An elicitation method for multiple linear regression models

This paper describes a method of quantifying subjective opinion about a normal linear regression model. Opinion about the regression coefficients and experimental error is elicited and modeled by a multivariate probability distribution (a Bayesian conjugate prior distribution). The distribution model is richly parameterized and various assessment tasks are used to estimate its parameters. These tasks include the revision of opinion in the light of hypothetical data, the assessment of credible intervals, and a task commonly performed in cue-weighting experiments. A new assessment task is also introduced. In addition, implementation of the method in an interactive computer program is described and the method is illustrated with a practical example.     

Nonmonotonic conditionals that behave like conditional probabilities above a threshold

I'll describe a range of systems for nonmonotonic conditionals that behave like conditional probabilities above a threshold. The rules that govern each system are probabilistically sound in that each rule holds when the conditionals are interpreted as conditional probabilities above a threshold level specific to that system. The well-known preferential and rational consequence relations turn out to be special cases in which the threshold level is 1. I'll describe systems that employ weaker rules appropriate to thresholds lower than 1, and compare them to these two standard systems.     

Naive Probability: Model‐Based Estimates of Unique Events

We describe a dual‐process theory of how individuals estimate the probabilities of unique events, such as Hillary Clinton becoming U.S. President. It postulates that uncertainty is a guide to improbability. In its computer implementation, an intuitive system 1 simulates evidence in mental models and forms analog non‐numerical representations of the magnitude of degrees of belief. This system has minimal computational power and combines evidence using a small repertoire of primitive operations. It resolves the uncertainty of divergent evidence for single events, for conjunctions of events, and for inclusive disjunctions of events, by taking a primitive average of non‐numerical probabilities. It computes conditional probabilities in a tractable way, treating the given event as evidence that may be relevant to the probability of the dependent event. A deliberative system 2 maps the resulting representations into numerical probabilities. With access to working memory, it carries out arithmetical operations in combining numerical estimates. Experiments corroborated the theory's predictions. Participants concurred in estimates of real possibilities. They violated the complete joint probability distribution in the predicted ways, when they made estimates about conjunctions: P(A), P(B), P(A and B), disjunctions: P(A), P(B), P(A or B or both), and conditional probabilities P(A), P(B), P(B|A). They were faster to estimate the probabilities of compound propositions when they had already estimated the probabilities of each of their components. We discuss the implications of these results for theories of probabilistic reasoning.     

Context affects the interpretation of low but not high numerical probabilities: A hypothesis testing account of subjective probability

Low numerical probabilities tend to be directionally ambiguous, meaning they can be interpreted either positively, suggesting the occurrence of the target event, or negatively, suggesting its non-occurrence. High numerical probabilities, however, are typically interpreted positively. We argue that the greater directional ambiguity of low numerical probabilities may make them more susceptible than high probabilities to contextual influences. Results from five experiments supported this premise, with perceived base rate affecting the interpretation of an event’s numerical posterior probability more when it was low than high. The effect is consistent with a confirmatory hypothesis testing process, with the relevant perceived base rate suggesting the directional hypothesis which people then test in a confirmatory manner.     

Temporal delays can facilitate causal attribution: Towards a general timeframe bias in causal induction

Does reasoning occur on the Wason selection task, or are card selections determined purely on the basis of heuristic processes? To answer this question two relevance-based theories of reasoning are compared: (1) the theory of Evans (1984, 1989; Evans &; Over, 1996), which takes the heuristic viewpoint, and (2) the theory of Sperber, Cara, and Girotto (1995), which takes the reasoning viewpoint. In three experiments, the effect of removing matching cards from the selection task array is examined. It is argued that the Sperber et al. theory makes clearer predictions about the results of these manipulations, which are confirmed, and that the Evans theory can only accommodate them if it allows the operation of reasoning processes. The results are also discussed in relation to Roth's (1979) account of the selection task, mental models theory, and information gain theory.     

Semantic coherence and fallacies in estimating joint probabilities

In three experiments on joint probability estimation, gist representations were manipulated with analogies, and the suboptimal strategy of ignoring relevant denominators was counteracted with training in using 2 × 2 tables to clarify joint probability estimates. The estimated probabilities of two events, as well as their conjunctive and disjunctive probabilities, were assessed against two benchmarks, logical fallacies and semantic coherence—a constellation of estimates consistent with the relationship among sets. Fuzzy‐trace theory (FTT) predicts that analogies will increase semantic coherence, and a table intervention affecting denominator neglect will both increase semantic coherence and reduce fallacies. In all three experiments, analogies increased semantic coherence. In both experiments training participants to use 2 × 2 tables, such tables reduced fallacies and increased semantic coherence. As the relations among sets in the problem materials progressed in cognitive complexity from identical sets, mutually exclusive sets, and subsets to overlapping sets, fallacies generally increased, and semantic coherence generally decreased. These findings indicate that denominator neglect is pervasive, but that it can be remedied with a straightforward intervention that clarifies relations among sets. Further, intuitive gist‐based probability estimation can be improved through the use of simple analogies. Copyright © 2009 John Wiley & Sons, Ltd.     

Single-case probabilities and the case of Monty Hall: Levy’s view

In Baumann (American Philosophical Quarterly 42: 71–79, 2005) I argued that reflections on a variation of the Monty Hall problem throws a very general skeptical light on the idea of single-case probabilities. Levy (Synthese, forthcoming, 2007) puts forward some interesting objections which I answer here.     

Verbal probabilities: a question of frame?

Verbal expressions of probability and uncertainty are of two kinds: positive (‘probable’, ‘possible’) and negative (‘not certain’, ‘doubtful’). Choice of term has implications for predictions and decisions. The present studies show that positive phrases are rated to be more optimistic (when the target outcome is positive), and more correct, when the target outcome actually occurs, even in cases where positive and negative phrases are perceived to convey the same probabilities (Experiments 1 and 2). Selection of phrase can be determined by linguistic frame. Positive quantifiers (‘some’, ‘several’) support positive probability phrases, whereas negative quantifiers (‘not all’) suggest negative phrases (Experiment 3). Positive frames induced by numeric frequencies (e.g. the number of students to be admitted) imply positive probability phrases, whereas negative frames (e.g. the number of students to be rejected) call for negative probability phrases (Experiment 4). It is concluded that choice of verbal phrase is based not only on level of probability, but also on situational and linguistic cues. Copyright © 2002 John Wiley & Sons, Ltd.     

A short proof of a theorem of Falmagne

Falmagne (J. Math. Psychol. 18 (1978) 52) proved the sufficiency of the Block-Marschak inequalities (in: I. Olkin, S. Ghurye, W. Hoefding, W. Madow, H. Mann (Eds.), Contributions to Probability and Statistics, Stanford University Press, Stanford, CA, 1960, pp. 97-132) and normalization equalities for a complete system of choice probabilities to be induced by rankings. Here, we give a considerably shorter proof of this result. Our approach combines Möbius inversion and network flows.