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.     

Shafer on conditional probability

Journal of Philosophical Logic -     

Induction as conditional probability judgment

Existing research on category-based induction has primarily focused on reasoning about blank properties, or predicates that are designed to elicit little prior knowledge. Here, we address reasoning about nonblank properties. We introduce a model of conditional probability that assumes that the conclusion prior probability is revised to the extent warranted by the evidence in the premise. The degree of revision is a function of the relevance of the premise category to the conclusion and the informativeness of the premise statement. An algebraic formulation with no free parameters accurately predicted conditional probabilities for single- and two-premise conditionals (Experiments 1 and 3), as well as problems involving negative evidence (Experiment 2).     

Construction of a conditional probability learning model and information theoretical evaluation of its efficiency

A learning model is introduced based on the simple idea that a learning subject has two hypotheses about the results (the outputs of the environment) caused by his actions and calculates the a posteriori probabilities of the hypotheses. Introducing a randomized decision rule of actions related to the probabilities, it is shown that the model is equivalent to Luce's beta model, while these models are derived from completely different considerations. Introducing an information theoretic measure and martingale theory, the asymptotic behavior of the model is examined. Using the information theoretic measure, the effect of memory function on the learning behavior is evaluated. Finally, the model is extended in such a way that the learning subject has more than two hypotheses or a continuum of hypotheses. The learning behaviors of the extended models are discussed.     

Contingency and behavior analysis

The concept of contingency is central to theoretical discussions of learned behavior and in the application of learning research to problems of social significance. This paper reviews three aspects of the contingency concept as it has been developed by behavior analysts. The first is the empirical analysis of contingency through experimental studies of both human and nonhuman behavior. The second is the synthesis of experimental studies in theoretical and conceptual frameworks to yield a more general account of contingency and to integrate the concept with other behavioral processes. The third aspect is one of practical considerations in the application of the contingency concept in both laboratory and applied settings.     

Resampling probability values for differences between two independent indices of ordinal variation and consensus

A permutation method is presented to calculate resampling probability values for differences between two independent indices of ordinal variation and consensus.     

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.     

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.     

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.     

Contingency bias in probability judgement may arise from ambiguity regarding additional causes

We tested whether preventive and generative reasoning processes are symmetrical by keeping the training and testing of preventive (inhibitory) and generative (excitatory) causal cues as similar as possible. In Experiment 1, we extinguished excitors and inhibitors in a blocking design, in which each extinguished cause was presented in compound with a novel cause, with the same outcome occurring following the compound and following the novel cause alone. With this novel extinction procedure, the inhibitory cues seemed more likely to lose their properties than the excitatory cues. In Experiment 2, we investigated blocking of excitatory and inhibitory causes and found similar blocking effects. Taken together, these results suggest that acquisition of excitation and inhibition is similar, but that inhibition is more liable to extinguish with our extinction procedure. In addition, we used a variable outcome, and this enabled us to test the predictions of an inferential reasoning account about what happens when the outcome level is at its minimum or maximum (De Houwer, Beckers, & Glautier, 2002). We discuss the predictions of this inferential account, Rescorla and Wagner's (1972) model, and a connectionist model—the auto-associator.     

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).     

Sequence learning in infancy: the independent contributions of conditional probability and pair frequency information

The ability to perceive sequences is fundamental to cognition. Previous studies have shown that infants can learn visual sequences as early as 2 months of age and it has been suggested that this ability is mediated by sensitivity to conditional probability information. Typically, conditional probability information has covaried with frequency information in these studies, raising the possibility that each type of information may have contributed independently to sequence learning. The current study explicitly investigated the independent contribution of each type of information. We habituated 2.5‐, 4.5‐, and 8.5‐month‐old infants to a sequence of looming visual shapes whose ordering was defined independently by specific conditional probability relations among pair elements and by the frequency of occurrence of such pairs. During test trials, we tested infants’ sensitivity to each type of information and found that both types of information independently influenced sequence learning by 4.5 months of age.     

Successive-beat analysis of cardiovascular orienting and conditional responses

We have shown that prominent changes in heart rate (HR) and blood pressure (BP) occur in dogs during classical defensive conditioning. These changes are usually analyzed as average HR and BP during control periods versus averages during conditional stimulus (CS) and unconditional stimulus (US) periods. A new method has been developed for evaluation of these functions, viz., analysis across many trials of successive beats beginning at onset of CS and working backward (control period) and forward (CS period). Nine awake dogs were continuously studied: six with HR and intra-arterial BP measurements and three with HR measurements only. A drop in HR occurs 1 to 2 beats after CS onset (often noted as a “dropped” beat on individual trials). In some dogs, on the third to fifth beat HR reaches its maximum during the CS period, thereafter falling slightly, but still above baseline; other dogs show progressively increasing HR throughout the whole CS period. At US onset (foreleg shock) HR usually rises rapidly to a peak greater than that at CS maximum. BP follows HR changes fairly closely, diastolic better than systolic. These results indicate that I) this method can reveal transients previously unsuspected, 2) a prominent initial sudden bradycardia followed by tachycardia commonly occurs in conditioning in dogs, 3) the latent period of prominent HR increase is usually quite short (1.3 to 1.6 secs.), and 4) BP follows HR directly during the early part of the CS.     

The effects of averaging subjective probability estimates between and within judges

The average probability estimate of J > 1 judges is generally better than its components. Two studies test 3 predictions regarding averaging that follow from theorems based on a cognitive model of the judges and idealizations of the judgment situation. Prediction 1 is that the average of conditionally pairwise independent estimates will be highly diagnostic, and Prediction 2 is that the average of dependent estimates (differing only by independent error terms) may be well calibrated. Prediction 3 contrasts between- and within-subject averaging. Results demonstrate the predictions' robustness by showing the extent to which they hold as the information conditions depart from the ideal and as J increases. Practical consequences are that (a) substantial improvement can be obtained with as few as 2-6 judges and (b) the decision maker can estimate the nature of the expected improvement by considering the information conditions.