The terms nested sets, partitive frequencies, inside-outside view, and dual processes add little but confusion to our original analysis (Gigerenzer & Hoffrage 1995; 1999). The idea of nested set was introduced because of an oversight; it simply rephrases two of our equations. Representation in terms of chances, in contrast, is a novel contribution yet consistent with our computational analysis System 1.dual process theory” is: Unless the two processes are defined, this distinction can account post hoc for almost everything. In contrast, (...) an ecological view of cognition helps to explain how insight is elicited from the outside (the external representation of information) and, more generally, how cognitive strategies match with environmental structures. (shrink)

People often struggle with Bayesian reasoning. However, research showed that people’s performance (and rationality) can be supported by the way of representing the statistical information. First, research showed that using natural frequencies instead of probabilities as format of statistical information increases people’s performance in Bayesian situations thoroughly. Second, research also yielded that people’s performance increases through using visualization. We build our paper on existing research in this field. The main aim is to analyse people’s strategies in (...) Bayesian situations that are still erroneous although statistical information is represented by natural frequencies and visualizations. In particular, we compare two pairs of visualization with similar numerical information (tree diagram vs. unit square and double-tree diagram vs. 2x2-table) concerning their impact on people’s erroneous strategies in Bayesian situations. For this aim, we conducted an experiment with 540 university students. The students were randomly assigned to four conditions that are defined by the four different visualizations of statistical information. The students were asked to indicate a fraction as response in four Bayesian situations. We documented the numerator and the denominator of the students’ responses representing a basic set and a subset in a Bayesian situation. Our results show that people’s erroneous strategies are highly dependent on the visualization. A central finding is that the visualization’s characteristic of making the nested-sets structure of a Bayesian situation transparent has a facilitating effect on people’s Bayesian reasoning. For example, compared to the unit square, a tree diagram does not explicitly visualize the set-subset relations that are relevant in a Bayesian situation. Accordingly, compared to a unit square, a tree diagram partly hinders people to find the correct denominator in a Bayesian situation, and, in particular, triggers to select a wrong numerator. By analyzing people’s erroneous strategies in Bayesian situations, we contribute to investigating approaches to facilitate Bayesian reasoning and to further develop the teaching of Bayesian reasoning. (shrink)

A diagnostic judgment of a teacher can be seen as an inference from manifest observable evidence on a student’s behavior to his or her latent traits. This can be described by a Bayesian model of in-ference: The teacher starts from a set of assumptions on the student (hypotheses), with subjective probabilities for each hypothesis (priors). Subsequently, he or she uses observed evidence (stu-dents’ responses to tasks) and knowledge on conditional probabilities of this evidence (likelihoods) to revise these assumptions. Many (...) systematic deviations from this model (biases, e.g. base-rate neglect, inverse fallacy) are reported in literature on Bayesian reasoning. In a teacher’s situation the information (hypotheses, priors, likelihoods) is usually not explicitly represented numerically (as in most research on Bayesian reasoning), but only by qualitative esti-mations in the mind of the teacher. In our study, we ask to which extent individuals (approximately) apply a rational Bayesian strategy or resort on other biased strategies of processing information for their diagnostic judgments. We explicitly pose this question with respect to non-numerical settings. To investigate this question, we developed a scenario that visually displays all relevant information (hypotheses, priors, likelihoods) in a graphically displayed hypothesis space (called “hypothegon”) – without recurring to numerical representations or mathematical procedures. 42 pre-service teach-ers were asked to judge the plausibility of different misconceptions of six students based on their responses to decimal comparison tasks (e.g. 3.39 > 3.4). Applying a Bayesian classification proce-dure, we identified three updating strategies: A Bayesian update strategy (BUS, processing all probabilities), a combined evidence strategy (CES, ignoring the prior probabilities but including all likelihoods), and a single evidence strategy (SES, only using the likelihood of the most probable hypothesis). In study 1, an instruction on the relevance of using all probabilities (priors and likelihoods) only weakly increased the processing of more information. In study 2, we found strong evidence that a visual explication of the prior-likelihood interaction led to an increase of processing the interaction of all relevant information. These results show that the phenomena found in general research on Bayesian reasoning in numerical settings extend to diagnostic judgments in non-numerical settings. (shrink)

A longstanding question is the extent to which "reasonable doubt" may be expressed simply in terms of a threshold degree of belief. In this context, we examine the extent to which learning about possible alternatives may alter one's beliefs about a target hypothesis, even when no new "evidence" linking them to the hypothesis is acquired. Imagine the following scenario: a crime has been committed and Alice, the police's main suspect has been brought to trial. There are several pieces of evidence (...) that raise the probability that Alice committed the crime. Her attorney's defense strategy is not to challenge this evidence, but instead to provide personal details about Alice's neighbour, Jane. While Jane is one of many people the police spoke to, they saw no reason to investigate her further. You now learn that Jane, too, had access to the shed where the murder weapon was stored, just like Alice. To what extent should this alter your beliefs about Alice's guilt? In this paper, we provide a formal description of the problem and a solution indicating circumstances under which learning about Jane will more or less impact beliefs about Alice. (shrink)

This paper discusses the Bayesian updating rules of ordinary and Jeffrey conditionalisation. Their justification has been a topic of interest for the last quarter century, and several strategies proposed. None has been accepted as conclusive, and it is argued here that this is for a good reason; for by extending the domain of the probability function to include propositions describing the agent's present and future degrees of belief one can systematically generate a class of counterexamples to the rules. (...) Dynamic Dutch Book and other arguments for them are examined critically. A concluding discussion attempts to put these results in perspective within the Bayesian approach. (shrink)

Chow ably defends classical significance testing by relating this method to venerable principles for inductive reasoning. Chow's success does not preclude the use of other approaches to statistical reasoning, which is fortunate not only for Bayesian rivals, but even for some fellow classicists.

Oaksford & Chater (O&C) focus on patterns of typical adult reasoning from a probabilistic perspective. We discuss implications of extending the probabilistic approach to lifespan development, considering the role of working memory, strategy use, and expertise. Explaining variations in human reasoning poses a challenge to Bayesian rational analysis, as it requires integrating knowledge about cognitive processes.

This article proposes two interpretations of the concept of belief hierarchies in Bayesian game theory: the behaviorist interpretation and the mentalist interpretation. On the former, belief hierarchies are derived from the players’ preferences over acts. On the latter, they are causal mechanisms that are responsible for the players’ choices and preferences over acts. The claim is that the epistemic program in game theory is potentially confronted with a dilemma regarding which interpretation should be adopted. If the behaviorist interpretation of (...) belief hierarchies is retained, then Bayesian game theory cannot account for the players’ reasoning that leads to the generation of specific belief hierarchies and thus to the implementation of a specific strategy profile in a game. If the mentalist interpretation is retained, then belief hierarchies appear to be unnecessarily and unrealistically complex objects and that as a result they fail to provide reasonable epistemic foundations for various solution concepts. In other words, neither interpretation seems to allow the epistemic program to fulfill all its aims. I suggest however that the mentalist interpretation can be arranged to escape this dilemma. (shrink)

This article proposes two interpretations of the concept of belief hierarchies in Bayesian game theory: the behaviorist interpretation and the mentalist interpretation. On the former, belief hierarchies are derived from the players’ preferences over acts. On the latter, they are causal mechanisms that are responsible for the players’ choices and preferences over acts. The claim is that the epistemic program in game theory is potentially confronted with a dilemma regarding which interpretation should be adopted. If the behaviorist interpretation of (...) belief hierarchies is retained, then Bayesian game theory cannot account for the players’ reasoning that leads to the generation of specific belief hierarchies and thus to the implementation of a specific strategy profile in a game. If the mentalist interpretation is retained, then belief hierarchies appear to be unnecessarily and unrealistically complex objects and that as a result they fail to provide reasonable epistemic foundations for various solution concepts. In other words, neither interpretation seems to allow the epistemic program to fulfill all its aims. I suggest however that the mentalist interpretation can be arranged to escape this dilemma. (shrink)

The report begins by outlining several case studies with varying levels of data, examining the role for extreme event risk analysis. The case studies include BA’s analysis of fire blight and New Zealand apples, bank operational risk and several technical failures. The report then surveys recent developments in methods relevant to evaluating extreme risks and evaluates their properties. These include methods for fraud detection in banks, formal extreme value theory, Bayesian approaches, qualitative reasoning, and adversary and advocacy models. (...) The document includes a supplementary report as an appendix, providing an overview of the quantification of bank operational risks. (shrink)

In this article, I explore a Bayesian approach to avalanche decision-making. I motivate this perspective by highlighting a version of the base-rate fallacy and show that a similar pattern applies to decision-making in avalanche-terrain. I then draw out three theoretical lessons from adopting a Bayesian approach and discuss these lessons critically. Lastly, I highlight a number of challenges for avalanche educators when incorporating the Bayesian perspective in their curriculum.

Molecular biologists use different kinds of reasoning strategies for different tasks, such as hypothesis formation, experimental design, and anomaly resolution. More specifically, the reasoning strategies discussed in this paper may be characterized as (1) abstraction-instantiation, in which an abstract skeletal model is instantiated to produce an experimental system; (2) the systematic scan, in which alternative hypotheses are systematically generated; and (3) modular anomaly resolution, in which components of a model are stated explicitly and methodically changed to (...) generate alternative changes to resolve an anomaly. This work grew out of close observation over a period of six months of an actively functioning molecular genetics laboratory. (shrink)

*PSA 2016, symposium on “Data in Time: Epistemology of Historical Data” organized by Sabina Leonelli, 5 November 2016* *See published version: "Radiocarbon Dating in Archaeology: Triangulation and Traceability" in Data Journeys in the Sciences (2020) - link below* Archaeologists put a premium on pressing “legacy data” into service, given the notoriously selective and destructive nature of their practices of data capture. Legacy data consist of material and records that been assembled over decades, sometimes centuries, often by means and for purposes (...) long since discredited or superseded. The primary strategies by which archaeologists put the data to work for new purposes are, I argue, secondary retrieval, recontextualization, and experimental modelling. I focus here on a particularly telling and complex example of secondary retrieval: the extraction of new data from old by means of radiocarbon dating. This is by no means a straightforward process of retrieving physical samples from legacy data to which 14C techniques can be applied that can, on their own, decisively settle chronological questions. When Libby’s post-war radiocarbon revolution got under way, it was expected to establish an absolute chronology that would render obsolete the local and relative chronologies on which archaeologists had long relied. Transformative though it has been, bringing these tools of physical dating to bear on archaeological problems has been a long, tortuous process, now described as proceeding through two subsequent radiocarbon revolutions. The second was an extended process of calibration by which 14C chronologies were corrected and refined, often against the very lines of evidence they were meant to displace. The most recent, a “pragmatic Bayesian” approach to archaeological dating, is motivated by concern that, no matter how much it is refined, radiocarbon dating cannot on its own resolve the chronological problems that archaeologists address; the challenge, its advocates argue, is to ‘fully integrate archaeological information with 14C dating in order to address archaeologically relevant timescales and episodes’. This is a genre of “robustness” reasoning that illustrates its epistemic risks as well as its appeal. As recent philosophical debate makes clear, it depends on appeals to the convergence of independent lines of evidence that may have more rhetorical than epistemic force and that may be spurious. Drawing on this philosophical literature I identify a set of conditions that must be met if these risks are to be avoided, all of which are an explicit focus of debate in cases of contestation about and reconciliation of chronologies based on legacy data. (shrink)

Deductive reasoning shares with other forms of thinking a reliance on strategies, as shown by the results of three experiments on the nature and development of control strategies to solve suppositional deductions. These puzzles are based on assertors who may or may not be telling the truth, and their assertions about their status as truthtellers and liars. The first experiment shows that reasoners make backward inferences as well as forward inferences, to short-cut their way through the alternatives, (...) and the generation of suppositions is a source of difficulty. The second experiment establishes that the elimination of the suppositional status of an individual does not render problems easier. The third experiment shows that reasoners can improve their reasoning accuracy and speed spontaneously, without feedback, and it clarifies the transfer of strategies and their development. We discuss the implications of these data for alternative theories of suppositional deduction and for the relationship between reasoning and other forms of thinking such as problem solving. (shrink)

This brief annotated bibliography is intended to help students get started with their research. It is not a substitute for personal investigation of the literature, and it is not a comprehensive bibliography on the subject. For those just beginning to study Bayesian reasoning, I suggest the starred items as good places to start your reading.

God in the Age of Science? is a critical examination of strategies for the philosophical defence of religious belief. Herman Philipse argues that the most promising for believers who want to be justified in accepting their creed in our scientific age is the Bayesian cumulative case strategy developed by Richard Swinburne, and goes on to present an in-depth analysis of this case for theism. Using a 'strategy of subsidiary arguments', Philipse concludes that theism cannot be stated meaningfully; that (...) if theism were meaningful, it would have no predictive power concerning existing evidence, so that Bayesian arguments cannot get started; and that if the Bayesian cumulative case strategy did work, one should conclude that atheism is more probable than theism. Philipse provides a careful, rigorous, and original critique of atheism in the world today. (shrink)

Delusions are one of the most classical symptoms described in schizophrenia. However, despite delusions are often emotionally charged, they have been investigated using tasks involving non-affective material, such as the Beads task. In this study we compared 30 patients with schizophrenia experiencing delusions with 32 matched controls in their pattern of responses to two versions of the Beads task within a Bayesian framework. The two versions of the Beads task consisted of one emotional and one neutral, both with ratios (...) of beads of 60:40 and 80:20, considered, respectively, as the “difficult” and “easy” variants of the task. Results indicate that patients showed a greater deviation from the normative model, especially in the 60:40 ratio, suggesting that more inaccurate probability estimations are more likely to occur under uncertainty conditions. Additionally, both patients and controls showed a greater deviation in the emotional version of the task, providing evidence of a reasoning bias modulated by the content of the stimuli. Finally, a positive correlation between patients’ deviation and delusional symptomatology was found. Impairments in the 60:40 ratio with emotional content was related to the amount of disruption in life caused by delusions. These results contribute to the understanding of how cognitive mechanisms interact with characteristics of the task in the context of delusional thinking. These findings might be used to inform improved intervention programs in the domain of inferential reasoning. (shrink)

Bayesianism is the position that scientific reasoning is probabilistic and that probabilities are adequately interpreted as an agent's actual subjective degrees of belief, measured by her betting behaviour. Confirmation is one important aspect of scientific reasoning. The thesis of this paper is the following: if scientific reasoning is at all probabilistic, the subjective interpretation has to be given up in order to get right confirmation—and thus scientific reasoning in general. The Bayesian approach to scientific (...) class='Hi'>reasoning Bayesian confirmation theory The example The less reliable the source of information, the higher the degree of Bayesian confirmation Measure sensitivity A more general version of the problem of old evidence Conditioning on the entailment relation The counterfactual strategy Generalizing the counterfactual strategy The desired result, and a necessary and sufficient condition for it Actual degrees of belief The common knock-down feature, or ‘anything goes’ The problem of prior probabilities. (shrink)

Consumers can have difficulty expressing their buying intentions on an explicit level. The most common explanation for this intention-action gap is that consumers have many cognitive biases that interfere with decision making. The current resource-rational approach to understanding human cognition, however, suggests that brain environment interactions lead consumers to minimize the expenditure of cognitive energy. This means that the consumer seeks as simple of a solution as possible for a problem requiring decision making. In addition, this resource-rational approach to decision (...) making emphasizes the role of the principle of Occam’s Razor, inductive inference, and Bayesian reasoning by illuminating the human-environment relationship. This paper introduces the Consumer Contextual Decision-Making Model (CCDMM). Based on the CCDMM, two hypothetical strategies of consumer decision making will be presented. First, the SIMilarity-Strategy (SIMS) is one in which most of a consumer’s decisions in a real-life context are based on prior beliefs about the role of a commodities specific to real-life situation being encountered. Because beliefs are based on previous experiences, consumers are already aware of the most likely consequences of their actions. At the same time, they do not waste time on developing contingencies for what, based on previous experience, is unlikely to happen. Second, the What-is-Out-there-in-the-World-Strategy (WOWS) is one in which prior beliefs do not work in a real-life situation, requiring consumers to update their beliefs. The principle argument being made is that most experimental consumer research describes decision making based on the WOWS, when participants cannot apply their previous knowledge and situation-based strategy to problems. The article analyses sensory and cognitive biases described by behavioral economists from a CCDMM perspective, followed by a description and explanation of the typical intention-action gap based on the model. Prior to a section dedicated to discussion, the neuroeconomic approach will be described along with the valuation network of the brain, which has evolved to solve problems that the human has previously encountered in an information-rich environment. The principles of brain functions will also be compared to CCDMM. Finally, different approaches and the future direction of consumer research from a contextual point of view will be presented. (shrink)

Stanovich & West interpret errors in syllogistic reasoning in terms of computational limitations. I argue that the variety of strategies used by reasoners in solving syllogisms requires us to consider also performance errors. Although reasoners' performance from one trial to another is quite consistent, it can be different, in line with the definition of performance errors. My argument has methodological implications for reasoning theories.

Ofsted requires UK schools to help students understand the working world and gain employability skills. However, the aims of education are much broader: Education should enable flourishing long after leaving school. Therefore, students’ career decisions should be conducive to long-term flourishing beyond career readiness and educational attainment. In this mixed-methods study, we asked a representative sample of UK adults to reflect on their career decision-making processes at school and at university. We also measured current levels of self-reported objective (e.g., financial (...) security) and subjective (e.g., subjective well-being) flourishing. The open-ended career decision reflections were coded for three moral reasoning strategies: virtue ethical, consequentialist, and deontological. Using correlations and structural equation modelling, we examined the association between the propensity for using each moral reasoning strategy in past career decisions and current flourishing. Virtue ethical moral reasoning in relation to career decision-making predicted aspects of flourishing most strongly and frequently. Consequentialist reasoning weakly and infrequently positively predicted aspects of flourishing. Deontological reasoning either did not predict flourishing at all, or negatively predicted flourishing. Our results suggest that the reasoning strategy behind career decisions people take in school or university is important to consider in UK careers provision, and current best practice. (shrink)

Morality, in the context of luxury counterfeit goods, has been widely discussed in existing literature as having a strong association with decreased purchase intention. However, drawing on moral disengagement theory, we argue that individuals are motivated to justify their immoral behaviors through guilt avoidance, thus increasing counterfeit purchase intention. This research demonstrates that consumers’ desire to purchase counterfeit luxuries hinges on two types of moral reasoning strategies: moral rationalization and moral decoupling. The empirical results show that each strategy (...) increases purchase intention, but respectively through moral judgment and perceived benefit. Implications for researchers and managers are discussed. (shrink)

This brief annotated bibliography is intended to help students get started with their research. It is not a substitute for personal investigation of the literature, and it is not a comprehensive bibliography on the subject. For those just beginning to study probabilistic confirmation theory and Bayesian reasoning, I suggest the starred items as good places to start your reading.

Three experiments investigated the effect of rarity on people's selection and interpretation of data in a variant of the pseudodiagnosticity task. For familiar (Experiment 1) but not for arbitrary (Experiment 3) materials, participants were more likely to select evidence so as to complete a likelihood ratio when the initial evidence they received was a single likelihood concerning a rare feature. This rarity effect with familiar materials was replicated in Experiment 2 where it was shown that participants were relatively insensitive to (...) explicit manipulations of the likely diagnosticity of rare evidence. In contrast to the effects for data selection, there was an effect of rarity on confidence ratings after receipt of a single likelihood for arbitrary but not for familiar materials. It is suggested that selecting diagnostic evidence necessitates explicit consideration of the alternative hypothesis and that consideration of the possible consequences of the evidence for the alternative weakens the rarity effect in confidence ratings. Paradoxically, although rarity effects in evidence selection and confidence ratings are in the spirit of Bayesian reasoning, the effect on confidence ratings appears to rely on participants thinking less about the alternative hypothesis. (shrink)

The Bayesian view of inference has become popular in philosophy in recent years. Scientific Reasoning: a Bayesian Approach, by Colin Howson and Peter Urbach, represents an articulate and persuasive defense of the Bayesian view. We focus on the theme of that book, and argue that there are difficulties with Bayesianism, and alternatives worth considering. One of the most serious drawbacks to Bayesianism is the subjectivity that pervades most versions of it. We argue that this is an (...) instance of a more general contemporary tendency to move away from claims of objectivity, and toward frankly subjective views. This results from a desire to find a deductive, incorrigible, basis for scientific inference. We claim that such a desire is doomed to frustration, but that does not spell the end of efforts to formalize inductive reasoning. (shrink)

This work examines the influence of reference dependence, including value selection bias and congruence effects, on diagnostic reasoning. Across two studies, we explored how dependence on the initial problem structure influences the ability to solve simplified precursors to the more traditional Bayesian reasoning problems. Analyses evaluated accuracy and types of response errors as a function of congruence between the problem presentation and question of interest, amount of information, need for computation, and individual differences in numerical abilities. Across (...) all problem variations, there was consistent and strong evidence of a value selection bias in that incorrect responses almost always conformed to values that were provided in the problem rather than other errors including those related to computation. The most consistent and unexpected error across all conditions in the first experiment was that people were often more likely to utilize the superordinate value as part of their solution rather than the anticipated reference class values. This resulted in a weakened effect of congruence, with relatively low accuracy even in congruent conditions, and a dominant response error of the superordinate value. Experiment 2 confirmed that the introduction of a new sample drew attention away from the provided reference class, increasing reliance on the overall sample size. This superordinate preference error, along with the benefit of repeating the PPV reference class within the question, demonstrated the importance of reference dependence based on the salience of information within the response prompt. Throughout, higher numerical skills were generally associated with higher accuracy, whether calculations were required or not. (shrink)

For more than 20 years, research has proven the beneficial effect of natural frequencies when it comes to solving Bayesian reasoning tasks (Gigerenzer & Hoffrage, 1995). In a recent meta-analysis, McDowell & Jacobs (2017) showed that presenting a task in natural frequency format increases performance rates to 24% compared to only 4% when the same task is presented in probability format. Nevertheless, on average three quarters of participants in their meta-analysis failed to obtain the correct solution for such (...) a task in frequency format. In this paper, we present an empirical study on what participants typically do wrong when confronted with natural frequencies. We found that many of them did not actually use natural frequencies for their calculations, but translated them back into complicated probabilities instead. This switch from the intuitive presentation format to a less intuitive calculation format will be discussed within the framework of psychological theories (e.g., the Einstellung effect). (shrink)

Many researchers want to unify probability and logic by defining logical probability or probabilistic logic reasonably. This paper tries to unify statistics and logic so that we can use both statistical probability and logical probability at the same time. For this purpose, this paper proposes the P–T probability framework, which is assembled with Shannon’s statistical probability framework for communication, Kolmogorov’s probability axioms for logical probability, and Zadeh’s membership functions used as truth functions. Two kinds of probabilities are connected by an (...) extended Bayes’ theorem, with which we can convert a likelihood function and a truth function from one to another. Hence, we can train truth functions (in logic) by sampling distributions (in statistics). This probability framework was developed in the author’s long-term studies on semantic information, statistical learning, and color vision. This paper first proposes the P–T probability framework and explains different probabilities in it by its applications to semantic information theory. Then, this framework and the semantic information methods are applied to statistical learning, statistical mechanics, hypothesis evaluation (including falsification), confirmation, and Bayesian reasoning. Theoretical applications illustrate the reasonability and practicability of this framework. This framework is helpful for interpretable AI. To interpret neural networks, we need further study. (shrink)

The Monty Hall Dilemma (MHD) is a two-step decision problem involving counterintuitive conditional probabilities. The first choice is made among three equally probable options, whereas the second choice takes place after the elimination of one of the non-selected options which does not hide the prize. Differing from most Bayesian problems, statistical information in the MHD has to be inferred, either by learning outcome probabilities or by reasoning from the presented sequence of events. This often leads to suboptimal decisions (...) and erroneous probability judgments. Specifically, decision makers commonly develop a wrong intuition that final probabilities are equally distributed, together with a preference for their first choice. Several studies have shown that repeated practice enhances sensitivity to the different reward probabilities, but does not facilitate correct Bayesian reasoning. However, modest improvements in probability judgments have been observed after guided explanations. To explain these dissociations, the present review focuses on two types of causes producing the observed biases: Emotional-based choice biases and cognitive limitations in understanding probabilistic information. Among the latter, we identify a crucial cause for the universal difficulty in overcoming the equiprobability illusion: Incomplete representation of prior and conditional probabilities. We conclude that repeated practice and/or high incentives can be effective for overcoming choice biases, but promoting an adequate partitioning of possibilities seems to be necessary for overcoming cognitive illusions and improving Bayesian reasoning. (shrink)

Bayesian reasoning, defined here as the updating of a posterior probability following new information, has historically been problematic for humans. Classic psychology experiments have tested human Bayesian reasoning through the use of word problems and have evaluated each participant’s performance against the normatively correct answer provided by Bayes’ theorem. The standard finding is of generally poor performance. Over the past two decades, though, progress has been made on how to improve Bayesian reasoning. Most notably, (...) research has demonstrated that the use of frequencies in a natural sampling framework —as opposed to single-event probabilities—can improve participants’ Bayesian estimates. Furthermore, pictorial aids and certain individual difference factors also can play significant roles in Bayesian reasoning success. The mechanics of how to build tasks which show these improvements is not under much debate. The explanations for why naturally-sampled frequencies and pictures help Bayesian reasoning remain hotly contested, however, with many researchers falling into ingrained “camps” organized around two dominant theoretical perspectives. The present paper evaluates the merits of these theoretical perspectives, including the weight of empirical evidence, theoretical coherence, and predictive power. By these criteria, the ecological rationality approach is clearly better than the heuristics and biases view. Progress in the study of Bayesian reasoning will depend on continued research that honestly, vigorously, and consistently engages across these different theoretical accounts rather than staying “siloed” within one particular perspective. The process of science requires an understanding of competing points of view, with the ultimate goal being integration. (shrink)

Previous research suggests that children can infer causal relations from patterns of events. However, what appear to be cases of causal inference may simply reduce to children recognizing relevant associations among events, and responding based on those associations. To examine this claim, in Experiments 1 and 2, children were introduced to a “blicket detector”, a machine that lit up and played music when certain objects were placed upon it. Children observed patterns of contingency between objects and the machine’s activation that (...) required them to use indirect evidence to make causal inferences. Critically, associative models either made no predictions, or made incorrect predictions about these inferences. In general, children were able to make these inferences, but some developmental differences between 3- and 4- year-olds were found. We suggest that children’s causal inferences are not based on recognizing associations, but rather that children develop a mechanism for Bayesian structure learning. Experiment 3 explicitly tests a prediction of this account. Children were asked to make an inference about ambiguous data based on the base-rate of certain events occurring. Fouryear-olds, but not 3-year-olds were able to make this inference. (shrink)

A key benefit of Bayesian reasoning is that it stipulates how to optimally integrate unreliable sources of information. The authors present evidence that humans use Bayesian inference to determine how much to trust advice from another person, based on information about that person's knowledge and strategy.

Previous research suggests that children can infer causal relations from patterns of events. However, what appear to be cases of causal inference may simply reduce to children recognizing relevant associations among events, and responding based on those associations. To examine this claim, in Experiments 1 and 2, children were introduced to a “blicket detector,” a machine that lit up and played music when certain objects were placed upon it. Children observed patterns of contingency between objects and the machine's activation that (...) required them to use indirect evidence to make causal inferences. Critically, associative models either made no predictions, or made incorrect predictions about these inferences. In general, children were able to make these inferences, but some developmental differences between 3- and 4-year-olds were found. We suggest that children's causal inferences are not based on recognizing associations, but rather that children develop a mechanism for Bayesian structure learning. Experiment 3 explicitly tests a prediction of this account. Children were asked to make an inference about ambiguous data based on the base rate of certain events occurring. Four-year-olds, but not 3-year-olds were able to make this inference. (shrink)

The formalism of operational statistics, a generalized approach to probability and statistics, provides a setting within which inference strategies can be studied with great clarity. This paper is concerned with the asymptotic behavior of the Bayesian inference strategy in this setting. We consider a sequence of posterior distributions, obtained from a prior as a result of successive conditionings by the events of an admissible sequence. We identify certain statistical hypotheses whose limiting posterior probabilities converge to one. We describe (...) these hypotheses, and show that when the prior is vague, they contain those probability models which represent the long-run relative frequencies of occurrence for the events in the sequence. (shrink)

We asked college students to make judgments about realistic moral situations presented as dilemmas (which asked for an either/or decision) vs. problems (which did not ask for such a decision) as well as when the situation explicitly included affectively salient language vs. non-affectively salient language. We report two main findings. The first is that there are four different types of cognitive strategy that subjects use in their responses: simple reasoning, intuitive judging, cautious reasoning, and empathic reasoning. We (...) give operational definitions of these types in terms of our observed data. In addition, the four types characterized strategies not only in the whole sample, but also in all of the subsamples in our study. The second finding is that the intuitive judging type comprised approximately 26% of our respondents, while about 74% of our respondents employed one of the three styles of reasoning named above. We think that these findings present an interesting challenge to models of moral cognition which predict that there is either a single, or a single most common, strategy – especially a strategy of relying upon one’s intuitions – that people use to think about moral situations. (shrink)

Scientific reasoning is—and ought to be—conducted in accordance with the axioms of probability. This Bayesian view—so called because of the central role it accords to a theorem first proved by Thomas Bayes in the late eighteenth ...