Conditional Probabilities, Potential Surprise, and the Conjunction Fallacy

Muth’s Rational Expectations Hypothesis, That “…The Subjective Probability Distributions Are Distributed Around an Objective Probability Distribution, for a given Information Set…” is an Oxymoron. It Can Be Corrected by Amending It to Read “…The Subjective Probability Distributions Are Distributed Around a Subjective Consensus Probability Distribution, for a given Information Set…”

In Service-Oriented Computing (SOC) environments, the trustworthiness of each service provider is critical for a service client when selecting one from a large pool of service providers. The trust value of a service provider is usually in the range of [0, 1] and is evaluated from the ratings given by service clients, which represent the subjective belief of service clients on the satisfaction of delivered services. So a trust value can be taken as a subjective probability, by 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 provider usually can invoke the services from other service providers forming a composite service. Thus, the global trust of a composite service should be evaluated based on both the subjective probability property of trust and complex invocation structures. In this paper, we first interpret the trust dependency caused by direct service invocations as conditional probability. Then, on the basis of trust dependency, we propose a Subjective probability based deductive (SELECTIVE) approach to evaluate the subjective global trustworthiness of a composite service. All these processes follow subjective probability theory and keep the subjective probability property of trust in evaluations. Our experimental results demonstrate that when compared with existing approaches our proposed SELECTIVE approach can yield more reasonable results.

I show that de Finetti’s coherence theorem is equivalent to the Hahn-Banach theorem and discuss some consequences of this result. First, the result unites two aspects of de Finetti’s thought in a nice way: a corollary of the result is that the coherence theorem implies the existence of a fair countable lottery, which de Finetti appealed to in his arguments against countable additivity. Another corollary of the result is the existence of sets that are not Lebesgue measurable. I offer a subjectivist interpretation of this corollary that is concordant with de Finetti’s views. I conclude by pointing out that my result shows that there is a sense in which de Finetti’s theory of subjective probability is necessarily nonconstructive. This raises questions about whether the coherence theorem can underwrite a legitimate theory of rational belief.

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.

In this paper, we present a brief version of de Finetti-Ramsey's subjective probability theory and provide a rigorous yet intuitively plausible explanation of expected utility using elementary mathematics.In a final section, we take up the case of some "Paradoxes in Expected Utility Theory" and try to reconcile them with the help of subjective probabilities.

Findings in recent research on the ‘conjunction fallacy’ have been taken as evidence that our minds are not designed to work by the rules of probability. This conclusion springs from the idea that norms should be content-blind—in the present case, the assumption that sound reasoning requires following the conjunction rule of probability theory. But content-blind norms overlook some of the intelligent ways in which humans deal with uncertainty, for instance, when drawing semantic and pragmatic inferences. In a series of studies, we first show that people infer nonmathematical meanings of the polysemous term ‘probability’ in the classic Linda conjunction problem. We then demonstrate that one can design contexts in which people infer mathematical meanings of the term and are therefore more likely to conform to the conjunction rule. Finally, we report evidence that the term ‘frequency’ narrows the spectrum of possible interpretations of ‘probability’ down to its mathematical meanings, and that this fact—rather than the presence or absence of ‘extensional cues’—accounts for the low proportion of violations of the conjunction rule when people are asked for frequency judgments. We conclude that a failure to recognize the human capacity for semantic and pragmatic inference can lead rational responses to be misclassified as fallacies. Copyright © 1999 John Wiley & Sons, Ltd.

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.

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.

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.

The subjective Bayesian approach to PRA: Theoretical and practical perspectives

Earthquake catalogue declustering is a critical preprocessing step in time-dependent seismicity analyses (Gardner and Knopoff, 1974; Reasenberg, 1985), yet its systematic influence on conditional earthquake probability estimates remains insufficiently quantified, particularly in tectonically complex continental collision zones such as the Himalayas (Bungum et al., 2017). Renewal-based recurrence models typically assume that declustered catalogues isolate tectonically driven mainshock recurrence by removing dependent events. However, recent advances in declustering theory demonstrate that methodological choices, ranging from fixed spatio-temporal windows to adaptive and stochastic approaches, can substantially modify inter-event time statistics and inferred recurrence memory (Zaliapin et al., 2008; Zaliapin & Ben-Zion, 2020; Teng & Baker, 2019). Despite these developments, the implications of declustering-induced variability for time-dependent conditional probabilities remain underexplored in active orogenic belts.In this study, we explicitly quantify how alternative declustering strategies influence time-dependent recurrence behavior and conditional rupture probabilities across selected Himalayan seismic source zones. Inter-event time series were constructed for moderate-to-large earthquakes (M ≥ 4.0) using both raw (non-declustered) and declustered catalogues derived from regional earthquake compilations. Declustering was performed using commonly applied fixed-window and adaptive approaches to capture epistemic variability associated with catalogue preprocessing. The resulting inter-event times were analyzed within renewal process models, including Brownian Passage Time (BPT), Lognormal, Weibull, and Gamma distributions, to estimate conditional probabilities as functions of elapsed time since the most recent major event.Results show that declustered catalogues consistently yield smoother initial probability gradients and delayed probability peaks relative to raw catalogues, reflecting reduced short-term temporal clustering in inter-event time distributions. These shifts correspond to systematic changes in inferred renewal memory parameters, with declustering suppressing short-term contagion effects while largely preserving long-term mean recurrence intervals. In the Himalayas, collision-driven aftershock swarms and spatially heterogeneous fault interactions amplify these effects, introducing substantial epistemic uncertainty in early-time conditional probabilities, which can locally exceed factors of two to three depending on the declustering strategy employed. In contrast, long-term probability remains comparatively robust across declustering scenarios, consistent with steady-state tectonic strain accumulation.These findings identify catalogue declustering as a dominant and often underappreciated source of uncertainty in time-dependent seismic probability modelling, reinforcing recent calls for ensemble-based and transparent pre-processing strategies in probabilistic seismic hazard workflows. This study advances a methodological framework for interpreting renewal-based conditional probabilities in clustered tectonic regimes. The Himalayas emerge as a natural laboratory where combined raw and declustered analyses can yield more resilient probabilistic interpretations.

We provide a new link between deductive and probabilistic reasoning fallacies. Illusory inferences from disjunction are a broad class of deductive fallacies traditionally explained by recourse to a matching procedure that looks for content overlap between premises. In two behavioral experiments, we show that this phenomenon is instead sensitive to real-world causal dependencies and not to exact content overlap. A group of participants rated the strength of the causal dependence between pairs of sentences. This measure is a near perfect predictor of fallacious reasoning by an independent group of participants in illusory inference tasks with the same materials. In light of these results, we argue that all extant accounts of these deductive fallacies require non-trivial adjustments. Crucially, these novel indirect illusory inferences from disjunction bear a structural similarity to seemingly unrelated probabilistic reasoning problems, in particular the conjunction fallacy from the heuristics and biases literature. This structural connection was entirely obscure in previous work on these deductive problems, due to the theoretical and empirical focus on content overlap. We argue that this structural parallelism provides arguments against the need for rich descriptions and individuating information in the conjunction fallacy, and we outline a unified theory of deductive illusory inferences from disjunction and the conjunction fallacy, in terms of Bayesian confirmation theory.

CHAPTER 3 - Conditional Probability and Conditional Expectation

The conditional mean value has applications in regression analysis and in financial mathematics, because they are used in it. We can find papers from recent years that use the conditional mean value in fuzzy cases. As the intuitionstic fuzzy sets are an extension of fuzzy sets, we will try to define a conditional mean value for the intuitionistic fuzzy case. The conditional mean value in crisp intuitionistic fuzzy events was first studied by V. Valenčáková in 2009. She used Gödel connectives. Her approach can only be used for special cases of intuitionistic fuzzy events, therefore, we want to define a conditional mean value for all elements of a family of intuitionistic fuzzy events. In this paper, we define the conditional mean value for intuitionistic fuzzy events using Lukasiewicz connectives. We use a Kolmogorov approach and the notions from a classical probability theory for construction. B. Riečan formulated a conditional intuitionistic fuzzy probability for intuitionistic fuzzy events using an intuitionistic fuzzy state in 2012. In classical cases, there exists a connection between the conditional probability and the conditional mean value, therefore we show a connection between the conditional intuitionistic fuzzy probability induced by the intuitionistic fuzzy state and the conditional intuitionistic fuzzy mean value.

People often overestimate probabilities of conjunctive events. The authors explored whether the accuracy of conjunctive probability estimates can be improved by increased experience with relevant constituent events and by using memory aids. The first experiment showed that increased experience with constituent events increased the correlation between the estimated and the objective conjunctive probabilities, but that it did not reduce overestimation of conjunctive probabilities. The second experiment showed that reducing cognitive load with memory aids for the constituent probabilities led to improved estimates of the conjunctive probabilities and to decreased overestimation of conjunctive probabilities. To explain the cognitive process underlying people’s probability estimates, the configural weighted average model was tested against the normative multiplicative model. The configural weighted average model generates conjunctive probabilities that systematically overestimate objective probabilities although the generated probabilities still correlate strongly with the objective probabilities. For the majority of participants this model was better than the multiplicative model in predicting the probability estimates. However, when memory aids were provided, the predictive accuracy of the multiplicative model increased. In sum, memory tools can improve people’s conjunctive probability estimates.