scholarly journals People’s conditional probability judgments follow probability theory (plus noise)

2016 ◽  
Vol 89 ◽  
pp. 106-133 ◽  
Author(s):  
Fintan Costello ◽  
Paul Watts
Keyword(s):  
Probability Theory ◽  
Conditional Probability ◽  
Probability Judgments

scholarly journals Disintegration and Bayesian inversion via string diagrams

2019 ◽  
Vol 29 (7) ◽  
pp. 938-971 ◽  
Author(s):  
Kenta Cho ◽  
Bart Jacobs
Keyword(s):  
Probability Theory ◽  
Conditional Probability ◽  
Opposite Direction ◽  
Bayesian Inversion ◽  
Conditional Probabilities ◽  
Discrete Probability ◽  
Joint State

AbstractThe notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability – via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

Quantum Models of Cognition and Decision

2015 ◽  
Author(s):  
Jerome R. Busemeyer ◽  
Zheng Wang ◽  
Emmanuel Pothos
Keyword(s):  
Decision Making ◽  
Probability Theory ◽  
Dynamic Systems ◽  
Quantum Probability ◽  
Quantum Dynamic ◽  
Classical Probability ◽  
New Formalism ◽  
Probability Judgments ◽  
Advantages And Disadvantages ◽  
Quantum Approach

Quantum probability theory provides a new formalism for constructing probabilistic and dynamic systems of cognition and decision. The purpose of this chapter is to introduce psychologists to this fascinating theory. This chapter is organized into six sections. First, some of the basic psychological principles supporting a quantum approach to cognition and decision are summarized; second, some notations and definitions needed to understand quantum probability theory are presented; third, a comparison of quantum and classical probability theories is presented; fourth, quantum probability theory is used to account for some paradoxical findings in the field of human probability judgments; fifth, a comparison of quantum and Markov dynamic theories is presented; and finally, a quantum dynamic model is used to account for some puzzling findings of decision-making research. The chapter concludes with a summary of advantages and disadvantages of a quantum probability theoretical framework for modeling cognition and decision.

Facilitating Normative Judgments of Conditional Probability: Frequency or Nested Sets?

2003 ◽  
Vol 50 (2) ◽  
pp. 97-106 ◽  
Author(s):  
Kimihiko Yamagishi
Keyword(s):  
Conditional Probability ◽  
Probabilistic Reasoning ◽  
Probability Judgment ◽  
Probability Judgments ◽  
Normative Reasoning ◽  
Roulette Wheel ◽  
Probability Task ◽  
Nested Sets ◽  
Probabilistic Structure ◽  
Inspection Machine

Abstract. Recent probability judgment research contrasts two opposing views. Some theorists have emphasized the role of frequency representations in facilitating probabilistic correctness; opponents have noted that visualizing the probabilistic structure of the task sufficiently facilitates normative reasoning. In the current experiment, the following conditional probability task, an isomorph of the “Problem of Three Prisoners” was tested. “A factory manufactures artificial gemstones. Each gemstone has a 1/3 chance of being blurred, a 1/3 chance of being cracked, and a 1/3 chance of being clear. An inspection machine removes all cracked gemstones, and retains all clear gemstones. However, the machine removes ½ of the blurred gemstones. What is the chance that a gemstone is blurred after the inspection?” A 2 × 2 design was administered. The first variable was the use of frequency instruction. The second manipulation was the use of a roulette-wheel diagram that illustrated a “nested-sets” relationship between the prior and the posterior probabilities. Results from two experiments showed that frequency alone had modest effects, while the nested-sets instruction achieved a superior facilitation of normative reasoning. The third experiment compared the roulette-wheel diagram to tree diagrams that also showed the nested-sets relationship. The roulette-wheel diagram outperformed the tree diagrams in facilitation of probabilistic reasoning. Implications for understanding the nature of intuitive probability judgments are discussed.

Some New Aspects of Slamming Probability Theory

1978 ◽  
Vol 22 (03) ◽  
pp. 186-192
Author(s):  
Harilaos N. Psaraftis
Keyword(s):  
Probability Theory ◽  
Poisson Process ◽  
Conditional Probability ◽  
Systematic Investigation ◽  
Unconditional Probability ◽  
Deck Wetness

A systematic investigation of some probabilistic aspects of slamming is presented. This investigation includes the assessment of the unconditional probability of slamming at a random instant of time; the estimation of the conditional probability of slamming at a given instant after a particular slam; and the consequent rejection of the hypothesis that slamming is a Poisson process. In addition, a procedure to approximate the distribution of slamming interarrival times2 is presented. Finally, new slamming statistics, obtainable from the theory of this work, are presented and compared with the existing slamming criteria. The theory of this paper can be readily applied to other seakeeping events such as deck wetness, keel emergence, and propeller racing.

Probability for Everyone—Even Philosophers

2017 ◽  
Author(s):  
Alan Hájek ◽  
Christopher Hitchcock
Keyword(s):  
Probability Theory ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Conditional Probability ◽  
Central Limit ◽  
Measure Theory ◽  
Bayes Theorem ◽  
The Central Limit Theorem ◽  
Large Numbers ◽  
Key Concepts

In this chapter the basics of probability theory are introduced, with particular attention to those topics that are most important for applications in philosophy. The formalism is described in two passes. The first presents finite probability, which suffices for most philosophical discussions of probability. The second presents measure theory, which is needed for applications involving infinities or limits. Key concepts such as conditional probability, probabilistic independence, random variables, and expectation are defined. In addition, several important theorems, including Bayes’ theorem, the weak and strong laws of large numbers, and the central limit theorem are defined. Along the way, several familiar puzzles or paradoxes involving probability are discussed.

Evidence for the Jeffrey Table

2019 ◽  
Vol 66 (3) ◽  
pp. 187-194
Author(s):  
Moyun Wang ◽  
Mingyi Zhu
Keyword(s):  
Conditional Probability ◽  
Fundamental Question ◽  
The Other ◽  
Adequate Account ◽  
Natural Languages ◽  
Probability Judgments ◽  
Low Level ◽  
De Finetti

Abstract. Conditionals statements are a common and necessary component in natural languages. The research reported in this paper is on a fundamental question about singular conditionals. Is there an adequate account of people’s truth, falsity, and credibility (probability) judgments about these conditionals when their antecedents are false? Two experiments examined people’s quantitative credibility ratings and qualitative truth and falsity judgments for singular conditionals, if p then q, given false antecedent, not-p, cases. The results demonstrate that, when relevant knowledge about the conditional probability of q given p, P( q|p), is available to participants in not-p cases, they tend to make credibility ratings based on P( q|p), and to make “true” (or “false”) judgments at a high (or low) level of these credibility ratings. These findings favor the Jeffrey table account of these conditionals over the other existing accounts, including that of the de Finetti table.

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