature death in January 2001 deprived us all of a genuinely broad theoretical economist with deep foundational interests. He was one of the few to be conversant with the intricacies of the theory of subjective probability, and he himself co-authored a formal reconstruction of the Impartial Observer Theorem (d'Aspremont and Gérard-Varet, 1991). At the time of publishing this alternative reconstruction, I cannot refrain from remembering his exceptionally active intelligence, as well as the fruitful interchanges we had.

Subjective probability theory with continuous acts

Subjective probability theory with continuous acts

Conditional Probabilities, Potential Surprise, and the Conjunction Fallacy

Tversky and Kahneman (1983) found that a relationship of positive conditional dependence between the components of a conjunction of two events increases the prevalence of the conjunction fallacy. Consistent with this finding, the results of two experiments reveal that dependence leads to higher estimates for the conjunctive probability and a higher incidence of the fallacy. However, contrary to the theoretical account proposed by Tversky and Kahneman, the actual magnitude of the conditional relationship does not directly affect the extent of the fallacy; all that is necessary is for a positive conditional relationship to exist. The pattern of results obtained can be accounted for in terms of Shackle's (1969) “potential surprise” theory of subjective probability. Surprise theory predicts that the impact of the conditional event will be at its maximum where the relationship is a negative one. Tversky and Kahneman's model, on the other hand, predicts the maximum effect when the relationship is positive. In all 12 scenarios tested, multiple regression analysis revealed that the standardized beta weight associated with the conditional event was greater when the relationship was a negative one. Thus the outcome was supportive of the surprise model rather than Tversky and Kahneman's account.

An integrated GIS and knowledge-based system as an aid for the geological analysis of sedimentary basins

The methods and advantages of integrating knowledge-based and geographical information system techniques for the analysis of provenance and diagenesis in sedimentary basins are demonstrated by examples from the Cheshire Basin located in north-west England. Approximate reasoning techniques to handle the vagueness and uncertainty inherent in a large amount of geological data, knowledge and reasoning are reviewed with particular emphasis on provenance analysis using subjective probability theory, Dempster-Shafer theory and fuzzy logic techniques. Object-oriented methods for analysing diagenetic history in user specified volumes of the basin are discussed.

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.

Money demand revisited: An operational subjective approach

SUMMARY This paper proposes a method of data analysis founded on the philosophy and understanding of uncertain knowledge developed by Bruno de Finetti. Specifically, the paper investigates the informational content of interest rates for the prediction of Ml. This empirical application replicates that of Cooley and Leroy (1981) and McAleer, Pagan, and Volker (1985), but the procedures and their interpretation follow the operational subjective approach. The issue of an autocorrelated error structure is recast in the operational subjective context. Methods are developed to assess the interest sensitivity of the demand for money in this context. This paper illustrates an operational method of data analysis in the context of linear models. The procedures advocated are founded on the subjective theory of probability originated by Bruno deFinetti (1974, 1975) and enhanced by L. J. Savage (1954). This foundational position denies the existence of a 'true' model and its concomitant parameters. The function of a model is its assistance in generating predictive distributions of observable unknown quantities. A model's performance at prediction is empirically observable. This proposed method of econometric analysis is illustrated with an applied example taking advantage of a previous, well-known application confronting the appropriateness of econometric practices. The example concerns money demand as measured by Ml with particular emphasis on the informational content of interest rates for the forecasting distribution of Ml. Cooley and Leroy (1981) examined the estimation of money demand, employing the Bayesian method of extreme bounds analysis developed in Leamer (1978). Although Bayesian in approach the method focuses on parameter estimation and retains the conviction of standard techniques in the reality of the model and in the reality of the model parameters. McAleer, Pagan, and Volker (1985) re-examine the same model and data from a frequentist approach, yielding an appropriate example with which to contrast the method presented here and its interpretation.

Support theory: A nonextensional representation of subjective probability.

This article presents a new theory of subjective probability according to which different descriptions of the same event can give rise to different judgments. The experimental evidence confirms the major predictions of the theory. First, judged probability increases by unpacking the focal hypothesis and decreases by unpacking the alternative hypothesis. Second, judged probabilities are complementary in the binary case and subadditive in the general case, contrary to both classical and revisionist models of belief. Third, subadditivity is more pronounced for probability judgments than for frequency judgments and is enhanced by compatible evidence. The theory provides a unified treatment of a wide range of empirical findings. It is extended to ordinal judgments and to the assessment of upper and lower probabilities. Both laypeople and experts are often called upon to evaluate the probability of uncertain events such as the outcome of a trial, the result of a medical operation, the success of a business venture, or the winner of a football game. Such assessments play an important role in deciding, respectively, whether to go to court, undergo surgery, invest in the venture, or bet on the home team. Uncertainty is usually expressed in verbal terms (e.g., unlikely or probable), but numerical estimates are also common. Weather forecasters, for example, often report the probability of rain (Murphy, 1985), and economists are sometimes required to estimate the chances of recession (Zarnowitz, 1985). The theoretical and practical significance of subjective probability has inspired psychologists, philosophers, and statisticians to investigate this notion from both descriptive and prescriptive standpoints.

Interpretative versus prescriptive fuzzy set theory

Both traditional fuzzy set theory and the theory of subjective probabilities postulate their formulas. The former because it does not accept a probabilistic interpretation of grades of membership, the latter because it makes no connection between subjective probabilities and relative frequencies. The interpretational theory of the TEE model ascribes a well-defined meaning to a membership value /spl mu//sub /spl lambda// in terms of probabilities which are limits of frequencies. /spl mu//sub /spl lambda// is interpreted as the estimate by the subject of the probability that a given object would be assigned the label X in an everyday situation of uncertainty. In contrast to the max-min formulas used in many applications of fuzzy sets, the postulated summation-to-1 formula of all fuzzy clustering algorithms follows from the interpretative TEE model. This agrees with the author's view that fuzzy set theory should be defined as a theory which allows partial membership values of an object in a class or cluster, not as a theory which uses specific mathematical operators. The operators must be derived from the well-defined interpretation of partial membership values. >

The meaning of probability in probabilistic safety analysis

Abstract A Probabilistic Safety Analysis expresses uncertainty about the possible future damaging consequences of complex installations, such as chemical or nuclear plants, in terms of probabilities. Often these probabilities are interpreted as measures of physical properties of the installation and how it is operated, and a PSA is seen as a means to carry out these measurements. This view is useful for making comparative statements about the riskiness of installations, particularly when comparing them with standards. It is argued here, however, that this interpretation of probability is inconsistent with all the standard philosophical theories of probability, and so a different interpretation of PSA is necessary. We suggest, alternatively, that by using the standard subjective theory of probability, PSA may be seen as a tool for argument, rather than an objective representation of truth. In this interpretation the problems of expert choice and model validation become less problematic.