Overcoming difficulties in Bayesian reasoning: A reply to Lewis and Keren (1999) and Mellers and McGraw (1999).

Bayesian reasoning can be improved by representing information in frequency formats rather than in probabilities. This thesis opens up applications in medicine, law, statistics education, and other fields. The beneficial effect is no longer in dispute, but rather its cause and its boundary conditions. C. Lewis and G. Keren (1999) argued that the effect of frequency formats is due to "joint statements" rather than to "frequency statements." However, they overlooked the fact that our thesis is about frequency formats, not just any kind of frequency statements. We show that joint statements alone cannot account for the effect. B. A. Mellers and A. P. McGraw (1999) proposed a boundary condition under which the beneficial effect is reduced. In a reanalysis of our original data, we found this reduction for the problem they used but not for any other problem. We conclude by summarizing results indicating that teaching frequency representations fosters insight into Bayesian reasoning. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

"The meaning(s) of conditionals: Conditional probabilities, mental models, and personal utilities": Correction to Oberauer and Wilhelm (2003).

Reports an error in "The meaning(s) of conditionals: Conditional probabilities, mental models, and personal utilities" by Klaus Oberauer and Oliver Wilhelm (Journal of Experimental Psychology: Learning, Memory, and Cognition, 2003[Jul], Vol 29[4], 680-693). On page 684, Table 4, all correlations should have been identified as having a p2003-06626-018.) Five experiments were conducted to test the hypothesis that people understand conditional statements ("if p then q") as indicating a high conditional probability P(q|p). Participants estimated the probability that a given conditional is true (Experiments 1A, 1B, and 3) or judged whether a conditional was true or false (Experiments 2 and 4) given information about the frequencies of the relevant truth table cases. Judgments were strongly influenced by the ratio of pq to p?q cases, supporting the conditional probability account. In Experiments 1A, 1B, and 3, judgments were also affected by the frequency of pq cases, consistent with a version of mental model theory. Experiments 3 and 4 extended the results to thematic conditionals and showed that the pragmatic utility associated with believing a statement also affected the degree of belief in conditionals but not in logically equivalent quantified statements. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

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. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

The meaning(s) of conditionals: Conditional probabilities, mental models, and personal utilities.

[Correction Notice: An erratum for this article was reported in Vol 29(6) of Journal of Experimental Psychology: Learning, Memory, and Cognition (see record 2007-16865-001). On page 684, Table 4, all correlations should have been identified as having a pp then q") as indicating a high conditional probability P(q|p). Participants estimated the probability that a given conditional is true (Experiments 1A, 1B, and 3) or judged whether a conditional was true or false (Experiments 2 and 4) given information about the frequencies of the relevant truth table cases. Judgments were strongly influenced by the ratio of pq to p?q cases, supporting the conditional probability account. In Experiments 1A, 1B, and 3, judgments were also affected by the frequency of pq cases, consistent with a version of mental model theory. Experiments 3 and 4 extended the results to thematic conditionals and showed that the pragmatic utility associated with believing a statement also affected the degree of belief in conditionals but not in logically equivalent quantified statements. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

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?q, ?pq, and ?p?q (where ? = not). Three hypotheses were compared: (a) that people equate the probability with that of the material conditional, 1 - P(p?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. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

How to improve Bayesian reasoning: Comment on Gigerenzer and Hoffrage (1995).

G. Gigerenzer and U. Hoffrage (1995) claimed that Bayesian inference problems, which have been notoriously difficult for laypeople to solve using base rates, hit rates, and false-alarm rates, become computationally simpler when information is presented with frequencies based on natural sampling. They made an evolutionary argument for the improved performance. The authors of the present article show that performance can improve with either probabilities or frequencies, depending on the rareness of the events and the type of information presented. When events are rare, probabilities are more difficult to understand than frequencies (i.e., 5 out of 1,000 vs. .005.). Furthermore, when the information is presented as joint and marginal events, nested sets become more apparent. Frequencies based on natural sampling have these desirable properties. The authors agree with Gigerenzer and Hoffrage that frequencies can improve Bayesian reasoning, but they attribute that improvement to the use of mental models that involve elements of nested sets. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Reinforcement of conversation "operants" in psychiatric patients.

Investigated the hypothesis that opinion statements and information statements could be conditioned to greater frequency in hospitalized psychiatric patients (N = 24) under natural conversation conditions. After a 10-min base-line period, each S's opinion and information statements were positively reinforced for a 20-min conditioning period by means of verbal agreement, enthusiasm, or paraphrase by E. Conditioning effects were achieved for both opinion and information statements. Results are interpreted as demonstrating that verbal "operants" can be conditioned in the verbal behavior of psychiatric patients under natural conversation conditions similar to those in psychotherapy. (French summary) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

How to improve Bayesian reasoning without instruction: Frequency formats.

Is the mind, by design, predisposed against performing Bayesian inference? Previous research on base rate neglect suggests that the mind lacks the appropriate cognitive algorithms. However, any claim against the existence of an algorithm, Bayesian or otherwise, is impossible to evaluate unless one specifies the information format in which it is designed to operate. The authors show that Bayesian algorithms are computationally simpler in frequency formats than in the probability formats used in previous research. Frequency formats correspond to the sequential way information is acquired in natural sampling, from animal foraging to neural networks. By analyzing several thousand solutions to Bayesian problems, the authors found that when information was presented in frequency formats, statistically naive participants derived up to 50% of all inferences by Bayesian algorithms. Non-Bayesian algorithms included simple versions of Fisherian and Neyman-Pearsonian inference. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Advances in statistical methods to map quantitative trait loci in outbred populations

Statistical methods to map quantitative trait loci (QTL) in outbred populations are reviewed, extensions and applications to human and plant genetic data are indicated, and areas for further research are identified. Simple and computationally inexpensive methods include (multiple) linear regression of phenotype on marker genotypes and regression of squared phenotypic differences among relative pairs on estimated proportions of identity-by-descent at a locus. These methods are less suited for genetic parameter estimation in outbred populations but allow the determination of test statistic distributions via simulation or data permutation; however, further inferences including confidence intervals of QTL location require the use of Monte Carlo or bootstrap sampling techniques. A method which is intermediate in computational requirements is residual maximum likelihood (REML) with a covariance matrix of random QTL effects conditional on information from multiple linked markers. Testing for the number of QTLs on a chromosome is difficult in a classical framework. The computationally most demanding methods are maximum likelihood and Bayesian analysis, which take account of the distribution of multilocus marker-QTL genotypes on a pedigree and permit investigators to fit different models of variation at the QTL. The Bayesian analysis includes the number of QTLs on a chromosome as an unknown.     

Trait-inferential relationships and judgments about the personalities of others.

Discusses the effects of trait-inferential relationships, defined in terms of perceived probabilities of joint occurrence, on the perception of the personalities of others. 216 male and female undergraduates served as judges predicted the targets' responses to personality statements, highly or lowly inferentially related to the target information. Judges made a greater number of trait-inference judgments, were more certain, and were more willing to make a prediction on high-inferential than on low-inferential statements. Female judges were more inferentially "accurate" and thus more certain of their predictions than male judges. Other aspects of the data suggest that judges may have experienced conflict on statements that were inferentially negatively related to the target information, but frequently endorsed by others. A secondary analysis of the data indirectly supported this interpretation. (French summary) (26 ref.) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Evaluating Three Criteria for Establishing Cue-Search Hierarchies in Inferential Judgment.

The authors identify and provide an integration of 3 criteria for establishing cue-search hierarchies in inferential judgment. Cues can be ranked by information value according to expected information gain (Bayesian criterion), cue-outcome correlation (correlational criterion), or ecological validity (accuracy criterion). All criteria significantly predicted information acquisition behavior; however, in 3 experiments, the most successful predictor was the correlational criterion (followed by the Bayesian). Although participants showed sensitivity to task constraints, searching for less information when it was more expensive (Experiment 1) and when under time constraints (Experiment 2), concomitant changes in the relative frequency of acquisition of cues with different information values were not observed. A rational analysis illustrates why such changes in the frequency of acquisition would be beneficial, and reasons for the failure to observe such behavior are discussed. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Measurement of attribution of causality in counselor behavior.

Describes a set of process scales (Locus of Control Content Scales, Locus of Control Counselor Style Scale, and Problem Solving Scale) developed for use in the study of group counseling. Three of the process scales were designed to rate counselors' references to locus of control, and a 4th scale recorded frequency of problem-solving statements. The locus of control scales differentiated between conveying attitudes about causality through the content of a counselor's statements vs through the style in which he or she interacted with the group. Results of 6 high school counselors working with 96 students indicate that counselors are not consistent in their use of style and content in expressing ideas about causality. Counselor statements tended to imply that students had the resources to solve their own problems (internal content) but provided the students with solutions rather than allowing them to decide on a course of action (external style). Appropriate reliability coefficients and suggestions for designing research using these process scales are given. (18 ref) (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

How can mental models theory account for content effects in conditional reasoning? A developmental perspective

This article proposes a modification to Johnson-Laird's mental models theory applied to the interpretation of conditional statement of the form 'if ... then'. The model suggests that this interpretation is based on the construction of mental models supplied by establishing a correspondence between the semantic spaces associated with the antecedent and consequent of the statements. The construction of the models and the interpretation of the statements would depend on the nature of the semantic spaces involved, the interpretative context and the subject's knowledge and processing capacity. Three experiments show that the interpretation of conditional rules depends, for example, on whether or not the conditional rule possess binary terms (e.g. boy/girl). The developmental approach makes it possible to reveal phenomena which tend to remain hidden in studies of adult functioning. We show that the model accounts for a number of the reasoning biases described in the literature as well as for the interpretation of various conditional forms which do not have a truth-functional meaning.     

Consequential Conditionals: Invited and Suppressed Inferences From Valued Outcomes.

Consequential conditionals are defined as "if P then Q" statements, where P is an action, and Q a predicted outcome of this action, which is either desirable or undesirable to the agent. Experiment 1 shows that desirable (viz. undesirable) outcomes invite an inference to the truth (viz. falsity) of their antecedent. Experiment 2 shows that the more extreme the outcome is, the stronger the invited inference is. Experiment 3 shows that modus ponens from premises "If A then C, A" can be suppressed with the introduction of a consequential conditional, "If C then Q," where Q is an undesirable outcome. Experiment 4 shows that the more undesirable Q is, the larger the suppression is. The authors discuss how these results can enrich current approaches of conditional inference on the basis of mental models, complementary necessary conditions, and conditional probabilities. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Bayesian Modal Updating using Complete Input and Incomplete Response Noisy Measurements

The problem of identification of the modal parameters of a structural model using complete input and incomplete response time histories is addressed. It is assumed that there exist both input error (due to input measurement noise) and output error (due to output measurement noise and modeling error). These errors are modeled by independent white noise processes, and contribute towards uncertainty in the identification of the modal parameters of the model. To explicitly treat these uncertainties, a Bayesian framework is adopted and a Bayesian time-domain methodology for modal updating based on an approximate conditional probability expansion is presented. The methodology allows one to obtain not only the optimal (most probable) values of the updated modal parameters but also their uncertainties, calculated from their joint probability distribution. Calculation of the uncertainties of the identified modal parameters is very important if one plans to proceed with the updating of a theoretical finite-element model based on these modal estimates. The proposed approach requires only one set of excitation and corresponding response data. It is found that the updated probability density function (PDF) can be well approximated by a Gaussian distribution centered at the optimal parameters at which the posterior PDF is maximized. Numerical examples using noisy simulated data are presented to illustrate the proposed method.     

Computation of the full likelihood function for estimating variance at a quantitative trait locus

The proportion of alleles identical by descent (IBD) determines the genetic covariance between relatives, and thus is crucial in estimating genetic variances of quantitative trait loci (QTL). However, IBD proportions at QTL are unobservable and must be inferred from marker information. The conventional method of QTL variance analysis maximizes the likelihood function by replacing the missing IBDs by their conditional expectations (the expectation method), while in fact the full likelihood function should take into account the conditional distribution of IBDs (the distribution method). The distribution method for families of more than two sibs has not been obvious because there are n(n - 1)/2 IBD variables in a family of size n, forming an n x n symmetrical matrix. In this paper, I use four binary variables, where each indicates the event that an allele from one of the four grandparents has passed to the individual. The IBD proportion between any two sibs is then expressed as a function of the indicators. Subsequently, the joint distribution of the IBD matrix is derived from the distribution of the indicator variables. Given the joint distribution of the unknown IBDs, a method to compute the full likelihood function is developed for families of arbitrary sizes.     

Inequality Constrained Analysis of Variance: A Bayesian Approach.

Researchers often have one or more theories or expectations with respect to the outcome of their empirical research. When researchers talk about the expected relations between variables if a certain theory is correct, their statements are often in terms of one or more parameters expected to be larger or smaller than one or more other parameters. Stated otherwise, their statements are often formulated using inequality constraints. In this article, a Bayesian approach to evaluate analysis of variance or analysis of covariance models with inequality constraints on the (adjusted) means is presented. This evaluation contains two issues: estimation of the parameters given the restrictions using the Gibbs sampler and model selection using Bayes factors in the case of competing theories. The article concludes with two illustrations: a one-way analysis of covariance and an analysis of a three-way table of ordered means. (PsycINFO Database Record (c) 2010 APA, all rights reserved)     

Remembering makes evidence compelling: Retrieval from memory can give rise to the illusion of truth.

The illusion of truth is traditionally described as the increase in perceived validity of statements when they are repeated (Hasher, Goldstein, & Toppino, 1977). However, subsequent work has demonstrated that the effect can arise due to the increased familiarity or fluency afforded by repetition and not necessarily to repetition per se. We examine the case of information retrieved from memory. Recently experienced information is expected to be subsequently reexperienced as more fluent and familiar than novel information (Jacoby, 1983; Jacoby & Dallas, 1981). Therefore, the possibility exists that information retrieved from memory, because it is subjectively reexperienced at retrieval, would be more fluent or familiar than when it was first learned and would thus lead to an increase in perceived validity. Using a method to indirectly poll the perceived truth of factual statements, our experiment demonstrated that information retrieved from memory does indeed give rise to an illusion of truth. The effect was larger than when statements were explicitly repeated twice and was of comparable size to when statements were repeated 4 times. We conclude that memory retrieval is a powerful method for increasing the perceived validity of statements (and subsequent illusion of truth) and that the illusion of truth is a robust effect that can be observed even without directly polling the factual statements in question. (PsycINFO Database Record (c) 2010 APA, all rights reserved)