Probability Learning Research Articles

Trial-by-Trial Fluctuations in the Event-Related Electroencephalogram Reflect Dynamic Changes in the Degree of Surprise

The P300 component of the human event-related brain potential has often been linked to the processing of rare, surprising events. However, the formal computational processes underlying the generation of the P300 are not well known. Here, we formulate a simple model of trial-by-trial learning of stimulus probabilities based on Information Theory. Specifically, we modeled the surprise associated with the occurrence of a visual stimulus to provide a formal quantification of the "subjective probability" associated with an event. Subjects performed a choice reaction time task, while we recorded their brain responses using electroencephalography (EEG). In each of 12 blocks, the probabilities of stimulus occurrence were changed, thereby creating sequences of trials with low, medium, and high predictability. Trial-by-trial variations in the P300 component were best explained by a model of stimulus-bound surprise. This model accounted for the data better than a categorical model that parametrically encoded the stimulus identity, or an alternative model of surprise based on the Kullback-Leibler divergence. The present data demonstrate that trial-by-trial changes in P300 can be explained by predictions made by an ideal observer keeping track of the probabilities of possible events. This provides evidence for theories proposing a direct link between the P300 component and the processing of surprising events. Furthermore, this study demonstrates how model-based analyses can be used to explain significant proportions of the trial-by-trial changes in human event-related EEG responses.

Social influence on predictions of simulated stock prices

AbstractHerding in financial markets refers to that investors are influenced by others. This study addresses the importance of consistency for herding. It is suggested that, in financial markets perceptions of consistency are based on repeated observations over time. Consistency may then be perceived as the agreement across time between investors' predictions. In addition, consistency may be related to variance over time in each investor's predictions. In an experiment using a Multiple Cue Probability Learning paradigm, 96 undergraduates made multi‐trial predictions of future stock prices given information about the current price and the predictions made by five fictitious others. Consistency was varied between the others' predictions (correlation) and within the others' predictions (variance). The results showed that the predictions were significantly influenced by the others' predictions when these were correlated. No effect of variance was observed. Hence, participants were influenced by the others when they were in agreement, regardless of whether they varied their predictions over trials or not. Copyright © 2008 John Wiley & Sons, Ltd.

Tuesday 22nd July 2008

Individual differences in learning nondeterministic relationships were investigated in samples with high mathematical proficiency. Probability Learning tasks were included within selection tests for admission to the French Air Transport Pilot Training. In Study 1 (N=401), two cues had positive and negative relationships with a target criterion. Four classes, homogeneous in mathematical performance, differed in learning the nondeterministic relationships. _ Fast-learners swiftly learned the relationships. _ Medium- and Slow-learners were slower but eventually succeeded. _ Nonlearners (12%) failed to learn even the positive relationship. These patterns replicated in Study 2 (N=448), including a third-irrelevant-cue. The irrelevant cue made the task more complex and degraded more the learning of the negative than the positive cue. Response time analyses confirmed that differences in the learning profiles went along with differences in the decision-making processes.

Intuitive statistics by 8-month-old infants

Human learners make inductive inferences based on small amounts of data: we generalize from samples to populations and vice versa. The academic discipline of statistics formalizes these intuitive statistical inferences. What is the origin of this ability? We report six experiments investigating whether 8-month-old infants are "intuitive statisticians." Our results showed that, given a sample, the infants were able to make inferences about the population from which the sample had been drawn. Conversely, given information about the entire population of relatively small size, the infants were able to make predictions about the sample. Our findings provide evidence that infants possess a powerful mechanism for inductive learning, either using heuristics or basic principles of probability. This ability to make inferences based on samples or information about the population develops early and in the absence of schooling or explicit teaching. Human infants may be rational learners from very early in development.

Choice as a function of reinforcer "hold": From probability learning to concurrent reinforcement.

Two procedures commonly used to study choice are concurrent reinforcement and probability learning. Under concurrent-reinforcement procedures, once a reinforcer is scheduled, it remains available indefinitely until collected. Therefore reinforcement becomes increasingly likely with passage of time or responses on other operanda. Under probability learning, reinforcer probabilities are constant and independent of passage of time or responses. Therefore a particular reinforcer is gained or not, on the basis of a single response, and potential reinforcers are not retained, as when betting at a roulette wheel. In the "real" world, continued availability of reinforcers often lies between these two extremes, with potential reinforcers being lost owing to competition, maturation, decay, and random scatter. The authors parametrically manipulated the likelihood of continued reinforcer availability, defined as hold, and examined the effects on pigeons' choices. Choices varied as power functions of obtained reinforcers under all values of hold. Stochastic models provided generally good descriptions of choice emissions with deviations from stochasticity systematically related to hold. Thus, a single set of principles accounted for choices across hold values that represent a wide range of real-world conditions.

Implicit learning of predictive relationships in three-element visual sequences by young and old adults.

Knowledge of sequential relationships enables future events to be anticipated and processed efficiently. Research with the serial reaction time task (SRTT) has shown that sequence learning often occurs implicitly without effort or awareness. Here, the authors report 4 experiments that use a triplet-learning task (TLT) to investigate sequence learning in young and older adults. In the TLT, people respond only to the last target event in a series of discrete, 3-event sequences or triplets. Target predictability is manipulated by varying the triplet frequency (joint probability) and/or the statistical relationships (conditional probabilities) among events within the triplets. Results reveal that both groups learned, though older adults showed less learning of both joint and conditional probabilities. Young people used the statistical information in both cues, but older adults relied primarily on information in the 2nd cue alone. The authors conclude that the TLT complements and extends the SRTT and other tasks by offering flexibility in the kinds of sequential statistical regularities that may be studied as well as by controlling event timing and eliminating motor response sequencing.

Illustrating Probability in Genetics with Hands-On Learning: Making the Math Real

Probability is an essential tool of genetics, and basic concepts of probability are usually included in introductory biology and beginning genetics courses. In spite of the central importance of probability for understanding and analyzing heredity, many students have difficulty with the subject. Part of this difficulty arises out of the quantitative nature of probability and the general problem that many students experience with quantification. Compounding the problem is the abstract nature of probability and statistics. In teaching introductory genetics to large numbers of college undergraduate students (200-250 per semester), we found that largely lecture-based instruction of probability concepts was ineffective; many students were disinterested in the topic and failed to understand and retain important concepts. McManus, Dunn, and Denig (2003) found that biology students who learned using hands-on manipulative activities had higher science achievement and science attitude scores than students who learned using traditional lecture, reading, and discussion activities. Considerable educational research demonstrates that cooperative learning within small groups leads to clear improvements in academic achievement, attitudes towards learning, and persistence compared with more traditional teaching methods in undergraduate science, math, engineering, and technology courses (Springer, Stanne & Donovan, 1999; Tanner, Chatman & Allen, 2003). To facilitate learning of probability for general genetics, we developed hands-on activities in which students in small groups actively and cooperatively learn probability. Teaching probability also can be facilitated through the use of electronic simulations of random events, which allow students to quickly collect large sets of empirical data to illustrate the predications of probability theory. A number of accessible and user-friendly Web-based tools are now available to simulate random events, such as tossing coins, rolling dice, and drawing cards from a deck (Table 1). Simulations of random events can also be produced through simple calculations performed by spreadsheets that are available on most personal computers. We have created four such spreadsheet-based simulation programs to accompany the probability exercises outlined in this article (Table 1, Figure 1); these are available free from the authors upon request. We believe that this combination of cooperative learning in small groups, with manipulative activities, and student-collected simulation data greatly enhances student interest in and understanding of basic concepts of probability. After we developed a laboratory exercise in probability based on manipulative activities and small group learning, approximately 70% of our students in an in-house survey agreed that this exercise was helpful in understanding concepts taught in the lecture portion of the course. The following laboratory exercise is appropriate for high school biology or advanced placement biology classes, as well as introductory college biology or genetics courses. Goals & Background We had four primary objectives for our students: 1. to understand the nature of probability and how it can be used to make predictions about the outcomes of genetic crosses 2. to understand and be able to apply the addition and multiplication rules of probability 3. to understand and be able to apply the binomial expansion to problems involving multiple outcomes 4. to know how to interpret probabilities associated with the chi-square test. [FIGURE 1 OMITTED] The addition and multiplication rules of probability are useful for calculating risk of inheriting genetic diseases as well as predicting the outcome of Mendelian crosses. For example, consider the following cross: AaBb x AaBb. One can quickly determine the probability of obtaining progeny with genotype aabb by using the multiplication rule. …

Choice strategies in multiple-cue probability learning.

Choice strategies for selecting among outcomes in multiple-cue probability learning were investigated using a simulated medical diagnosis task. Expected choice probabilities (the proportion of times each outcome was selected given each cue pattern) under alternative choice strategies were constructed from corresponding observed judged probabilities (of each outcome given each cue pattern) and compared with observed choice probabilities. Most of the participants were inferred to have responded by using a deterministic strategy, in which the outcome with the higher judged probability is consistently chosen, rather than a probabilistic strategy, in which an outcome is chosen with a probability equal to its judged probability. Extended practice in the learning environment did not affect choice strategy selection, contrary to reports from previous studies, results of which may instead be attributable to changes with practice in the variability and extremity of the perceived probabilities on which the choices were based.

Computer-based graphical displays for enhancing mental animation and improving reasoning in novice learning of probability

OUR RESEARCH SUGGESTS static and animated graphics can lead to more animated thinking and more correct problem solving in computer-based probability learning. Pilot software modules were developed for graduate online statistics courses and representation research. A study with novice graduate student statisticians compared problem solving in five graphic versions: text, static visual, static motion cues, computer animated, and interactive computer animated. Groups were also compared on transfer problems with static graphics without motion cues. Level of animation in thinking was assessed as number of images and movement symbols in notes. All groups provided with graphic maps had more correct solutions than the text group. Displaying static motion cues, computer animated, and interactive animated maps resulted in more correct solutions and animation in notes than just text or static visuals without motion cues. Graphic maps with static motion cues or computer animated overlay resulted in equally more correct solutions and greater animation in notes. Graphic maps with static motion cues better prepared learners for solving less animated and more difficult problems. Imagery and movement in notes were significant predictors of correct training and transfer problem solutions.

What do we learn from binding features? Evidence for multilevel feature integration.

Four experiments were conducted to investigate the relationship between the binding of visual features (as measured by their aftereffects on subsequent binding) and the learning of feature-conjunction probabilities. Both binding and learning effects were obtained, but they did not interact. Interestingly, (shape-color) binding effects disappeared with increasing practice, presumably because of the fact that only 1 of the features involved was relevant to the task. However, this instability was only observed for arbitrary, not highly overlearned combinations of simple geometric features and not for real objects (colored pictures of a banana and strawberry), where binding effects were strong and resistant to practice. These findings suggest that learning has no direct impact on the strength or resistance of bindings or on speed with which features are bound; however, learning does affect the amount of attention particular feature dimensions attract, which again can influence which features are considered in binding.

Gender differences in the use of high-variance strategies in tournament competition

We report on a series of experiments in which participants first completed a simple game of chance and then competed in a tournament based on a multiple cue probability learning task. The results show that men made riskier choices in the simple game of chance, but women adopted the high-variance strategy more frequently in the tournament. However, after controlling for differences in forecasting skill, we find no significant differences in the rates at which men and women adopted a high-variance strategy. Although altering the difficulty of the forecasting task produces differences in the rate at which participants selected the high-variance strategy, it did not produce a significant difference in the rate at which men and women selected the strategy. Thus, this paper suggests that although women are more risk-averse than men, women are no less likely to adopt a high-variance strategy in a tournament competition.

Prior belief and polarity in multicue learning

We report two experiments in which participants are trained using a multicue probability learning (MCPL) task, which attempts to simulate the acquisition of expert judgement by experience in the real world. Participants were asked to predict performance in certain occupations given a profile of personality test results with trial-by-trial outcome feedback. Only some cues were relevant, and the polarity of the cues (positive or negative predictors) was unspecified. In addition, 25% of random noise was added to the feedback to simulate real world uncertainty. The main factor of interest was that the role of prior belief (determined in a separate study of stereotypes) interfered with the learning process. Experiment 1 failed to find any influence of prior belief in the cues that were irrelevant to the criterion being trained. However, in Experiment 2 people learned to use the relevant cues better when their effect conformed with rather than conflicted with prior belief Both experiments showed strong effects of cue polarity, with positive predictors much more easily learned. The results are discussed with reference to the cognitive processes involved in MCPL and closely related tasks.

An fMRI study of reward-related probability learning

The human striatum has been implicated in processing reward-related information. More recently, activity in the striatum, particularly the caudate nucleus, has been observed when a contingency between behavior and reward exists, suggesting a role for the caudate in reinforcement-based learning. Using a gambling paradigm, in which affective feedback (reward and punishment) followed simple, random guesses on a trial by trial basis, we sought to investigate the role of the caudate nucleus as reward-related learning progressed. Participants were instructed to make a guess regarding the value of a presented card (if the value of the card was higher or lower than 5). They were told that five different cues would be presented prior to making a guess, and that each cue indicated the probability that the card would be high or low. The goal was to learn the contingencies and maximize the reward attained. Accuracy, as measured by participant's choices, improved throughout the experiment for cues that strongly predicted reward, while no change was observed for unpredictable cues. Event-related fMRI revealed that activity in the caudate nucleus was more robust during the early phases of learning, irrespective of contingencies, suggesting involvement of this region during the initial stages of trial and error learning. Further, the reward feedback signal in the caudate nucleus for well-learned cues decreased as learning progressed, suggesting an evolving adaptation of reward feedback expectancy as a behavior–outcome contingency becomes more predictable.

Missing information in multiple-cue probability learning

In a multiple-cue probability learning task, participants learned to use six discrete symptoms (i.e., cues) to diagnose which of three possible flu strains a hypothetical patient suffered from. For some patients, information regarding the status of certain symptoms was not available. Various possible ways in which the missing cue information might be processed were distinguished and tested in a series of three experiments (Ns = 80, 109, and 61). The results suggest that the judged probability of the outcome variable (i.e., flu strain) was assessed by "filling in" the missing cue information with a mean value based on previous observations. The predictions of other methods of processing missing cue information are inconsistent with the data.

Aging and Probabilistic Learning in Single- and Multiple-Cue Tasks

The objective of the study was to examine some of the conditions under which elderly people are able to learn probabilistic inverse relationships and when this type of learning is no longer possible. Two kind of tasks were used: (a) two single-cue learning tasks with either direct or inverse relationships (the Single-Cue Probability Learning paradigm), and (b) three two-cue learning tasks, one with two direct relationships, one with a combination of direct and inverse relationships, and one with two inverse relationships (the Multiple-Cue Probability Learning paradigm). Four groups of participants were included in the study: young adults (18- to 25-year-olds), adults (40- to 50-year-olds), elderly people (65- to 74-year-olds), and very elderly people (75- to 90-year-olds). It was shown that (a) older adults are able to reject the direct relationship “default” hypothesis and select the inverse relationship hypothesis when outcome feedback contradicting the default hypothesis is given, and provided that the learning setting be a very simple one, involving only one cue; (b) some older adults are able to select the inverse relationship hypothesis provided that the learning setting be a simple one, involving only two inverse relationship cues; and (c) very few older adults are able to select the inverse relationship hypothesis when the learning setting is a complex one, involving two cues with both direct and inverse relationships with the criterion. These results led to the revision of the “gradual decrease of cognitive flexibility in older adults” hypothesis proposed by Chasseigne, Mullet, and Stewart (Acta Psychologic`a, 103, 229–238, 1997).

The Use of Cognitive Feedback by Experts and Novices in a Judgment Task

Cognitive feedback (CFB) has been shown to increase performance in Multiple Cue Probability Learning (MCPL) tasks in both real life and experimental situations. Our experiment focuses on investigating how experts and novices behave differently in applying CFB, which is presented in forms of relative importance weights of each of the cues and the display of function forms (Weaver and Steward, 2000). We believe this study provided some interesting insight into how much domain knowledge is necessary to understand and apply CFB. The study examined how well novices and experts are able to process and use different types of feedback (i.e. CFB and Outcome Feedback) and hence determined which type is more useful for novices or for those with prior knowledge of the task. A total of 20 participants, 10 experts and 10 novices, were given a series of baseball statistics and asked to predict how many wins the team would have in a season based on these statistics. Performance measures (e.g. relative weighting accuracy, Lens Model measures), mental workload and judgment strategy were measured. Domain Knowledge was statistically significant for all weights except one less important cue, “stolen bases.” In other words, domain knowledge impacts weighting accuracy greatly. Domain knowledge was also found to have a significant effect on judgment performance (i.e. Lens Model measures), such as accuracy and knowledge. Mental workload was significantly affected by CFB, but none of the other dependent variables were.

Explicit and implicit processes in multicue judgment.

In two experiments, a multicue probability learning task was used to train participants in relating judgments to a criterion, on the basis of several cues that could or could not be relevant. The outcome feedback had 25% added noise to simulate real-world experience-based learning. Judgmental strategies acquired were measured by individual multiple linear regression analyses of a test phase (with no feedback) and were compared with self-ratings of cue relevance. In a third experiment, participants were instructed explicitly on cue relevance, with no training phase. The pattern of results suggested that both implicit and explicit cognitive processes influenced judgments and that they may have been sensitive to different task manipulations in the learning phase. On more complex tasks, despite weak explicit learning, explicit processes continued to influence judgments, producing a decrement in performance. These findings explain why studies of expert judgment often show only moderate levels of self-insight, since people have only partial access to the processes determining their judgments.

Task difficulty, incentive effects, and the selection of high-variance strategies: an experimental examination of tournament behavior

Previous research on tournaments shows two important results. First, increasing the gap between the payoffs to winners and losers will increase effort and performance. Second, agents with lower capabilities will adopt strategies that increase the variance of their performance. This paper argues that these results are not independent but vary with the difficulty of the task. The results from a series of experiments using a multiple cue probability learning forecasting task confirm that wider payments do not lead to increased performance when the task is simple. In addition, agents in the simple task select high-variance strategies more frequently. The increased use of high-variance strategies in the simple task occurs because agents with low capabilities use the strategy more frequently. This implies that the difficulty of the task must be taken into consideration when designing tournaments.

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.

The Illusion of Linearity: Expanding the Evidence towards Probabilistic Reasoning.

Previous research has shown that – due to the extensive attention spent to proportional reasoning in mathematics education – many students have a strong tendency to apply linear or proportional models anywhere, even in situations where they are not applicable. This phenomenon is sometimes referred to as the ‘illusion of linearity’. For example, in geometry it is known that many students believe that if the sides of a figure are doubled, the area is doubled too. In this article, the empirical evidence for this phenomenon is expanded to the domain of probabilistic reasoning. First, we elaborate on the notion of chance and provide some reasons for expecting the over generalization of linear models in the domain of probability too. Afterwards, a number of well-known and less-known probabilistic misconceptions are described and analysed, showing that they have one remarkable characteristic in common: they can be interpreted in terms oft he improper application of linear relations. Finally, we report on an empirical investigation aimed at identifying the ability of 10th and12th grade students to compare the probabilities of two binomial chance situations. It appears that before instruction in probability, students have a good capability of comparing two events qualitatively, but at the same time they incorrectly quantify this qualitative insight as if the variables in the problem were linked by a linear relationship. Remarkably, these errors persist after instruction in probability. The potential of this study for improving the teaching and learning of probability, as well as suggestions for further research, are discussed.