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.     

Alleviating inflation of conditional predictions

Previous studies indicated that conditional predictions—the assessed probability that a certain outcome will occur given a certain condition—tend to be markedly inflated. Five experiments tested the effects of manipulations that were expected to alleviate this inflation by inducing participants to engage in analytic processing. Rewarding participants for accurate predictions proved ineffective. A training procedure in which participants assessed the likelihood of each of several outcomes before assessing the probability of a target outcome was partly effective in reducing overestimation. Most effective was the requirement to work in dyads and to come to an agreement about the assessed likelihood. Working in dyads helped alleviate prediction inflation even after participants made their individual predictions alone, and its debiasing effect also transferred to the estimates that were made individually on a new set of stimuli. The results were discussed in terms of the factors that make prediction inflation resistant to change.     

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.     

On Holland's Dutch identity conjecture

The manifest probabilities of observed examinee response patterns resulting from marginalization with respect to the latent ability distribution produce the marginal likelihood function in item response theory. Under the conditions that the posterior distribution of examinee ability given some test response pattern is normal and the item logit functions are linear, Holland (1990a) gives a quadratic form for the log-manifest probabilities by using the Dutch Identity. Further, Holland conjectures that this special quadratic form is a limiting one for all smooth unidimensional item response models as test length tends to infinity. The purpose of this paper is to give three counterexamples to demonstrate that Holland's Dutch Identity conjecture does not hold in general. The counterexamples suggest that only under strong assumptions can it be true that the limits of log-manifest probabilities are quadratic. Three propositions giving sets of such strong conditions are given.     

Ordinal data,ordered scale points,and order statistics

This paper concerns ordinal responses. An ordered Dirichlet distribution describes prior and posterior beliefs about the cumulative probabilities of response categories. Associating the response categories with intervals of a latent random variable then induces a distribution on the order statistics of that variable. The psychometrician can use the asymptotic theory of order statistics to learn how distributional assumptions about the latent variable effect inference. An example relates the skewness of a latent variable to the proportional odds and proportional hazards models of McCullagh [1980].     

What is learned from experience in a probabilistic environment?

Three experiments explored what is learned from experience in a probabilistic environment. The task was a simulated medical decision‐making task with each patient having one of two test results and one of two diseases. The test result was highly predictive of the disease for all participants. The base rate of the test result was varied between participants to produce different inverse conditional probabilities of the test result given the disease across conditions. Participants trained using feedback to predict a patient's disease from a test result showed the classic confusion of the inverse error, substituting the forward conditional probability for the inverse conditional probability when tested on it. Additional training on the base rate of the test result did little to improve performance. Training on the joint probabilities, however, produced good performance on either conditional probability. The pattern of results demonstrated that experience with the environment is not always sufficient for good performance. That natural sampling leads to good performance was not supported. Further, because participants not trained on joint probabilities did, however, know them but still committed the confusion of the inverse error, the hypothesis that having joint probabilities would facilitate performance was not supported. The pattern of results supported the conclusion that people learn all the necessary information from experience in a probabilistic environment, but depending upon what the experience was, it may interfere with their ability to recall to memory the appropriate sample set necessary for estimating or using the inverse conditional probability. Copyright © 2004 John Wiley & Sons, Ltd.