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.     

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.     

Causal Probability

Examples growing out of the Newcomb problem have convinced many people that decision theory should proceed in terms of some kind of causal probability. I endorse this view and define and investigate a variety of causal probability. My definition is related to Skyrms' definition, but proceeds in terms of objective probabilities rather than subjective probabilities and avoids taking causal dependence as a primitive concept.     

Effects of traits and states subjective probability of task success and performance

In a laboratory experiment with 113 male university students, probability of succeeding at an arithmetic task was set at either .25, .50, or .75. This objective probability of success was linealy and positively related to subjective probability of success and to level of performance. No significant main effects of the trait of generalized expectancy of task success were found either on subjective probability of success or on performance. However, the more specific trait of expectancy of success at numerical tasks, with measured ability held constant, did show significant positive effects on subjective probability of success and performance. These results suggest that persons with high self-appraisals in a particular skill area form higher subjective probabilities of success and perform at higher levels on tasks calling for those skills than do persons with low self-appraisals, even though they do not differ in measured ability.Research reported here is based on a portion of a dissertation prepared at at the University of Minnesota in partial fulfillment of the requirements for the doctoral degree. Grateful appreciation is extended to John P. Campbell, who served as thesis advisor.     

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.     

The nature of probability

The thesis of this article is that the nature of probability is centered on its formal properties, not on any of its standard interpretations. Section 2 is a survey of Bayesian applications. Section 3 focuses on two examples from physics that seem as completely objective as other physical concepts. Section 4 compares the conflict between subjective Bayesians and objectivists about probability to the earlier strident conflict in physics about the nature of force. Section 5 outlines a pragmatic approach to the various interpretations of probability. Finally, Sect. 6 argues that the essential formal nature of probability is expressed in the standard axioms, but more explicit attention should be given to the concept of randomness.     

Decision Theory and Vocational Counseling: Important Concepts and Questions

Ignorance of specifics of career decision-making processes has prevented development and use of more effective vocational counseling procedures. Simply giving clients vocational information and assuming rational use of it is criticized. Contemporary decision theories suggest several relevant variables. Two variables, subjective probabilities (individual's self-estimates of success) and utilities (desirabilities held for outcomes or alternatives), are discussed in relation to presenting clients with objective probability information concerning future plans. Some relevant research is reviewed and two questions are discussed: (1) How do different methods of presenting information influence subjective probabilities? (2) How are utilities influenced by objective probability data?     

Bayesian decision theory,subjective and objective probabilities,and acceptance of empirical hypotheses

It is argued that we need a richer version of Bayesian decision theory, admitting both subjective and objective probabilities and providing rational criteria for choice of our prior probabilities. We also need a theory of tentative acceptance of empirical hypotheses. There is a discussion of subjective and of objective probabilities and of the relationship between them, as well as a discussion of the criteria used in choosing our prior probabilities, such as the principles of indifference and of maximum entropy, and the simplicity ranking of alternative hypotheses.     

Decision time,subjective probability,and task difficulty

This study analyzed the relationships between decision time, subjective probability, and task difficulty in the context of a probability assessment task involving memory search. The results indicate that decision time and subjective probability do not yield identical functions. Also, decision time increases as subjective task difficulty increases. A similar relationship obtains between decision time and a measure of objective task difficulty. These latter two findings are inconsistent with Hogarth’s (1975) prediction of a nonmonotonic relationship between decision time and task difficulty.     

Conditioned Assessment of Subjective Probabilities: Identifying the Benefits of Decomposition

Conditioned assessment of subjective probability decomposes an assessment task by using a related event to define a series of conditional probabilities. These component assessments are subsequently aggregated mathematically. Two experiments compared conditioned and unconditioned (direct) assessments of 32 general knowledge questions by graduate business students. Reliability was estimated through repeat administration of the questionnaire and accuracy was measured with a quadratic scoring rule. Conditioned assessments were more reliable and more accurate than direct assessments, confirming previous research on other decomposition approaches. Conditioned assessment appears to be effective because the mathematical combination of conditional probabilities reduces random response errors and reduces the tendency to produce overly extreme probability judgments. Implications for further research and application are considered.     

Bayesian confusions surrounding simplicity and likelihood in perceptual organization

In the study of perceptual organization, the Occamian simplicity principle (which promotes efficiency) and the Helmholtzian likelihood principle (which promotes veridicality) have been claimed to be equivalent. Proposed models of these principles may well yield similar outcomes (especially in everyday situations), but as argued here, claims that the principles are equivalent confused subjective probabilities (which are used in Bayesian models of the Occamian simplicity principle) and objective probabilities (which are needed in Bayesian models of the Helmholtzian likelihood principle). Furthermore, Occamian counterparts of Bayesian priors and conditionals have led to another confusion, which seems to have been triggered by a dual role of regularity in perception. This confusion is discussed by contrasting complete and incomplete Occamian approaches to perceptual organization.     

A Logic For Inductive Probabilistic Reasoning

Inductive probabilistic reasoning is understood as the application of inference patterns that use statistical background information to assign (subjective) probabilities to single events. The simplest such inference pattern is direct inference: from “70% of As are Bs” and “a is an A” infer that a is a B with probability 0.7. Direct inference is generalized by Jeffrey’s rule and the principle of cross-entropy minimization. To adequately formalize inductive probabilistic reasoning is an interesting topic for artificial intelligence, as an autonomous system acting in a complex environment may have to base its actions on a probabilistic model of its environment, and the probabilities needed to form this model can often be obtained by combining statistical background information with particular observations made, i.e., by inductive probabilistic reasoning. In this paper a formal framework for inductive probabilistic reasoning is developed: syntactically it consists of an extension of the language of first-order predicate logic that allows to express statements about both statistical and subjective probabilities. Semantics for this representation language are developed that give rise to two distinct entailment relations: a relation ⊨ that models strict, probabilistically valid, inferences, and a relation that models inductive probabilistic inferences. The inductive entailment relation is obtained by implementing cross-entropy minimization in a preferred model semantics. A main objective of our approach is to ensure that for both entailment relations complete proof systems exist. This is achieved by allowing probability distributions in our semantic models that use non-standard probability values. A number of results are presented that show that in several important aspects the resulting logic behaves just like a logic based on real-valued probabilities alone.     

The Binary Additivity of Subjective Probability Does not Indicate the Binary Complementarity of Perceived Certainty

People's numeric probability estimates for 2 mutually exclusive and exhaustive events commonly sum to 1.0, which seems to indicate the full complementarity of subjective certainty in the 2 events (i.e., increases in certainty for one event are accompanied by decreases in certainty for the other). In this article, however, a distinction is made between the additivity of probability estimates and the complementarity of internal perceptions of certainty. In Experiment 1, responses on a verbal measure of certainty provide evidence of binary noncomplementarity in the perceived likelihoods of possible scenario outcomes, and a comparison of verbal and numeric certainty estimates suggests that numeric probabilities overestimated the complementarity of people's certainty. Experiment 2 used a choice task to detect binary noncomplementarity. Soliciting numeric probability estimates prior to the choice task changed the participants' choices in a direction consistent with complementarity. Possible mechanisms yielding (non)complementarity are discussed.     

主观概率判断的支持理论

支持理论是一个关于主观概率判断的非外延性理论,它有以下几个主要观点：1.主观概率判断受到描述的影响,具有描述的依赖性;2.主观概率判断的结果是判断者对中心假设的相对支持的反映;3.主观概率在二元判断中表现出二元互补性,在多元判断中表现出次可加性;4.主观概率判断存在分解效应和促进效应     

Subjective sampling distributions and the additivity of estimates

Abstract.— Previous studies of sampling distributions have been conducted almost exclusively under the assumption that persons behave in accordance with the "fundamental convention" of probability, i.e. that the sum of all probability estimates will equal 1. When this assumption was tested by asking subjects to give "unrestricted" probability estimates of all possible outcomes of samples from a given population, a general tendency of overestimation made the sum of all probabilities exceed 1 to a considerable extent. The subjective sampling distributions appeared to be unaffected by sample size ( N=5 or 10) and number of outcomes, and were flatter than the corresponding "objective" sampling distributions.     

Age-related changes in frequency of mind-wandering and task-related interferences during memory encoding and their impact on retrieval

During the performance of cognitive tasks such as memory encoding, attention can become decoupled from the external environment and instead focused on internal thoughts related to the appraisal of the current task (task-related interferences; TRI), or personal thoughts unrelated to the task at hand (mind-wandering; MW). However, the association between the frequency of these thoughts experienced at encoding and retrieval accuracy in young and older adults remains poorly understood. In this study young and older adults encoded lists of words using one of two encoding tasks: judging whether words are man-made/natural (objective task), or whether they are pleasant/neutral (subjective task). We measured the frequency of TRI and MW at encoding, and related them to retrieval accuracy in both age groups. We found that encoding task influenced the type of internal thoughts experienced by young, but not older, adults: young exhibited greater MW in the subjective vs the objective task, and greater TRI in the objective vs subjective encoding task. Second, across both tasks we found marked age-related decreases in both MW and TRI at encoding, and frequency of these thoughts negatively impacted memory retrieval in young adults only. We discuss these findings in relation to current theories of ageing, attention and memory.     

The impact of age on useful visual field deterioration and risk evaluation in a simulated driving task

The primary goal of this study is to evaluate whether the driver can estimate his performance and the deterioration of his state of alertness during a simulated driving task. The second goal is to study the relation between useful visual field (UVF) deterioration and the capacity to estimate performance in a visual task and the decrease of level of alertness as a function of age. In our experiment, two groups of subjects: 10 drivers between 21 and 34 years old and nine drivers between 46 and 57 years old were required to follow a vehicle in a simulated road traffic situation for 2 hours. In addition, the driver had to detect the change of colour of a signal located in the central part of his visual field or a peripheral signal appeared on the rear light of one of the vehicles in the traffic. The analysis of data collected during this visual task confirms that UVF deteriorates with the duration of the driving task and with the driver's age. The analysis of subjective data related to the state of alertness highlights an effect both of age and of the moment when this self-evaluation was carried out. However, self-evaluation of the subject's performance does not depend on driver's age. Finally, the study shows that the correlation between objective data (performance of visual task) and subjective data (state of drowsiness and self-evaluation of performance of the visual task) is low, and the implications with regard to road safety are discussed.     

A random support model of the calibration of subjective probabilities

A stochastic model of the calibration of subjective probabilities based on support theory (Rottenstreich and Tversky, 1997, Tversky and Koehler, 1994) is presented. This model extends support theory—a general representation of probability judgment—to the domain of calibration, the analysis of the correspondence between subjective and objective probability. The random support model can account for the common finding of overconfidence, and also predicts the form of the relationship between overconfidence and item difficulty (the “hard–easy effect”). The parameters of the model have natural psychological interpretations, such as discriminability between correct and incorrect hypotheses, and extremity of judgment. The random support model can be distinguished from other stochastic models of calibration by: (a) using fewer parameters, (b) eliminating the use of variable cutoffs by mapping underlying support directly into judged probability, (c) allowing validation of model parameters with independent assessments of support, and (d) applying to a wide variety of tasks by framing probability judgment in the integrative context of support theory.     

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.     

Children's comprehension of visual formal features in television programs

A review of previous studies on children's comprehension of visual formal features did not warrant predictions about children's understanding of the formal features as used in three items from Sesamstraat, the Dutch version of Sesame Street. Therefore, a study was designed in which 45 children in the age range of 4–6 years watched the items and were interviewed. In the first item, a split screen was used to visualize simultaneity of actions. The second item used a subjective camera to suggest the construction of a home video by one of the characters. Version 1 of this item did not show the character while making home video, whereas Version 2 did show the character while filming. The third item contained a dissolve (Version 1) or a cut (Version 2) to indicate the end of a dream. A general conclusion was that although older children (mean AGE=5.9 years) understood the visual formal features better than younger children (mean AGE=4.4 years), the extent to which children in both age groups understood the visual formal features appeared to vary between items and versions. Most children in both age groups understood the split screen in the first item. The younger age group did not understand the subjective camera if the making of the home video was not shown. Both the younger and the older age group understood the version with the cut better than the original version with the dissolve. In the discussion, the question was addressed what the findings of this study and previous studies teach us about children's understanding of visual formal features in general.