Partition Priming in Judgments of Imprecise Probabilities

We elicit numerical expectations for late-onset dementia in the Health and Retirement Study. Our elicitation distinguishes between precise and imprecise probabilities, while accounting for rounding of reports. Respondents quantify imprecision using probability intervals. Nearly half of respondents hold imprecise dementia probabilities, while almost a third of precise-probability respondents round their reports. We provide the first empirical evidence on dementia-risk perceptions among dementia-free older Americans and novel evidence about imprecise probabilities in a nationally-representative sample. We show, in a specific framework, that failing to account for imprecise or rounded probabilities can yield incorrect predictions of long-term care insurance purchase decisions.

We elicit numerical expectations for late-onset dementia and long-term-care (LTC) outcomes in the US Health and Retirement Study. We provide the first empirical evidence on dementia-risk perceptions among dementia-free older Americans and establish important patterns regarding imprecision of subjective probabilities. Our elicitation distinguishes between precise and imprecise probabilities, while accounting for rounding of reports. Imprecise-probability respondents quantify imprecision using probability intervals. Nearly half of respondents hold imprecise dementia and LTC probabilities, while almost a third of precise-probability respondents round their reports. These proportions decrease substantially when LTC expectations are conditioned on hypothetical knowledge of the dementia state. Among rounding and imprecise-probability respondents, our elicitation yields two measures: an initial rounded or approximated response and a post-probe response, which we interpret as the respondent's true point or interval probability. We study the mapping between the two measures and find that respondents initially tend to over-report small probabilities and under-report large probabilities. Using a specific framework for study of LTC insurance choice with uncertain dementia state, we illustrate the dangers of ignoring imprecise or rounded probabilities for modeling and prediction of insurance demand.

This special issue contains a selection of articles from the Second International Symposium on Imprecise Probabilities and Their Applications (ISIPTA ’01) [4]. The symposium was held at Cornell University, USA, in June 2001, and was chaired by Gert de Cooman, Terrence L. Fine, and Teddy Seidenfeld. It was a successor of the first successful ISIPTA meeting [3]. This was held in 1999, and was intended to promote interaction and communication between researchers in a great diversity of fields but with common interests in imprecise probabilities. ISIPTA ’01 was no exception in this respect: the 47 papers presented there covered a wide and heterogeneous range of topics, such as algorithms, belief change, civil engineering, pattern recognition, coherence, combination of uncertainties, conditioning, credal nets, decision making, economics and finance, elicitation, foundations, independence, mathematical models for uncertainty, probabilistic logic, reliability, and statistical inference. This heterogeneity as well as lively discussions during the meeting, turned ISIPTA ’01 into successful attempt to bring together an interdisciplinary community of researchers actively involved in advancing and promoting a generalized view of probability theories. Shortly after ISIPTA ’01, in February 2002, a few active members in this community decided to found SIPTA, the Society for Imprecise Probability: Theories and Applications. This society manages the organization of the ISIPTA meetings, nowadays a well-established international forum for discussing imprecise probabilities. And, being aware that the field of imprecise probabilities is not always easily accessible, due to its heterogeneity and the related variety of different Flanguages_ used to talk about imprecise probabilities, the Society has also recently started organizing the SIPTA Schools in Imprecise Probabilities, the next of which is due to be held in July 2006 in Madrid, Spain. At this point it is useful to be more precise about the subject of the ISIPTA meetings, namely, imprecise probabilities. The name Fimprecise probability_ was intended from the beginnings in a very wide sense, as a generic term for the many mathematical or statistical models that measure chance or uncertainty without sharp numerical probabilities. These models can be qualitative, such as comparative probability models and partial preference orderings, or quantitative, such as interval probabilities, belief functions, upper and lower expectations or previsions, just to name a few. More important, though, is that imprecise probability models are needed in inference problems characterized by scarce, vague or conflicting information. These

Reports an error in "Bridging the gap between subjective probability and probability judgments: The quantum sequential sampler" by Jiaqi Huang, Jerome R. Busemeyer, Zo Ebelt and Emmanuel M. Pothos (Psychological Review, 2025[Jul], Vol 132[4], 916-955; see record 2025-25219-001). The authors noticed a typo in Equation 17. In the matrix shown, the entry in the fourth row should be "(0 0 β+ … . .)" instead of "(0 β+ … . .)". A zero is missing from the fourth row. The online version of this article has been corrected. (The following abstract of the original article appeared in record 2025-25219-001.) One of the most important challenges in decision theory has been how to reconcile the normative expectations from Bayesian theory with the apparent fallacies that are common in probabilistic reasoning. Recently, Bayesian models have been driven by the insight that apparent fallacies are due to sampling errors or biases in estimating (Bayesian) probabilities. An alternative way to explain apparent fallacies is by invoking different probability rules, specifically the probability rules from quantum theory. Arguably, quantum cognitive models offer a more unified explanation for a large body of findings, problematic from a baseline classical perspective. This work addresses two major corresponding theoretical challenges: first, a framework is needed which incorporates both Bayesian and quantum influences, recognizing the fact that there is evidence for both in human behavior. Second, there is empirical evidence which goes beyond any current Bayesian and quantum model. We develop a model for probabilistic reasoning, seamlessly integrating both Bayesian and quantum models of reasoning and augmented by a sequential sampling process, which maps subjective probabilistic estimates to observable responses. Our model, called the Quantum Sequential Sampler, is compared to the currently leading Bayesian model, the Bayesian Sampler (J. Zhu et al., 2020) using a new experiment, producing one of the largest data sets in probabilistic reasoning to this day. The Quantum Sequential Sampler embodies several new components, which we argue offer a more theoretically accurate approach to probabilistic reasoning. Moreover, our empirical tests revealed a new, surprising systematic overestimation of probabilities. (PsycInfo Database Record (c) 2026 APA, all rights reserved).

R. Lucas's Research Program: Far Too Narrow with Too Many Special Assumptions

One of the major challenges for product lifecycle management systems is the lack of integrated decision support tools to help decision-making with available information in collaborative enterprise networks. Uncertainties are inherent in such networks due to lack of perfect knowledge or conflicting information. In this paper, a robust decision support approach based on imprecise probabilities is proposed. Robust Bayesian belief networks with interval probabilities are used to estimate imprecise posterior probabilities in probabilistic inference. This generic approach is demonstrated with decision-makings in design for closed-loop supply chain. The ultimate goal of robust intelligent decision support systems is to enhance the effective use of information available in collaborative engineering environments.

Two studies examined the relationship between memory recall and subjective probability judgement in an attempt to understand the mechanisms involved in depressive future thinking. In study 1, the relationship between general impression memory (GIM) latencies and subjective probability, and the relationship between specific event memory (SEM) latencies and subjective probability, were examined in normal mood individuals. Results demonstrated that GIM latencies were correlated with subjective probability. That is, participants who were faster to say they had experienced events in the past also thought themselves more likely to experience similar events in the future. In contrast, SEMs were not significantly correlated with subjective probability judgements. Study 2 tested the prediction that the same mechanism is operative in depression, but in this instance it is differential accessibility of negative memories which biases judgements for future events. As predicted, depressed patients, relative to controls, recalled more negative events and rated negative events as more likely to happen to them in the future. However, subjective probability judgements were not associated with the recall of either GIMs or SEMs in depressed patients, but GIMs were correlated with subjective probability judgements in the control participants. Copyright © 2000 John Wiley & Sons, Ltd.

Since engineering design requires decision making under uncertainty, the degree to which good decisions can be made depends upon the degree to which the decision maker has expressive and accurate representations of his or her uncertain beliefs. Whereas traditional decision analysis uses precise probability distributions to represent uncertain beliefs, recent research has examined the effects of relaxing this assumption of precision. A specific example of this is the theory of imprecise probability. Imprecise probabilities are more expressive than precise probabilities, but they are also more computationally expensive to propagate through mathematical models. The probability box (p-box) is an alternative representation that is both more expressive than precise probabilities, and less computationally expensive than general imprecise probabilities. In this paper, we introduce a method for propagating p-boxes through black box models. Based on two example models, a new method, called p-box convolution sampling (PCS), is compared with three other p-box propagation methods. It is found that, although PCS is less expensive than the alternatives, it is still relatively expensive and therefore only justifiable when the expected benefits are large. Several directions for further improving the efficiency of PCS are discussed.

The main goal of information fusion can be seen as improving human or automatic decision-making by exploiting diversities in information from multiple sources. High-level information fusion aims specifically at decision support regarding situations, often expressed as “achieving situation awareness”. A crucial issue for decision making based on such support is trust that can be defined as “accepted dependence”, where dependence or dependability is an overall term for many other concepts, e.g., reliability. This position paper reports on ongoing and planned research concerning imprecise probability as an approach to improved dependability in high-level information fusion. We elaborate on high-level information fusion from a generic perspective and a partial mapping from a taxonomy of dependability to high-level information fusion is presented. Three application domains: defense, manufacturing, and precision agriculture, where experiments are planned to be implemented are depicted. We conclude that high-level information fusion as an application-oriented research area, where precise probability (Bayesian theory) is commonly adopted, provides an excellent evaluation ground for imprecise probability.

The SIPTA (Society for Imprecise Probability,Theory and Application) view of Keynes’s contributions to imprecise probability and application form a one-to-one, onto mapping from the claims of Henry E. Kyburg to what is accepted as being what Keynes’s contribution was. Thus, if one is familiar with Kyburg’s assessment of Keynes’s contributions, then one also knows what SIPTA’s views of Keynes’s contributions are. There is an extremely severe problem here, however, because Kyburg never read beyond chapter 6 in Keynes’s A Treatise on Probability. Kyburg was, therefore, ignorant about the many technical and mathematical contributions made to the theory of imprecise probability by Keynes, such as upper and lower probabilities, non additive probabilities,sub additive probabilities, interval valued estimates, decision weights, etc, in Parts II, III, IV and V of the A Treatise on Probability. Non additive probability appears in Keynes’s applications of non additive probability in his analysis of uncertainty in his theory of the liquidity preference approach to the rate of interest in the General Theory, where uncertainty was defined as an inverse function of the evidential weight of the argument in footnote 1 on page 148 of chapter 12. Keynes greatly emphasized the relative importance of his 'non-numerical' (interval) probabilities and weight of argument in his correspondence with Townshend in 1937-38. Kyburg’s view of Keynes’s contribution was to argue, based on Keynes’s verbal discussions on pp.30-34 of chapter III of Keynes’s A Treatise on Probability, that Keynes had been the first to (a) define a partial order and (b) the first to argue that all probabilities were not comparable and measurable. Kyburg also acknowledged Keynes’s contribution in chapter 6 concerning the weight of the argument as it related to the Bayesian refusal to give any importance to questions dealing with the relative strength of the evidence of different probabilities, i.e. whether the evidentiary base supporting a probability estimate was composed of strong evidence or made up of weak evidence. We will conclude that the above assessment made by Kyburg is identical to the SIPTA assessment made from 1999 (Kyburg’s 1999 assessment for SIPTA) to the last SIPTA meeting in 2019.

Adam Smith as an Example of Samuelson’s 1952 Point That Mathematics Can Be Written Out in the English Language: On Smith’s Anti-Utilitarianism Based on the Differences Between Precise and Imprecise Probability Presented in 1776 in the Wealth of Nations

Sensitivity analysis measures how changes in system inputs affect outputs. Previously, a large amount of sensitivity analysis research was relevant to the precise probability that is regarded as an ideal condition of engineering. Due to insufficient test samples and the low accuracy of test data, system reliability with hybrid uncertainty is difficult to be described as a precise value. As a profusion of highly integrated electromechanical equipment is applied in modern life, it is impossible to apply sufficient resources to eliminate the stochastic property of every component, which necessitates the identification of highly sensitive components to efficiently reduce imprecision. Hence, based on the theory of imprecise probability, imprecise sensitivity analysis has become a popular research topic in the last decade. In this paper, a method for uncertain system reliability and imprecise sensitivity analysis is proposed based on a Bayesian network, a probability box and the pinching method. The feasibility and accuracy of the combined method are fully verified through the evaluation and analysis of a numerical example and a case study of an electromechanical system, and the highly sensitive components that heavily influence the imprecision of system outputs are accurately identified.

Introduction

Bayesian approaches have been demonstrated as effective methods for reliability analysis of complex systems with small-amount data, which integrate prior information and sample data using Bayes’ theorem. However, there is an assumption that precise prior probability distributions are available for unknown parameters, yet these prior distributions are sometimes unavailable in practical engineering. A possible way to avoiding this assumption is to generalize Bayesian reliability analysis approach by using imprecise probability theory. In this paper, we adopt a set of imprecise Dirichlet distributions as priors to quantify uncertainty of unknown parameters and extend traditional Bayesian reliability analysis approach by introducing an imprecise Dirichlet model (IDM). When the prior information is rare, the result of imprecise Bayesian analysis method is too rough to support engineering decision-making, so we proposed an optimization model to reduce the imprecision of the new method. Spindles are crucial for machine tools and reliability data related to spindles of new-developed machine tools are often rare. We can then use the imprecise Bayesian reliability analysis method to assess its reliability. In this paper, we mainly investigate the reliability assessment of a motorized spindle to illustrate the effectiveness of the proposed method.

Psychological theories of subjective probability judgments assume that accumulated evidence (support) mediates the relation between the description of a to-be-judged event (hypothesis) and the judgment. To scale a probability from the support for a hypothesis, these psychological theories make a strong independence assumption. This assumption is stated in the form of the product rule, in which the support garnered for a particular hypothesis is independent of the support for the alternative hypothesis. In the study reported here, I asked participants to judge the likelihood of a bicyclist winning a simulated race. Results showed that the independence assumption was systematically violated. Observed judgments suggest that when a probability judgment is made, the comparability or similarity of the hypotheses on one dimension increases the weight that judges allocate to differences on the other dimensions. These results speak against the simple scalability-processing assumption of support theory, and they illustrate the need for a theory of judgment processes that describes how the similarity between hypotheses shapes judgment.