Partition-edit-count: naive extensional reasoning in judgment of conditional probability

The authors provide evidence that people typically evaluate conditional probabilities by subjectively partitioning the sample space into n interchangeable events, editing out events that can be eliminated on the basis of conditioning information, counting remaining events, then reporting probabilities as a ratio of the number of focal to total events. Participants' responses to conditional probability problems were influenced by irrelevant information (Study 1), small variations in problem wording (Study 2), and grouping of events (Study 3), as predicted by the partition-edit-count model. Informal protocol analysis also supports the authors' interpretation. A 4th study extends this account from situations where events are treated as interchangeable (chance and ignorance) to situations where participants have information they can use to distinguish among events (uncertainty).     

Shafer on conditional probability

Journal of Philosophical Logic -     

Conditionals and conditional probability

The authors report 3 experiments in which participants were invited to judge the probability of statements of the form if p then q given frequency information about the cases pq, p not q, not pq, and not p not q (where not = not). Three hypotheses were compared: (a) that people equate the probability with that of the material conditional, 1 - P(p not q); (b) that people assign the conditional probability, P(q/p); and (c) that people assign the conjunctive probability P(pq). The experimental evidence allowed rejection of the 1st hypothesis but provided some support for the 2nd and 3rd hypotheses. Individual difference analyses showed that half of the participants used conditional probability and that most of the remaining participants used conjunctive probability as the basis of their judgments.     

Coherent probability from incoherent judgment

People often have knowledge about the chances of events but are unable to express their knowledge in the form of coherent probabilities. This study proposed to correct incoherent judgment via an optimization procedure that seeks the (coherent) probability distribution nearest to a judge's estimates of chance. This method was applied to the chances of simple and complex meteorological events, as estimated by college undergraduates. No judge responded coherently, but the optimization method found close (coherent) approximations to their estimates. Moreover, the approximations were reliably more accurate than the original estimates, as measured by the quadratic scoring rule. Methods for correcting incoherence facilitate the analysis of expected utility and allow human judgment to be more easily exploited in the construction of expert systems.     

Languages and designs for probability judgment

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.     

Representativeness and fallacies of probability judgment

Representativeness is the name given to the heuristic people often employ when they judge the probability of a sample by how well it represents certain salient features of the population from which it was drawn. The representativeness heuristic has also been used to account for how people judge the probability that a given population is the source of some sample. The latter probability, however, depends on other factors (e.g., the population's prior probability) as well as on the sample characteristics. A review of existing evidence suggests that the ignoring of such factors, a central finding of the heuristics approach to judgment under uncertainty, is a phenomenon which is conceptually distinct from the representativeness heuristic. These factors (base rates, sample size, and predictability) do not always exert the proper influence on people's first-order probability judgments, but they are not ignored when people make second-order (i.e., confidence) judgments. Other fallacies and biases in subjective evaluations of probability are, however, direct causal results of the employment of representativeness. For example, representativeness may be applied to the wrong features. Most devastating, perhaps, is that subjective probability judgments obey a logic of representativeness judgments, even though probability ought to obey an altogether different logic. Yet although the role of representativeness judgments in probability estimation leaves a lot to be desired, it is hard to envision prediction and inference completely unaided by representativeness.     

The influence of hierarchy on probability judgment

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.     

Using the conditional grade-of-membership model to assess judgment accuracy

Consider the case whereJ instruments are used to classify each ofI objects relative toK nominal categories. The conditional grade-of-membership (GoM) model provides a method of estimating the classification probabilities of each instrument (or judge) when the objects being classified consist of both pure types that lie exclusively in one ofK nominal categories, and mixtures that lie in more than one category. Classification probabilities are identifiable whenever the sample of GoM vectors includes pure types from each category. When additional, relatively mild, assumptions are made about judgment accuracy, the identifiable correct classification probabilities are the greatest lower bounds among all solutions that might correspond to the observed multinomial process, even when the unobserved GoM vectors do not include pure types from each category. Estimation using the conditional GoM model is illustrated on a simulated data set. Further simulations show that the estimates of the classification probabilities are relatively accurate, even when the sample contains only a small percentage of approximately pure objects.The authors thank Max A. Woodbury, Kenneth G. Manton and H. Dennis Tolley for their help and four anonymous Psychometrika reviewers (including an associate editor) for their beneficial expository and technical suggestions. This work was supported by the Dean's Fund for Summer Research, Owen Graduate School of Management, Vanderbilt University.     

A quantum theoretical explanation for probability judgment errors

A quantum probability model is introduced and used to explain human probability judgment errors including the conjunction and disjunction fallacies, averaging effects, unpacking effects, and order effects on inference. On the one hand, quantum theory is similar to other categorization and memory models of cognition in that it relies on vector spaces defined by features and similarities between vectors to determine probability judgments. On the other hand, quantum probability theory is a generalization of Bayesian probability theory because it is based on a set of (von Neumann) axioms that relax some of the classic (Kolmogorov) axioms. The quantum model is compared and contrasted with other competing explanations for these judgment errors, including the anchoring and adjustment model for probability judgments. In the quantum model, a new fundamental concept in cognition is advanced--the compatibility versus incompatibility of questions and the effect this can have on the sequential order of judgments. We conclude that quantum information-processing principles provide a viable and promising new way to understand human judgment and reasoning.     

Learning conditional frequencies in a probability learning task

The study was prompted by a theoretical discussion of probability learning by Estes (1976). In three separate experiments, subjects were presented with frequency information in the form of wins and losses among 3 teams, and later predicted future wins and losses. Frequencies were devised so that conditional win frequencies for a pair of teams were either inconsistent or consistent with marginal win frequencies for each team. In experiment 1, when subjects predicted future events on the basis of known past frequencies, predictions were generally based on conditional frequencies. In experiment 2 six blocks of observations were presented, with predictions following each block. What little learning did occur was in the direction of the conditional frequencies. Subjects in experiment 3 were able to learn conditional frequencies when given explicit instructions to do so. Results were discussed in terms of a two-stage hypothesis generation model that might operate within the framework of an associative theory of probability learning.     

On the provenance of judgments of conditional probability

In standard treatments of probability, is defined as the ratio of to , provided that . This account of conditional probability suggests a psychological question, namely, whether estimates of arise in the mind via implicit calculation of . We tested this hypothesis (Experiment 1) by presenting brief visual scenes composed of forms, and collecting estimates of relevant probabilities. Direct estimates of conditional probability were not well predicted by . Direct estimates were also closer to the objective probabilities defined by the stimuli, compared to estimates computed from the foregoing ratio. The hypothesis that arises from the ratio fared better (Experiment 2). In a third experiment, the same hypotheses were evaluated in the context of subjective estimates of the chance of future events.     

Contingency and its two indices within conditional probability analysis

FOUR THEORETICAL BASES FOR DETECTING A CONTINGENCY BETWEEN BEHAVIOR AND CONSEQUENT STIMULI ARE CONSIDERED: contiguity, correlation, conditional probability, and logical implication. It is argued that conditional probability analysis is statistically the most powerful of these options, in part due to its provision of two indices of contingency: a forward time probability that reinforcement follows behavior and a backward time probability that behavior precedes reinforcement. Evidence is cited that both indices appear to bear on the learning of a variety of animals, although they are unequally salient to human adults and to artificial neural networks designed to solve time-series functions. It is hypothesized that humans may acquire the capacity to detect contingency in the progressive sequence: contiguity, correlation, forward time conditional probability, backward time conditional probability, and ultimately logical implication.     

Evidential diversity and premise probability in young children's inductive judgment

A familiar adage in the philosophy of science is that general hypotheses are better supported by varied evidence than by uniform evidence. Several studies suggest that young children do not respect this principle, and thus suffer from a defect in their inductive methodology. We argue that the diversity principle does not have the normative status that psychologists attribute to it, and should be replaced by a simple rule of probability. We then report experiments designed to detect conformity to the latter rule in children's inductive judgment. The results suggest that young children in both the United States and Taiwan are sensitive to the constraints imposed by the rule on judgments of probability and evidential strength. We conclude with a suggested reinterpretation of the thesis that children's inductive methodology qualifies them as “little scientists.”     

Aging and conditional probability judgments: a global matching approach

Age differences in bias in conditional probability judgments were investigated based on predictions derived from the Minerva-Decision Making model (M. R. P. Dougherty, C. F. Gettys, & E. E. Ogden, 1999), a global matching model of likelihood judgment. In this study, 248 younger and older adults completed frequency judgment and conditional probability judgment tasks. Age differences in the frequency judgment task are interpreted as an age-related deficit in memory encoding. Older adults' stronger biases in the probability judgment task point to age differences in criterion setting. Age-related biases were eliminated when age groups were equated on memory encoding by means of study time manipulation. The authors conclude that older adults' stronger judgment biases are a function of memory impairment.