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*Correspondence:

This article was submitted to Cognition, a section of the journal Frontiers in Psychology.

Edited by: Gorka Navarrete, Universidad Diego Portales, Chile

Reviewed by: Johan Kwisthout, Radboud University Nijmegen, Netherlands; Karin Binder, University of Regensburg, Germany

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One of the oldest hypotheses in cognitive psychology is that controlled information integration^{1}

Considering each input in isolation, without modifying the adjustments contingently on other inputs to the judgment, invites

This simplistic and probably not overly controversial model of controlled integration immediately has important consequences for our abilities to make judgments, some of which are well-known, some of which may still need to be further digested. At a general level, the most fundamental constraint on people's ability to comprehend and control their environment is this tendency to view it in terms of an “additive caricature,” as if they “looked at the world through a straw,” appreciating each factor in isolation, but with limited ability to capture the interactions and dynamics of the entire system. In more prosaic terms, a wealth of evidence suggests that multiple-cue judgments are typically well described by simple linear additive models (Brehmer,

There are important exceptions where people transcend this imprisonment in a linear additive mental universe also without external computational aids, in particular, an ability to use a prior input to “contextualize” the meaning of an immediately following input. For example, for a lottery, like a 0.10 chance of winning $100 and $0 otherwise, people have little difficulty with contextualizing the outcome in view of the preceding probability; that is, to discount the “appeal” of the positive outcome of receiving $100 by the fact that the probability of ever seeing it is low. Likewise, people often have little difficulty with understanding normalized probability ratios and appreciate that, say, “30 chances in 100” and “300 chances in 1000” describe comparable states of uncertainty, something that again requires that one input is contextualized by another^{2}

This contrasts with the requirements for multiplication implied by many rules of probability theory. We have therefore argued that additive combination may be an important—and often neglected—constraint on people's ability to reason with probability. Nilsson et al. (

Juslin et al. (^{3}

A strong example of problems with probability integration comes from studies of experienced bettors that have played on soccer games at least a couple of times each month for a period of 10 years or more (Nilsson and Andersson,

Bayes' theorem in its odds format is,

Although apparently simple, the adjustment of the probability required in view of the evidence depends not only on the evidence attended at the moment, but on the prior probability (e.g., when the likelihood ratio is 2, you should adjust the prior probability of ^{4}^{5}

A first example of a simplifying condition is natural frequencies (Gigerenzer and Hoffrage,

The “cure” to base-rate neglect suggested by this view is, of course, to replace multiplicative integration with additive integration. An immediate implication is that people should have very little problem with certain kinds of “Bayesian updating;” for example, with updating their prior belief about the mean in a population after observing a new sample from the population. “Bayesian updating” here amounts to a (sample-size) weighted average between the “prior mean” and the “sample mean,” a task that people should be able to learn quite easily.

An example directly related to Bayes' theorem is provided in Juslin et al. (^{6}

The performance is summarized in Figure

A caveat is that although these results demonstrate limits on

The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

^{1}Controlled processes refer to cognitive processes that are slow, conscious, intentional, and constrained by attention, in contrast to automatic processes that are rapid, not constrained by attention, and can be triggered also directly by stimulus properties (Schneider and Shiffrin,

^{2}This ability is not perfect as illustrated by the phenomenon of denominator neglect (Reyna and Brainerd,

^{3}A linear additive model captures many properties of the data, such that people do appreciate the qualitative effect of the base-rate, flexibly change their weighting as a function of contextual cues, and that the judgments are typically less extreme as compared to Bayes' theorem, but until we have a theory of how contextual cues affect the weight of the base-rate, we have limited ability to predict a priori how the base-rate will be used in a specific situation.

^{4}With prior odds 1 and likelihood ratio 2, the posterior odds is 2 (Equation 1); an adjustment from a prior probability of 0.5 to a posterior probability of 0.67. With prior odds 10, the corresponding adjustment will be from 0.91 to 0.95.

^{5}More specifically, when the base-rate is extreme, as in the “mammography problem” (e.g., Gigerenzer and Hoffrage,

^{6}Here is an example of a medical diagnosis task: The probability that a person randomly selected from the population of all Swedes has the disease is 2%. The probability of receiving a positive test result given that one has the disease is 96%. The probability of receiving a positive test result if one does not have the disease is 8%. What is the probability that a randomly selected person with a positive test result has the disease? Correct answer: 20%.