Optimal data selection: revision, review, and reevaluation

A Normative Account of Confirmation Bias During Reinforcement Learning

Reinforcement learning involves updating estimates of the value of states and actions on the basis of experience. Previous work has shown that in humans, reinforcement learning exhibits a confirmatory bias: when the value of a chosen option is being updated, estimates are revised more radically following positive than negative reward prediction errors, but the converse is observed when updating the unchosen option value estimate. Here, we simulate performance on a multi-arm bandit task to examine the consequences of a confirmatory bias for reward harvesting. We report a paradoxical finding: that confirmatory biases allow the agent to maximize reward relative to an unbiased updating rule. This principle holds over a wide range of experimental settings and is most influential when decisions are corrupted by noise. We show that this occurs because on average, confirmatory biases lead to overestimating the value of more valuable bandits and underestimating the value of less valuable bandits, rendering decisions overall more robust in the face of noise. Our results show how apparently suboptimal learning rules can in fact be reward maximizing if decisions are made with finite computational precision.

Normative Principles for Decision-Making in Natural Environments

The decisions we make are shaped by a lifetime of learning. Past experience guides the way that we encode information in neural systems for perception and valuation, and determines the information we retrieve when making decisions. Distinct literatures have discussed how lifelong learning and local context shape decisions made about sensory signals, propositional information, or economic prospects. Here, we build bridges between these literatures, arguing for common principles of adaptive rationality in perception, cognition, and economic choice. We discuss how a single common framework, based on normative principles of efficient coding and Bayesian inference, can help us understand a myriad of human decision biases, including sensory illusions, adaptive aftereffects, choice history biases, central tendency effects, anchoring effects, contrast effects, framing effects, congruency effects, reference-dependent valuation, nonlinear utility functions, and discretization heuristics. We describe a simple computational framework for explaining these phenomena. Expected final online publication date for the Annual Review of Psychology, Volume 73 is January 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

Information seeking criteria: artificial intelligence, economics, psychology, and neuroscience

There has been an enormous amount of interest in how the brain seeks information. The study of this issue is a rapidly growing field in neuroscience. Information seeking is to make informative choices among multiple alternatives. A central issue in information seeking is how the value of information is assessed in order to choose informative alternatives. This issue has been studied in psychology, economics, and artificial intelligence. The present review is focused on information assessment and summarizes the psychological and computational criteria with which humans and computers assess information. Based on the summary, neurophysiological findings are discussed. In addition, a computational view of the relationships between these criteria is presented.

Individual differences in strategy use and performance during fault diagnosis

Background

Research on causal reasoning often uses group-level data analyses that downplay individual differences and simple reasoning problems that are unrepresentative of everyday reasoning. In three empirical studies, we used an individual differences approach to investigate the cognitive processes people used in fault diagnosis, which is a complex diagnostic reasoning task. After first showing how high-level fault diagnosis strategies can be composed of simpler causal inferences, we discussed how two of these strategies—elimination and inference to the best explanation (IBE)—allow normative performance, which minimizes the number of diagnostic tests, whereas backtracking strategies are less efficient. We then investigated whether the use of normative strategies was infrequent and associated with greater fluid intelligence and positive thinking dispositions and whether normative strategies used slow, analytic processing while non-normative strategies used fast, heuristic processing.

Results

Across three studies and 279 participants, uses of elimination and IBE were infrequent, and most participants used inefficient backtracking strategies. Fluid intelligence positively predicted elimination and IBE use but not backtracking use. Positive thinking dispositions predicted avoidance of backtracking. After classifying participants into groups that consistently used elimination, IBE, and backtracking, we found that participants who used elimination and IBE made fewer, but slower, diagnostic tests compared to backtracking users.

Conclusions

Participants’ fault diagnosis performance showed wide individual differences. Use of normative strategies was predicted by greater fluid intelligence and more open-minded and engaged thinking dispositions. Elimination and IBE users made the slow, efficient responses typical of analytic processing. Backtracking users made the fast, inefficient responses suggestive of heuristic processing.

Strategies for selecting and evaluating information

Within the domain of psychology, Optimal Experimental Design (OED) principles have been used to model how people seek and evaluate information. Despite proving valuable as computational-level methods to account for people's behaviour, their descriptive and explanatory powers remain largely unexplored. In a series of experiments, we used a naturalistic crime investigation scenario to examine how people evaluate queries, as well as outcomes, in probabilistic contexts. We aimed to uncover the psychological strategies that people use, not just to assess whether they deviated from OED principles. In addition, we explored the adaptiveness of the identified strategies across both one-shot and stepwise information search tasks. We found that people do not always evaluate queries strictly in OED terms and use distinct strategies, such as by identifying a leading contender at the outset. Moreover, we identified aspects of zero-sum thinking and risk aversion that interact with people's information search strategies. Our findings have implications for building a descriptive account of information seeking and evaluation, accounting for factors that currently lie outside the realm of information-theoretic OED measures, such as context and the learner's own preferences.

An Active Inference Approach to Modeling Structure Learning: Concept Learning as an Example Case

Within computational neuroscience, the algorithmic and neural basis of structure learning remains poorly understood. Concept learning is one primary example, which requires both a type of internal model expansion process (adding novel hidden states that explain new observations), and a model reduction process (merging different states into one underlying cause and thus reducing model complexity via meta-learning). Although various algorithmic models of concept learning have been proposed within machine learning and cognitive science, many are limited to various degrees by an inability to generalize, the need for very large amounts of training data, and/or insufficiently established biological plausibility. Using concept learning as an example case, we introduce a novel approach for modeling structure learning-and specifically state-space expansion and reduction-within the active inference framework and its accompanying neural process theory. Our aim is to demonstrate its potential to facilitate a novel line of active inference research in this area. The approach we lay out is based on the idea that a generative model can be equipped with extra (hidden state or cause) "slots" that can be engaged when an agent learns about novel concepts. This can be combined with a Bayesian model reduction process, in which any concept learning-associated with these slots-can be reset in favor of a simpler model with higher model evidence. We use simulations to illustrate this model's ability to add new concepts to its state space (with relatively few observations) and increase the granularity of the concepts it currently possesses. We also simulate the predicted neural basis of these processes. We further show that it can accomplish a simple form of "one-shot" generalization to new stimuli. Although deliberately simple, these simulation results highlight ways in which active inference could offer useful resources in developing neurocomputational models of structure learning. They provide a template for how future active inference research could apply this approach to real-world structure learning problems and assess the added utility it may offer.

Solving the Diagnostic Challenge: A Patient-Centered Approach

Arriving at an agreed-on and valid explanation for a clinical problem is important to patients as well as to clinicians. Current theories of how clinicians arrive at diagnoses, such as the threshold approach and the hypothetico-deductive model, do not accurately describe the diagnostic process in general practice. The problem space in general practice is so large and the prior probability of each disease being present is so small that it is not realistic to limit the diagnostic process to testing specific diagnoses on the clinician's list of possibilities. Here, new evidence is discussed about how patients and clinicians collaborate in specific ways, in particular, via a process that can be termed inductive foraging, which may lead to information that triggers a diagnostic routine. Navigating the diagnostic challenge and using patient-centered consulting are not separate tasks but rather synergistic.

Generalized Information Theory Meets Human Cognition: Introducing a Unified Framework to Model Uncertainty and Information Search

Searching for information is critical in many situations. In medicine, for instance, careful choice of a diagnostic test can help narrow down the range of plausible diseases that the patient might have. In a probabilistic framework, test selection is often modeled by assuming that people's goal is to reduce uncertainty about possible states of the world. In cognitive science, psychology, and medical decision making, Shannon entropy is the most prominent and most widely used model to formalize probabilistic uncertainty and the reduction thereof. However, a variety of alternative entropy metrics (Hartley, Quadratic, Tsallis, Rényi, and more) are popular in the social and the natural sciences, computer science, and philosophy of science. Particular entropy measures have been predominant in particular research areas, and it is often an open issue whether these divergences emerge from different theoretical and practical goals or are merely due to historical accident. Cutting across disciplinary boundaries, we show that several entropy and entropy reduction measures arise as special cases in a unified formalism, the Sharma-Mittal framework. Using mathematical results, computer simulations, and analyses of published behavioral data, we discuss four key questions: How do various entropy models relate to each other? What insights can be obtained by considering diverse entropy models within a unified framework? What is the psychological plausibility of different entropy models? What new questions and insights for research on human information acquisition follow? Our work provides several new pathways for theoretical and empirical research, reconciling apparently conflicting approaches and empirical findings within a comprehensive and unified information-theoretic formalism.

Causal and noncausal conditionals: An integrated model of interpretation and reasoning

We present an integrated model for the understanding of and the reasoning from conditional statements. Central assumptions from several approaches are integrated into a causal path model. According to the model, the cognitive availability of exceptions to a conditional reduces the subjective conditional probability of the consequent, given the antecedent. This conditional probability determines people's degree of belief in the conditional, which in turn affects their willingness to accept logically valid inferences. In addition to this indirect pathway, the model contains a direct pathway: Availability of exceptional situations directly reduces the endorsement of valid inferences. We tested the integrated model with three experiments using conditional statements embedded in pseudonaturalistic cover stories. An explicitly mentioned causal link between antecedent and consequent was either present (causal conditionals) or absent (arbitrary conditionals). The model was supported for the causal but not for the arbitrary conditional statements.

Active Inference, Curiosity and Insight

This article offers a formal account of curiosity and insight in terms of active (Bayesian) inference. It deals with the dual problem of inferring states of the world and learning its statistical structure. In contrast to current trends in machine learning (e.g., deep learning), we focus on how people attain insight and understanding using just a handful of observations, which are solicited through curious behavior. We use simulations of abstract rule learning and approximate Bayesian inference to show that minimizing (expected) variational free energy leads to active sampling of novel contingencies. This epistemic behavior closes explanatory gaps in generative models of the world, thereby reducing uncertainty and satisfying curiosity. We then move from epistemic learning to model selection or structure learning to show how abductive processes emerge when agents test plausible hypotheses about symmetries (i.e., invariances or rules) in their generative models. The ensuing Bayesian model reduction evinces mechanisms associated with sleep and has all the hallmarks of "aha" moments. This formulation moves toward a computational account of consciousness in the pre-Cartesian sense of sharable knowledge (i.e., con: "together"; scire: "to know").

Predicting Short-Term Remembering as Boundedly Optimal Strategy Choice

It is known that, on average, people adapt their choice of memory strategy to the subjective utility of interaction. What is not known is whether an individual's choices are boundedly optimal. Two experiments are reported that test the hypothesis that an individual's decisions about the distribution of remembering between internal and external resources are boundedly optimal where optimality is defined relative to experience, cognitive constraints, and reward. The theory makes predictions that are tested against data, not fitted to it. The experiments use a no-choice/choice utility learning paradigm where the no-choice phase is used to elicit a profile of each participant's performance across the strategy space and the choice phase is used to test predicted choices within this space. They show that the majority of individuals select strategies that are boundedly optimal. Further, individual differences in what people choose to do are successfully predicted by the analysis. Two issues are discussed: (a) the performance of the minority of participants who did not find boundedly optimal adaptations, and (b) the possibility that individuals anticipate what, with practice, will become a bounded optimal strategy, rather than what is boundedly optimal during training.

Imaging deductive reasoning and the new paradigm

There has been a great expansion of research into human reasoning at all of Marr’s explanatory levels. There is a tendency for this work to progress within a level largely ignoring the others which can lead to slippage between levels (Chater et al., 2003). It is argued that recent brain imaging research on deductive reasoning—implementational level—has largely ignored the new paradigm in reasoning—computational level (Over, 2009). Consequently, recent imaging results are reviewed with the focus on how they relate to the new paradigm. The imaging results are drawn primarily from a recent meta-analysis by Prado et al. (2011) but further imaging results are also reviewed where relevant. Three main observations are made. First, the main function of the core brain region identified is most likely elaborative, defeasible reasoning not deductive reasoning. Second, the subtraction methodology and the meta-analytic approach may remove all traces of content specific System 1 processes thought to underpin much human reasoning. Third, interpreting the function of the brain regions activated by a task depends on theories of the function that a task engages. When there are multiple interpretations of that function, interpreting what an active brain region is doing is not clear cut. It is concluded that there is a need to more tightly connect brain activation to function, which could be achieved using formalized computational level models and a parametric variation approach.

Act first, think later: the presence and absence of inferential planning in problem solving

Planning is fundamental to successful problem solving, yet individuals sometimes fail to plan even one step ahead when it lies within their competence to do so. In this article, we report two experiments in which we explored variants of a ball-weighing puzzle, a problem that has only two steps, yet nonetheless yields performance consistent with a failure to plan. The results fit a computational model in which a solver's attempts are determined by two heuristics: maximization of the apparent progress made toward the problem goal and minimization of the problem space in which attempts are sought. The effectiveness of these heuristics was determined by lookahead, defined operationally as the number of steps evaluated in a planned move. Where move outcomes cannot be visualized but must be inferred, planning is constrained to the point where some individuals apply zero lookahead, which with n-ball problems yields seemingly irrational unequal weighs. Applying general-purpose heuristics with or without lookahead accounts for a range of rational and irrational phenomena found with insight and noninsight problems.

The nature of thinking, shallow and deep

Because the criteria for success differ across various domains of life, no single normative standard will ever work for all types of thinking. One method for dealing with this apparent dilemma is to propose that the mind is made up of a large number of specialized modules. This review describes how this multi-modular framework for the mind overcomes several critical conceptual and theoretical challenges to our understanding of human thinking, and hopefully clarifies what are (and are not) some of the implications based on this framework. In particular, an evolutionarily informed "deep rationality" conception of human thinking can guide psychological research out of clusters of ad hoc models which currently occupy some fields. First, the idea of deep rationality helps theoretical frameworks in terms of orienting themselves with regard to time scale references, which can alter the nature of rationality assessments. Second, the functional domains of deep rationality can be hypothesized (non-exhaustively) to include the areas of self-protection, status, affiliation, mate acquisition, mate retention, kin care, and disease avoidance. Thus, although there is no single normative standard of rationality across all of human cognition, there are sensible and objective standards by which we can evaluate multiple, fundamental, domain-specific motives underlying human cognition and behavior. This review concludes with two examples to illustrate the implications of this framework. The first example, decisions about having a child, illustrates how competing models can be understood by realizing that different fundamental motives guiding people's thinking can sometimes be in conflict. The second example is that of personifications within modern financial markets (e.g., in the form of corporations), which are entities specifically constructed to have just one fundamental motive. This single focus is the source of both the strengths and flaws in how such entities behave.

Missing the dog that failed to bark in the nighttime: on the overestimation of occurrences over non-occurrences in hypothesis testing

In three studies, we investigated whether and to what extent the evaluation of two mutually exclusive hypotheses is affected by a feature-positive effect, wherein present clues are weighted more than absent clues. Participants (N = 126) were presented with abstract problems concerning the most likely provenance of a card that was drawn from one of two decks. We factored the correct response (the hypothesis favored by the consideration of all clues) and the ratio of present-to-absent features in each set of observations. Furthermore, across the studies, we manipulated the presentation format of the features' probabilities by providing the probability distributions of occurrences (Study 1), non-occurrences (Study 3) or both (Study 2). In all studies, both participant preference and accuracy were mostly determined by an over-reliance on present features. Moreover, across participants, both confidence in the responses and the informativeness of the present clues correlated positively with the number of responses given in line with an exclusive consideration of present features. These results were mostly independent of both the rarity of the absent clues and the presentation format. We concluded that the feature-positive effect influences hypothesis evaluation, and we discussed the implications for confirmation bias.

The “is-ought fallacy” fallacy

Mere facts about how the world is cannot determine how we ought to think or behave. Elqayam & Evans (E&E) argue that this “is-ought fallacy” undercuts the use of rational analysis in explaining how people reason, by ourselves and with others. But this presumed application of the “is-ought” fallacy is itself fallacious. Rational analysis seeks to explain how people do reason, for example in laboratory experiments, not how they ought to reason. Thus, no ought is derived from an is; and rational analysis is unchallenged by E&E's arguments.

Bayesian inference for the information gain model

One of the most popular paradigms to use for studying human reasoning involves the Wason card selection task. In this task, the participant is presented with four cards and a conditional rule (e.g., “If there is an A on one side of the card, there is always a 2 on the other side”). Participants are asked which cards should be turned to verify whether or not the rule holds. In this simple task, participants consistently provide answers that are incorrect according to formal logic. To account for these errors, several models have been proposed, one of the most prominent being the information gain model (Oaksford & Chater, Psychological Review, 101, 608–631, 1994). This model is based on the assumption that people independently select cards based on the expected information gain of turning a particular card. In this article, we present two estimation methods to fit the information gain model: a maximum likelihood procedure (programmed in R) and a Bayesian procedure (programmed in WinBUGS). We compare the two procedures and illustrate the flexibility of the Bayesian hierarchical procedure by applying it to data from a meta-analysis of the Wason task (Oaksford & Chater, Psychological Review, 101, 608–631, 1994). We also show that the goodness of fit of the information gain model can be assessed by inspecting the posterior predictives of the model. These Bayesian procedures make it easy to apply the information gain model to empirical data. Supplemental materials may be downloaded along with this article from www.springerlink.com.

The imaginary fundamentalists: The unshocking truth about Bayesian cognitive science

If Bayesian Fundamentalism existed, Jones & Love's (J&L's) arguments would provide a necessary corrective. But it does not. Bayesian cognitive science is deeply concerned with characterizing algorithms and representations, and, ultimately, implementations in neural circuits; it pays close attention to environmental structure and the constraints of behavioral data, when available; and it rigorously compares multiple models, both within and across papers. J&L's recommendation of Bayesian Enlightenment corresponds to past, present, and, we hope, future practice in Bayesian cognitive science.

The effects of problem content and scientific background on information search and the assessment and valuation of correlations

The effects of problem contents and one's scientific background on the detection of correlations and the assessment of their strength were studied using a task that required active data search, assessment of the strength of a correlation, and monetary valuation of the correlation's predictive utility. Participants (N = 72) who were trained either in the natural sciences or in the social sciences and humanities explored data sets differing in contents and actual strength of correlation. Data search was consistent across all variables: Participants drew relatively small samples whose relative sizes would favor the detection of a correlation, if one existed. In contrast, the assessment of the correlation strength and the valuation of its predictive utility were strongly related not only to its objective strength, but also to the correspondence between problem contents and one's scientific background: When the two matched, correlations were judged to be stronger and more valuable than when they did not.

Broadening the study of inductive reasoning: confirmation judgments with uncertain evidence

Although evidence in real life is often uncertain, the psychology of inductive reasoning has, so far, been confined to certain evidence. The present study extends previous research by investigating whether people properly estimate the impact of uncertain evidence on a given hypothesis. Two experiments are reported, in which the uncertainty of evidence is explicitly (by means of numerical values) versus implicitly (by means of ambiguous pictures) manipulated. The results show that people's judgments are highly correlated with those predicted by normatively sound Bayesian measures of impact. This sensitivity to the degree of evidential uncertainty supports the centrality of inductive reasoning in cognition and opens the path to the study of this issue in more naturalistic settings.

Conditional and syllogistic deductive tasks dissociate functionally during premise integration

Deduction allows us to draw consequences from previous knowledge. Deductive reasoning can be applied to several types of problem, for example, conditional, syllogistic, and relational. It has been assumed that the same cognitive operations underlie solutions to them all; however, this hypothesis remains to be tested empirically. We used event-related fMRI, in the same group of subjects, to compare reasoning-related activity associated with conditional and syllogistic deductive problems. Furthermore, we assessed reasoning-related activity for the two main stages of deduction, namely encoding of premises and their integration. Encoding syllogistic premises for reasoning was associated with activation of BA 44/45 more than encoding them for literal recall. During integration, left fronto-lateral cortex (BA 44/45, 6) and basal ganglia activated with both conditional and syllogistic reasoning. Besides that, integration of syllogistic problems additionally was associated with activation of left parietal (BA 7) and left ventro-lateral frontal cortex (BA 47). This difference suggests a dissociation between conditional and syllogistic reasoning at the integration stage. Our finding indicates that the integration of conditional and syllogistic reasoning is carried out by means of different, but partly overlapping, sets of anatomical regions and by inference, cognitive processes. The involvement of BA 44/45 during both encoding (syllogisms) and premise integration (syllogisms and conditionals) suggests a central role in deductive reasoning for syntactic manipulations and formal/linguistic representations.

Recruitment of intuitive versus analytic thinking strategies affects the role of working memory in a gambling task

Previous studies found mixed results concerning the role of working memory (WM) in the gambling task (GT). Here, we aimed at reconciling inconsistencies by showing that the standard version of the task can be solved using intuitive strategies operating automatically, while more complex versions require analytic strategies drawing on executive functions. In Study 1, where good performance on the GT could be achieved using intuitive strategies, participants performed well both with and without a concurrent WM load. In Study 2, where analytical strategies were required to solve a more complex version of the GT, participants without WM load performed well, while participants with WM load performed poorly. In Study 3, where the complexity of the GT was further increased, participants in both conditions performed poorly. In addition to the standard performance measure, we used participants' subjective expected utility, showing that it differs from the standard measure in some important aspects.

Preferences for different questions when testing hypotheses in an abstract task: positivity does play a role, asymmetry does not

Previous studies on hypothesis-testing behaviour have reported systematic preferences for posing positive questions (i.e., inquiries about features that are consistent with the truth of the hypothesis) and different types of asymmetric questions (i.e., questions where the hypothesis confirming and the hypothesis disconfirming responses have different evidential strength). Both tendencies can contribute - in some circumstances - to confirmation biases (i.e., the improper acceptance or maintenance of an incorrect hypothesis). The empirical support for asymmetric testing is, however, scarce and partly contradictory, and the relative strength of positive testing and asymmetric testing has not been empirically compared. In four studies where subjects were asked to select (Experiment 1) or evaluate (Experiments 2-4) questions for controlling an abstract hypothesis, we orthogonally balanced the positivity/negativity of questions by their symmetry/asymmetry (Experiments 1-3), or by the type of asymmetry (confirmatory vs disconfirmatory; Experiment 4). In all Experiments participants strongly preferred positive to negative questions. Their choices were on the other hand mostly unaffected by symmetry and asymmetry in general, or - more specifically - by different types of asymmetry. Other results indicated that participants were sensitive to the diagnosticity of the questions (Experiments 1-3), and that they preferred testing features with a high probability under the focal hypothesis (Experiment 4). In the discussion we argue that recourse to asymmetric testing - observed in some previous studies using more contextualized problems - probably depends on context-related motivations and prior knowledge. In abstract tasks, where that knowledge is not available, more simple strategies - such as positive testing - are prevalent.

Bayesian learning theory applied to human cognition

Probabilistic models based on Bayes' rule are an increasingly popular approach to understanding human cognition. Bayesian models allow immense representational latitude and complexity. Because they use normative Bayesian mathematics to process those representations, they define optimal performance on a given task. This article focuses on key mechanisms of Bayesian information processing, and provides numerous examples illustrating Bayesian approaches to the study of human cognition. We start by providing an overview of Bayesian modeling and Bayesian networks. We then describe three types of information processing operations-inference, parameter learning, and structure learning-in both Bayesian networks and human cognition. This is followed by a discussion of the important roles of prior knowledge and of active learning. We conclude by outlining some challenges for Bayesian models of human cognition that will need to be addressed by future research. WIREs Cogn Sci 2011 2 8-21 DOI: 10.1002/wcs.80 For further resources related to this article, please visit the WIREs website.

Weighting of contingency information in causal judgement: evidence of hypothesis dependence and use of a positive-test strategy

Contingency is an important cue to causation. Research shows that people unequally weight the cells of a 2 x 2 contingency table as follows: cause-present/effect-present (A) > cause-present/effect-absent (B) > cause-absent/effect-present (C) > cause-absent/effect-absent (D). Although some models of causal judgement can accommodate that fact, most of them assume that the weighting of information is invariant as a function of whether one is assessing a hypothesized generative versus preventive relationship. An experiment was conducted that tested the hypothesis-independence assumption against the predictions of a novel weighted-positive-test-strategy account, which predicts hypothesis dependence in cell weighting. Supporting that account, judgements of hypothesized generative causes showed the standard A > B > C > D inequality, but judgements of hypothesized preventive causes showed the predicted B > A > D > C inequality. The findings reveal that cell weighting in causal judgement is both unequal and hypothesis dependent.

Précis of bayesian rationality: The probabilistic approach to human reasoning

According to Aristotle, humans are the rational animal. The borderline between rationality and irrationality is fundamental to many aspects of human life including the law, mental health, and language interpretation. But what is it to be rational? One answer, deeply embedded in the Western intellectual tradition since ancient Greece, is that rationality concerns reasoning according to the rules of logic--the formal theory that specifies the inferential connections that hold with certainty between propositions. Piaget viewed logical reasoning as defining the end-point of cognitive development; and contemporary psychology of reasoning has focussed on comparing human reasoning against logical standards. Bayesian Rationality argues that rationality is defined instead by the ability to reason about uncertainty. Although people are typically poor at numerical reasoning about probability, human thought is sensitive to subtle patterns of qualitative Bayesian, probabilistic reasoning. In Chapters 1-4 of Bayesian Rationality (Oaksford & Chater 2007), the case is made that cognition in general, and human everyday reasoning in particular, is best viewed as solving probabilistic, rather than logical, inference problems. In Chapters 5-7 the psychology of "deductive" reasoning is tackled head-on: It is argued that purportedly "logical" reasoning problems, revealing apparently irrational behaviour, are better understood from a probabilistic point of view. Data from conditional reasoning, Wason's selection task, and syllogistic inference are captured by recasting these problems probabilistically. The probabilistic approach makes a variety of novel predictions which have been experimentally confirmed. The book considers the implications of this work, and the wider "probabilistic turn" in cognitive science and artificial intelligence, for understanding human rationality.

How do individuals reason in the Wason card selection task?

The probabilistic approach to human reasoning is exemplified by the information gain model for the Wason card selection task. Although the model is elegant and original, several key aspects of the model warrant further discussion, particularly those concerning the scope of the task and the choice process of individuals.

Increased sensitivity to differentially diagnostic answers using familiar materials: implications for confirmation bias

Researchers have recently pointed out that neither biased testing nor biased evaluation of hypotheses necessitates confirmation bias--defined here as systematic overconfidence in a focal hypothesis--but certain testing/evaluation combinations do. One such combination is (1) a tendency to ask about features that are either very likely or very unlikely under the focal hypothesis (extremity bias) and (2) a tendency to treat confirming and disconfirming answers as more similar in terms of their diagnosticity (or informativeness) than they really are. However, in previous research showing the second tendency, materials that are highly abstract and unfamiliar have been used. Two experiments demonstrated that using familiar materials led participants to distinguish much better between the differential diagnosticity of confirming and disconfirming answers. The conditions under which confirmation bias is a serious concern might be quite limited.

A Bayesian view of covariation assessment

When participants assess the relationship between two variables, each with levels of presence and absence, the two most robust phenomena are that: (a) observing the joint presence of the variables has the largest impact on judgment and observing joint absence has the smallest impact, and (b) participants' prior beliefs about the variables' relationship influence judgment. Both phenomena represent departures from the traditional normative model (the phi coefficient or related measures) and have therefore been interpreted as systematic errors. However, both phenomena are consistent with a Bayesian approach to the task. From a Bayesian perspective: (a) joint presence is normatively more informative than joint absence if the presence of variables is rarer than their absence, and (b) failing to incorporate prior beliefs is a normative error. Empirical evidence is reported showing that joint absence is seen as more informative than joint presence when it is clear that absence of the variables, rather than their presence, is rare.

Finding useful questions: on Bayesian diagnosticity, probability, impact, and information gain

Several norms for how people should assess a question's usefulness have been proposed, notably Bayesian diagnosticity, information gain (mutual information), Kullback-Liebler distance, probability gain (error minimization), and impact (absolute change). Several probabilistic models of previous experiments on categorization, covariation assessment, medical diagnosis, and the selection task are shown to not discriminate among these norms as descriptive models of human intuitions and behavior. Computational optimization found situations in which information gain, probability gain, and impact strongly contradict Bayesian diagnosticity. In these situations, diagnosticity's claims are normatively inferior. Results of a new experiment strongly contradict the predictions of Bayesian diagnosticity. Normative theoretical concerns also argue against use of diagnosticity. It is concluded that Bayesian diagnosticity is normatively flawed and empirically unjustified.

Framing effects in inference tasks--and why they are normatively defensible

Framing effects occur when logically equivalent redescriptions of objects or outcomes lead to different behaviors, and, traditionally, such effects have been seen as irrational. However, recent evidence has shown that a speaker's choice among logically equivalent attribute frames can implicitly convey (or "leak") normatively relevant information about the speaker's reference point, among other things. In a reinterpretion of data published elsewhere, in this article it is shown that some common effects in inference tasks (covariation assessment and hypothesis testing) can also be seen as framing effects, thereby expanding the domain of framing. It is also shown that these framing effects are normatively defensible because normatively relevant information about event rarity is leaked through the description of data and through the phrasing of hypotheses, thereby broadening the information leakage approach to explaining framing effects. Information leakage can also explain why framing effects in such inference tasks disappear under certain conditions.

Negations and natural sampling in data selection: Ecological versus heuristic explanations of matching bias

Matching bias occurs when people ignore negations when testing a hypothesis--for example, if A, then not 2--and select possible data types that are named in the hypothesis (i.e., A and 2; Evans & Lynch, 1973). There are two explanations of this bias: the heuristic account and the contrast class account. The latter is part of Oaksford and Chater's (1994) ecological approach to data selection. On this account, a contrast set (i.e., birds that are not ravens) has a higher probability than the original set (i.e., birds that are ravens). This article reports two experiments in which these accounts make divergent predictions. The same materials were used as those in Yama (2001), who found more support for the heuristic approach. Experiment 1 replicated Yama with Western participants. Experiment 2 used a procedure introduced by Oaksford and Wakefield (2003). Rather than present participants with one of each of the four possible data types all at once, 50 were presented one at a time. The proportions of each data type reflected the relevant probabilities. The results supported the ecological approach, showing that people constructed contrast sets that strongly influenced their data selection behavior. The results were not consistent with the heuristic approach.

Rational models as theories - not standards - of behavior

When people's behavior in laboratory tasks systematically deviates from a rational model, the implication is that real-world performance could be improved by changing the behavior. However, recent studies suggest that behavioral violations of rational models are at least sometimes the result of strategies that are well adapted to the real world (and not necessarily to the laboratory task). Thus, even if one accepts that certain behavior in the laboratory is irrational, compelling evidence that real-world behavior ought to change accordingly is often lacking. It is suggested here that rational models be seen as theories, and not standards, of behavior.

When wrong predictions provide more support than right ones

Correct predictions of rare events are normatively more supportive of a theory or hypothesis than correct predictions of common ones. In other words, correct bold predictions provide more support than do correct timid predictions. Are lay hypothesis testers sensitive to the boldness of predictions? Results reported here show that participants were very sensitive to boldness, often finding incorrect bold predictions more supportive than correct timid ones. Participants were willing to tolerate inaccurate predictions only when predictions were bold. This finding was demonstrated in the context of competing forecasters and in the context of competing scientific theories. The results support recent views of human inference that postulate that lay hypothesis testers are sensitive to the rarity of data. Furthermore, a normative (Bayesian) account can explain the present results and provides an alternative interpretation of similar results that have been explained using a purely descriptive model.