## How to improve Bayesian reasoning without instruction: Frequency formats (1995)

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Venue: | Psychological Review |

Citations: | 275 - 22 self |

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@ARTICLE{Gigerenzer95howto,

author = {Gerd Gigerenzer and Ulrich Hoffrage},

title = {How to improve Bayesian reasoning without instruction: Frequency formats},

journal = {Psychological Review},

year = {1995},

volume = {102},

pages = {684--704}

}

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### Abstract

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. Is the mind, by design, predisposed against performing Bayesian inference? The classical probabilists of the Enlightenment, including Condorcet, Poisson, and Laplace, equated probability theory with the common sense of educated people, who were known then as “hommes éclairés.” Laplace (1814/1951) declared that “the theory of probability is at bottom nothing more than good sense reduced to a calculus which evaluates that which good minds know by a sort of instinct,

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1851 | Nonparametric Statistics - Savage - 1957 |

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- Tversky, Kahneman
- 1974
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- Marr
- 1982
(Show Context)
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- Ericsson, Simon
- 1993
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- Nisbett, &Wilson
- 1977
(Show Context)
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- 1990
(Show Context)
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- Stephens, Krebs
- 1986
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- 1996
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- Simon
- 1956
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- 1973
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- Allais
- 1953
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- 1975
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- 1982
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- Edwards
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- 1982
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- 1983
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- Casscells, Schoenberger, et al.
- 1978
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- 1995
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- 1993
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- M, Imagery
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- Medin, Wattenmaker, et al.
- 1987
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