Studies in subjective probability III: The unimportance of alternatives

In four experiments, student subjects were asked to estimate probabilities for a list of two to ten exhaustive, non-chance events, covering a variety of situations, both of prediction and diagnosis. Only in the two-alternative case a majority gave estimates which add up to unity (or 100%). As the number of alternatives increased, the total probability increased far beyond 100%, indicating a non-distributional probability concept. For instance, the probability that a person has committed murder was considered to be quite independent of his being one of three or one of five suspects. Even when subjects were told that the list should be extended with two additional alternatives and were allowed to correct their earlier estimates, few thought it necessary to do so, and corrections went equally in both directions.     

Studies in subjective probability l: Prediction of random events

When students are asked to predict the outcome of a random event, where all alternatives are equally probable (lotteries), they tend to choose central, "representative" values, and avoid extreme ones. In ten informal experiments, it is shown how this pattern of choices is influenced by various procedural and structural changes in the basic task. The results show that guessing behavior can be described as a kind of absolute judgment, subject to grouping, anchoring and context effects. Of the two general prediction heuristics originally proposed by Kahneman & Tversky (1973), "representativeness" applies better than "availability". In fact, a major strategy of guessing is apparently to eschew numbers with prominent, "non-random" properties, which at the same time are highly available to the subjects.     

Studies in subjective probability IV: probabilities, confidence, and luck

Probably because of formal advantages, probabilities are often regarded as more basic than other dimensions of attitudes towards uncertain events (beliefs, confidences, doubts, statements of hope and fear, good and bad luck etc.). In a series of experiments, some of these concepts were empirically compared by asking students to give their views on a variety of uncertain events, ranging from future examinations to lotteries. Confidence turned out to be closely related to perceived chance, but very imperfectly to the subjective probability of the event in question, except when all outcomes are judged equally due to chance. Judgments of good and bad luck were still more independent of the probabilities involved, even in a chance situation, It is concluded that subjective probability plays a secondary role in assessments of confidence as well as of luck, and is poorly suited as a common measure of the varieties of subjective uncertainty. A final experiment suggests the subjective and statistical conceptions of uncertainty to have partially opposing connotations. "An uncertain future" seems to be subjectively interpreted as an open future, restricted possibilities of prediction.     

Studies in subjective probability V: Chance vs. structure in visual patterns

Student subjects were instructed to produce maximally non-random, aesthetically attractive and random appearing dot patterns of two different levels of complexity (3 and 5 dots). When the patterns were grouped according to similarity, random patterns were found to belong to greater equivalence groups than non-random patterns. They were also estimated to be far more probable to arise from a chance process, regardless of number of elements. The aesthetically attractive patterns were placed between random and non-random patterns in terms of probability estimates. With increasing pattern complexity, patterns produced according to different instructions became increasingly distinguishable. It is argued that there may be an optimal complexity level for discovering "structure" or unique configurations in chance patterns (cf. the constellations of stars).     

Deterministic probability: neither chance nor credence

Some have argued that chance and determinism are compatible in order to account for the objectivity of probabilities in theories that are compatible with determinism, like Classical Statistical Mechanics (CSM) and Evolutionary Theory (ET). Contrarily, some have argued that chance and determinism are incompatible, and so such probabilities are subjective. In this paper, I argue that both of these positions are unsatisfactory. I argue that the probabilities of theories like CSM and ET are not chances, but also that they are not subjective probabilities either. Rather, they are a third type of probability, which I call counterfactual probability. The main distinguishing feature of counterfactual-probability is the role it plays in conveying important counterfactual information in explanations. This distinguishes counterfactual probability from chance as a second concept of objective probability.     

Conceptual fallacies in subjective probability

Subjective probability considered as a logic of partial belief succumbs to three fundamental fallacies. These concern the representation of preference via expectation, the measurability of partial belief, and the normalization of belief.     

The cognitive substrate of subjective probability

The prominent cognitive theories of probability judgment were primarily developed to explain cognitive biases rather than to account for the cognitive processes in probability judgment. In this article the authors compare 3 major theories of the processes and representations in probability judgment: the representativeness heuristic, implemented as prototype similarity, relative likelihood, or evidential support accumulation (ESAM; D. J. Koehler, C. M. White, & R. Grondin, 2003); cue-based relative frequency; and exemplar memory, implemented by probabilities from exemplars (PROBEX; P. Juslin & M. Persson, 2002). Three experiments with different task structures consistently demonstrate that exemplar memory is the best account of the data whereas the results are inconsistent with extant formulations of the representativeness heuristic and cue-based relative frequency.     

Developmental change in subjective probability during adolescence

The developmental change in subjective probability during adolescence, an important period for establishing the probability concept, was investigated. 75 Japanese adolescents, from 12 to 23 yr. of age, were asked to make probability judgments for a lottery under 15 conditions. Analysis showed that with increase in age their subjective probability came closer to the objective probability. Discussion of these results took into consideration recent studies on the development of the concept of probability.     

The assessment of subjective probability distributions: A training experiment

For this experiment 22 Ss were especially trained to assess subjective probability distributions for proportions concerning student characteristics. Two different assessment techniques were used. There were four experimental sessions. From the second session on, Ss received feedback about their performance in the previous session(s). While Ss' associated by far too much confidence with their assessments at the beginning, they learned to assess more realistic distributions. Furthermore, discrepancies between techniques were reduced.Some considerations for a more effective training procedure are made and the concept of giving feedback only according to the tightness of the assessed distribution is critically discussed.     

Eliciting subjective probability distributions on continuous variables

Five procedures for assessing subjective probability distributions over continuous variables were compared using almanac questions as stimuli. The procedures varied on the uncertainty measures used (probabilities, odds, and odds on a logarithmic scale) and the type of response required from the subjects (uncertainty measure or value of the unknown quantity). The results showed the often used fractile procedures were inferior to procedures requiring probabilities or odds as the response from subjects. The results are also discussed in terms of the “anchoring and adjustment” hypothesis.     

Development of the concept of comparative subjective probability

This study investigates which formal principles govern subjective probability, and whether the validity of these principles depends on age. Two types of tasks were administered to 144 subjects from 3;8 to 19 years: a gambling task (with objective probabilities) and a sporting task (without objective probabilities). Six formal principles of the mathematical concept of qualitative probability (a nonnumerical concept based on ordinal scale properties) were tested. Results indicate that these principles are valid as principles of subjective probability for all age groups. Only the youngest age group (4 years or younger) had a smaller degree of confirmation.     

An evidential support accumulation model of subjective probability

A model of cue-based probability judgment is developed within the framework of support theory. Cue diagnosticity is evaluated from experience as represented by error-free frequency counts. When presented with a pattern of cues, the diagnostic implications of each cue are assessed independently and then summed to arrive at an assessment of the support for a hypothesis, with greater weight placed on present than on absent cues. The model can also accommodate adjustment of support in light of the baserate or prior probability of a hypothesis. Support for alternatives packed together in a "residual" hypothesis is discounted; fewer cues are consulted in assessing support for alternatives as support for the focal hypothesis increases. Results of fitting this and several alternative models to data from four new multiple-cue probability learning experiments are reported.     

Transition structure versus commitment in sequential subjective probability revision

When Ss revise subjective probabilities, in the light of new evidence, a common finding is that they are conservative with respect to Bayes' theorem; revisions are too small. One kind of hypothesis to account for this is ‘model specific’, assuming a breakdown in an otherwise potentially Bayesian process. The other kind assumes that statistically irrelevant, task-specific information is processed. An example of the latter is the commitment hypothesis, assuming a commitment building up to the indications of early evidence, causing Ss to lag behind Bayes' theorem in their later judgements. Evidence is presented suggesting that Ss are not necessarily sensitive to mere sub-sets of a sequence, but that this form of suboptimality may result from overall sequence structure; specifically from a bias against long runs of like evidence. This would fit with findings from other areas of research, and would suggest that there is a general form of suboptimality operating which is relevant to all sequential processing tasks.     

Violations of coherence in subjective probability: a representational and assessment processes account

Coherent judgment is a cardinal feature of rational cognition. Six experiments revealed systematic violations of coherence in probability judgment in which participants assigned different probabilities to mathematically equiprobable events. Experiments 1-5 revealed a strict refocusing effect: Compared to an occurrence frame, a non-occurrence frame resulted in higher estimates if base-rate evidence favored occurrence, lower estimates if evidence favored non-occurrence, and similar estimates if evidence supported indifference. Moreover, Experiments 5 and 6 revealed a pessimistic bias in which the less favorable of two equiprobable events was assigned greater probability. The findings support a Representational and Assessment Processes account (RAP) in which subjective probability is influenced by the perceived compatibility between representations of focal events and representations of evidence.