The Bayesian model is used in psychology as the reference for the study of dynamic probability judgment. The main limit induced by this model is that it confines the study of revision of degrees of belief to the sole situations of revision in which the universe is static (revising situations). However, it may happen that individuals have to revise their degrees of belief when the message they learn specifies a change of direction in the universe, which is considered (...) as changing with time (updating situations). We analyze the main results of the experimental literature with regard to elementary qualitative properties of these two situations of revision. First, the order effect phenomenon is confronted with the commutative property. Second, an apparent new phenomenon is presented: the redundancy effect that is confronted with the idempotence property. Finally, results obtained in this kind of experimental situations are reinterpreted in the light of pragmatic analysis. (shrink)

Isaac Levi and I have different views of probability and decision making. Here, without addressing the merits, I will try to answer some questions recently asked by Levi (1985) about what my view is, and how it relates to his.

Tversky and Koehler's support theory attempts to explain why probability judgments are affected by the manner in which formally similar events are described. Support theory suggests that as the explicitness of a description increases, an event will be judged to be more likely. In the present experiment, experienced decision-makers from large, international accounting firms were given case-specific information about an audit client and asked to provide a series of judgments regarding the perceived likelihood of events. Unpacking a hypothesis into (...) four formally equivalent hypotheses more than doubled its perceived likelihood. These results are largely consistent with support theory, extending its generalizability to applied contexts involving business professionals. Further, the paper finds that task-specific factors, a previously untested area, can also affect support theory interpretations. (shrink)

Bayesians often confuse insistence that probability judgment ought to be indeterminate (which is incompatible with Bayesian ideals) with recognition of the presence of imprecision in the determination or measurement of personal probabilities (which is compatible with these ideals). The confusion is discussed and illustrated by remarks in a recent essay by R. C. Jeffrey.

We advance a model of human probability judgment and apply it to the design of an extrapolation algorithm. Such an algorithm examines a person's judgment about the likelihood of various statements and is then able to predict the same person's judgments about new statements. The algorithm is tested against judgments produced by thirty undergraduates asked to assign probabilities to statements about mammals.

We studied how people attribute action outcomes to their own actions under conditions of uncertainty. Participants chose between left and right keypresses to produce an action effect , while a computer player made a simultaneous keypress decision. In each trial, a random generator determined which of the players controlled the action effect at varying probabilities, and participants then judged which player had produced it. Participants’ effect control ranged from 20% to 80%, varied blockwise, and they could use trial-by-trial feedback to (...) optimize the accuracy of their agency judgments. Participants tended to attribute action effects to themselves , probably reflecting a rational guessing strategy of always naming the more likely player. However, participants systematically neglected information favoring the computer player as the agent, even under conditions where this bias could only harm judgment accuracy. We conclude that agency biases have both rational and irrational components. (shrink)

Theories of subjective probability are viewed as formal languages for analyzing evidence and expressing degrees of belief. This article focuses on two probability langauges, the Bayesian language and the language of belief functions (Shafer, 1976). We describe and compare the semantics (i.e., the meaning of the scale) and the syntax (i.e., the formal calculus) of these languages. We also investigate some of the designs for probability judgment afforded by the two languages.

Empirical studies have demonstrated that uncertainty about event probabilities, also known as ambiguity or second-order uncertainty, can affect decision makers’ choice preferences. Despite the importance of second-order uncertainty in decision making, almost no effort has been directed towards the development of methods that evaluate the accuracy of second-order probabilities. In this paper, we describe conditions under which strictly proper scoring rules can be used to assess the accuracy of second-order probability judgments. We investigate the effectiveness of using a particular (...) strictly proper scoring rule the ranked probability score - to discourage biased assessments of second-order uncertainty. (shrink)

The purpose of the study was to investigate how numeracy and acuity of the approximate number system (ANS) relate to the calibration and coherence of probability judgments. Based on the literature on number cognition, a first hypothesis was that those with lower numeracy would maintain a less linear use of the probability scale, contributing to overconfidence and nonlinear calibration curves. A second hypothesis was that also poorer acuity of the ANS would be associated with overconfidence and non-linearity. A (...) third hypothesis, in line with dual-systems theory (e.g., Kahneman and Frederick, 2002 ) was that people higher in numeracy should have better access to the normative probability rules, allowing them to decrease the rate of conjunction fallacies. Data from 213 participants sampled from the Swedish population showed that: (i) in line with the first hypothesis, overconfidence and the linearity of the calibration curves were related to numeracy, where people higher in numeracy were well calibrated with zero overconfidence. (ii) ANS was not associated with overconfidence and non-linearity, disconfirming the second hypothesis. (iii) The rate of conjunction fallacies was slightly, but to a statistically significant degree decreased by numeracy, but still high at all numeracy levels. An unexpected finding was that participants with better ANS acuity gave more realistic estimates of their performance relative to others. (shrink)

Research suggests that moral evaluations change during adulthood. Older adults (75+) tend to judge accidentally harmful acts more severely than younger adults do, and this age‐related difference is in part due to the greater negligence older adults attribute to the accidental harmdoers. Across two studies (N = 254), we find support for this claim and report the novel discovery that older adults’ increased attribution of negligence, in turn, is associated with a higher perceived likelihood that the accident would occur. We (...) propose that, because older adults perceive accidents as more likely than younger adults do, they condemn the agents and their actions more and even infer that the agents’ omission to exercise due care is intentional. These findings refine our understanding of the cognitive processes underpinning moral judgment in older adulthood and highlight the role of subjective probability judgments in negligence attribution. (shrink)

Consider the task of predicting which soccer team will win the next World Cup. The bookmakers may judge Brazil to be the team most likely to win, but also judge it most likely that a European rather than a Latin American team will win. This is an example of a non-aligned hierarchy structure: the most probable event at the subordinate level (Brazil wins) appears to be inconsistent with the most probable event at the superordinate level (a European team wins). In (...) this paper we exploit such structures to investigate how people make predictions based on uncertain hierarchical knowledge. We distinguish between aligned and non-aligned environments, and conjecture that people assume alignment. Participants were exposed to a non-aligned training set in which the most probable superordinate category predicted one outcome, whereas the most probable subordinate category predicted a different outcome. In the test phase participants allowed their initial probability judgments about category membership to shift their final ratings of the probability of the outcome, even though all judgments were made on the basis of the same statistical data. In effect people were primed to focus on the most likely path in an inference tree, and neglect alternative paths. These results highlight the importance of the level at which statistical data are represented, and suggest that when faced with hierarchical inference problems people adopt a simplifying heuristic that assumes alignment. (shrink)

Probabilistic reasoning is heavily investigated in decision research. Violations of probability theory have been demonstrated numerously, for instance, the tendency to overestimate the joint probab...

We investigate Scottish end users' and professional forecasters' risk perception in relation to the 5-point European Avalanche Danger Scale by eliciting numerical estimates of the probability of triggering an avalanche. Our main findings are that neither end users nor professional forecasters interpret the avalanche danger scale as intended, that is, in an exponential fashion. Second, we find that numerical interpretations by end users and professional forecasters have high variance, but are similar, in that both groups tend to overestimate the (...) probability of triggering an avalanche and underestimate the relative risk increase. Finally, we find significant differences in the perceived probability of triggering an avalanche relative to a low or moderate avalanche danger level, and in the numerical interpretation of verbal probability terms depending on whether respondents provide their estimates using a frequency or a percentage chance format. We summarize our findings by identifying important lessons to improve avalanche risk understanding and its communication. (shrink)

Richard Jeffrey is beyond dispute one of the most distinguished and influential philosophers working in the field of decision theory and the theory of knowledge. His work is distinctive in showing the interplay of epistemological concerns with probability and utility theory. Not only has he made use of standard probabilistic and decision theoretic tools to clarify concepts of evidential support and informed choice, he has also proposed significant modifications of the standard Bayesian position in order that it provide a (...) better fit with actual human experience. Probability logic is viewed not as a source of judgment but as a framework for explaining the implications of probabilistic judgments and their mutual compatability. This collection of essays spans a period of some 35 years and includes what have become some of the classic works in the literature. There is also one completely new piece, while in many instances Jeffrey includes afterthoughts on the older essays. (shrink)

Explanation and prediction play an important role in medical decision making, particularly for diagnostic and treatment decisions. For the most part, explanations as well as predictions are derived from causal knowledge and have to be made under uncertainty. In cognitive psychology, these phenomena have been approached from two directions. On the one hand, there is research on knowledge representation and inferential reasoning (Holland et al. 1986; Anderson 1990). On the other hand, there is research on heuristics and biases in judgments (...) under uncertainty (Kahneman et al. 1982; Rehm & Gadenne 1990). However, no exact models have yet been developed to describe the process that leads from a given knowledge base to judgments of probabilities, i.e., from causal knowledge about the world to subjective expressions of the uncertainty associated with explanations or predictions. (shrink)

We describe a computational model of two central aspects of people's probabilistic reasoning: descriptive probability estimation and inferential probability judgment. This model assumes that people's reasoning follows standard frequentist probability theory, but it is subject to random noise. This random noise has a regressive effect in descriptive probability estimation, moving probability estimates away from normative probabilities and toward the center of the probability scale. This random noise has an anti-regressive effect in inferential judgement, (...) however. These regressive and anti-regressive effects explain various reliable and systematic biases seen in people's descriptive probability estimation and inferential probability judgment. This model predicts that these contrary effects will tend to cancel out in tasks that involve both descriptive estimation and inferential judgement, leading to unbiased responses in those tasks. We test this model by applying it to one such task, described by Gallistel et al. ). Participants' median responses in this task were unbiased, agreeing with normative probability theory over the full range of responses. Our model captures the pattern of unbiased responses in this task, while simultaneously explaining systematic biases away from normatively correct probabilities seen in other tasks. (shrink)

Nonmonotonic reasoning is often claimed to mimic human common sense reasoning. Only a few studies, though, investigated this claim empirically. In the present paper four psychological experiments are reported, that investigate three rules of system p, namely the and, the left logical equivalence, and the or rule. The actual inferences of the subjects are compared with the coherent normative upper and lower probability bounds derived from a non-infinitesimal probability semantics of system p. We found a relatively good agreement (...) of human reasoning and principles of nonmonotonic reasoning according to the coherence interpretation of system p. Contrary to the results reported in the “heuristics and biases” tradition, the subjects committed relatively few upper bound violations (conjunction fallacies). More lower than upper bound violations were observed. When the premises were presented in terms of intervals higher mean lower bounds were observed as when the premises were presented in terms of point percentages. (shrink)

Subjects in experiments use real-life strategies that differ significantly from those assumed by experimenters. First, true randomness is rare in both natural and constructed environments. Second, communication follows conventions which depend on the game-theoretic aspects of situations. Third, in the common rhetorical stance of storytelling, people do not tell about the representative but about unusual, exceptional, and rare cases.

Examining the social and political upheavals that characterized the collapse of public judgment in early modern Europe, Liberating Judgment offers a unique account of the achievement of liberal democracy and self-government. The book argues that the work of John Locke instills a civic judgment that avoids the excesses of corrosive skepticism and dogmatic fanaticism, which lead to either political acquiescence or irresolvable conflict. Locke changes the way political power is assessed by replacing deteriorating vocabularies of legitimacy with (...) a new language of justification informed by a conception of probability. For Locke, the coherence and viability of liberal self-government rests not on unassailable principles or institutions, but on the capacity of citizens to embrace probable judgment. The book explores the breakdown of the medieval understanding of knowledge and opinion, and considers how Montaigne's skepticism and Descartes' rationalism--interconnected responses to the crisis--involved a pragmatic submission to absolute rule. Locke endorses this response early on, but moves away from it when he encounters a notion of reasonableness based on probable judgment. In his mature writings, Locke instructs his readers to govern their faculties and intellectual yearnings in accordance with this new standard as well as a vocabulary of justification that might cultivate a self-government of free and equal individuals. The success of Locke's arguments depends upon citizens' willingness to take up the labor of judgment in situations where absolute certainty cannot be achieved. (shrink)

Attributing the stability‐based theory to Hume explains his equation of degree of belief with degree of evidence in his treatment of philosophical probability. In his discussion of the fourth kind of unphilosophical probability, Hume uncovers contradictions that arise from accidental or rash generalizations; his response, that stability can be restored by appeal to higher‐order generalizations or general rules, facilitates his analysis of causation. Hume's first three kinds of unphilosophical probability involve variation in degrees of confidence that parallels (...) variation in moral judgments due to the operation of sympathy. Such variation and fluctuation in sentiment and degree of confidence feel like contradictions in that they give rise to uneasiness or discomfort that we seek to relieve. Hume thus identifies psychological pressures that motivate corrections to variable judgments about probability and virtue and vice, resulting in more stable judgments that have normative standing. (shrink)

A diagnostic judgment of a teacher can be seen as an inference from manifest observable evidence on a student’s behavior to his or her latent traits. This can be described by a Bayesian model of in-ference: The teacher starts from a set of assumptions on the student (hypotheses), with subjective probabilities for each hypothesis (priors). Subsequently, he or she uses observed evidence (stu-dents’ responses to tasks) and knowledge on conditional probabilities of this evidence (likelihoods) to revise these assumptions. Many (...) systematic deviations from this model (biases, e.g. base-rate neglect, inverse fallacy) are reported in literature on Bayesian reasoning. In a teacher’s situation the information (hypotheses, priors, likelihoods) is usually not explicitly represented numerically (as in most research on Bayesian reasoning), but only by qualitative esti-mations in the mind of the teacher. In our study, we ask to which extent individuals (approximately) apply a rational Bayesian strategy or resort on other biased strategies of processing information for their diagnostic judgments. We explicitly pose this question with respect to non-numerical settings. To investigate this question, we developed a scenario that visually displays all relevant information (hypotheses, priors, likelihoods) in a graphically displayed hypothesis space (called “hypothegon”) – without recurring to numerical representations or mathematical procedures. 42 pre-service teach-ers were asked to judge the plausibility of different misconceptions of six students based on their responses to decimal comparison tasks (e.g. 3.39 > 3.4). Applying a Bayesian classification proce-dure, we identified three updating strategies: A Bayesian update strategy (BUS, processing all probabilities), a combined evidence strategy (CES, ignoring the prior probabilities but including all likelihoods), and a single evidence strategy (SES, only using the likelihood of the most probable hypothesis). In study 1, an instruction on the relevance of using all probabilities (priors and likelihoods) only weakly increased the processing of more information. In study 2, we found strong evidence that a visual explication of the prior-likelihood interaction led to an increase of processing the interaction of all relevant information. These results show that the phenomena found in general research on Bayesian reasoning in numerical settings extend to diagnostic judgments in non-numerical settings. (shrink)

Abductive reasoning assigns special status to the explanatory power of a hypothesis. But how do people make explanatory judgments? Our study clarifies this issue by asking: How does the explanatory power of a hypothesis cohere with other cognitive factors? How does probabilistic information affect explanatory judgments? In order to answer these questions, we conducted an experiment with 671 participants. Their task was to make judgments about a potentially explanatory hypothesis and its cognitive virtues. In the responses, we isolated three constructs: (...) Explanatory Value, Rational Acceptability, and Entailment. Explanatory judgments strongly cohered with judgments of causal relevance and with a sense of understanding. Furthermore, we found that Explanatory Value was sensitive to manipulations of statistical relevance relations between hypothesis and evidence, but not to explicit information about the prior probability of the hypothesis. These results indicate that probabilistic information about statistical relevance is a strong determinant of Explanatory Value. More generally, our study suggests that abductive and probabilistic reasoning are two distinct modes of inference. (shrink)

We propose a theory that relates perceived evidence to numerical probability judgment. The most successful prior account of this relation is Support Theory, advanced in Tversky and Koehler. Support Theory, however, implies additive probability estimates for binary partitions. In contrast, superadditivity has been documented in Macchi, Osherson, and Krantz, and both sub- and superadditivity appear in the experiments reported here. Nonadditivity suggests asymmetry in the processing of focal and nonfocal hypotheses, even within binary partitions. We extend Support (...) Theory by revising its basic equation to allow such asymmetry, and compare the two equations’ ability to predict numerical assessments of probability from scaled estimates of evidence for and against a given proposition. Both between- and within-subject experimental designs are employed for this purpose. We find that the revised equation is more accurate than the original Support Theory equation. The implications of asymmetric processing on qualitative assessments of chance are also briefly discussed. (shrink)