How probability theory explains the conjunction fallacy

The conjunction fallacy occurs when people judge a conjunctive statement B‐and‐A to be more probable than a constituent B, in contrast to the law of probability that P(B ∧ A) cannot exceed P(B) or P(A). Researchers see this fallacy as demonstrating that people do not follow probability theory when judging conjunctive probability. This paper shows that the conjunction fallacy can be explained by the standard probability theory equation for conjunction if we assume random variation in the constituent probabilities used in that equation. The mathematical structure of this equation is such that random variation will be most likely to produce the fallacy when one constituent has high probability and the other low, when there is positive conditional support between the constituents, when there are two rather than three constituents, and when people rank probabilities rather than give numerical estimates. The conjunction fallacy has been found to occur most frequently in exactly these situations. Copyright © 2008 John Wiley & Sons, Ltd.     

基于非逻辑机制的条件推理模式：P—Q映射模型及其实证检验

研究结合数学分析方法,提出了基于非逻辑机制的条件推理模型：P-Q映射模型。并根据这个模型,对人们在不同命题类型奈件下的推理行为进行了预测。预测结果显示,当推理前提为LH和HL型命题时,基于P-Q映射模型的预测结果与基于条件概率模型的预测结果完全一致。但当推理前提为LL和HH型命题时,两种模型给出的预测结果存在差异。实验结果表明,当前提命题为LL和HH型命题时,被试的条件推理行为与P-Q映射模型的预言完全一致。     

不同概率表征的贝叶斯推理试题形式结构的同质性研究

研究从先验概率、概率表征、推理任务等方面探讨了经典贝叶斯推理研究中存在的不足,试图在＂知识和试题双重模型＂框架下,探索现实和标准贝叶斯试题的形式结构的同质性,结果表明：1）自然频次表征比百分比表征的贝叶斯推算题正确率高,这是因为试题的形式结构不同,与概率表征无关;2）贝叶斯判断题与贝叶斯推算题的试题形式结构存在显著差异;3）贝叶斯推算题中,概率词表征与其它两种表征的试题形式结构存在显著差异,其实质是贝叶斯判断。     

Learning in reverse: Eight-month-old infants track backward transitional probabilities

Numerous recent studies suggest that human learners, including both infants and adults, readily track sequential statistics computed between adjacent elements. One such statistic, transitional probability, is typically calculated as the likelihood that one element predicts another. However, little is known about whether listeners are sensitive to the directionality of this computation. To address this issue, we tested 8-month-old infants in a word segmentation task, using fluent speech drawn from an unfamiliar natural language. Critically, test items were distinguished solely by their backward transitional probabilities. The results provide the first evidence that infants track backward statistics in fluent speech.     

How strong is the comparison‐to‐the‐strongest heuristic?: An empirical test of the comparison‐ to‐the‐strongest heuristic in probability judgment1,2

Abstract: Windschitl and Wells (1998 ) proposed that the pairwise comparison between the focal and strongest alternative outcomes plays an important role in probability judgment. However, their studies did not control the effects of alternative outcomes other than the strongest one. This article tested whether only the strongest alternative outcome would affect probability judgment, using a variable selection method in a multiple regression analysis. Study 1 reanalyzed the results of Windschitl and Young (2001 ) and showed that only the strongest alternative outcome affects probability judgment. In Study 2, a new experiment was conducted to modify the methodological problems in Study 1. The results of Study 2 were identical to those of Study 1. All these results consistently supported the comparison hypothesis.     

Delay, Probability, and Social Discounting in A Public Goods Game

A human social discount function measures the value to a person of a reward to another person at a given social distance. Just as delay discounting is a hyperbolic function of delay, and probability discounting is a hyperbolic function of odds-against, social discounting is a hyperbolic function of social distance. Experiment 1 obtained individual social, delay, and probability discount functions for a hypothetical $75 reward; participants also indicated how much of an initial $100 endowment they would contribute to a common investment in a public good. Steepness of discounting correlated, across participants, among all three discount dimensions. However, only social and probability discounting were correlated with the public-good contribution; high public-good contributors were more altruistic and also less risk averse than low contributors. Experiment 2 obtained social discount functions with hypothetical $75 rewards and delay discount functions with hypothetical $1,000 rewards, as well as public-good contributions. The results replicated those of Experiment 1; steepness of the two forms of discounting correlated with each other across participants but only social discounting correlated with the public-good contribution. Most participants in Experiment 2 predicted that the average contribution would be lower than their own contribution.     

Linda问题:"齐当别"抉择模型的解释

大量有关人类归因判断的研究表明,人类经常违反理性概率公理.Tversky和Kahneman(1983)使用Linda问题等特定场景的研究发现,人们系统性地表现出违反理性推断标准,判断合取事件发生概率大于其组成事件发生概率,称之为合取谬误,并用人们使用代表性启发式判断概率来解释该现象产生的原因.然而使用启发式观点对合取谬误现象进行解释过于模糊不清.该文首先介绍了合取谬误现象及其解释模型,然后应用Li(1994,2004)提出的不确定情形下决策理论--"齐当别"抉择模型对Linda问题中合取谬误产生的原因进行了新的解释.     

An impossibility theorem for verisimilitude

In this paper, we show that Arrow’s well-known impossibility theorem is instrumental in bringing the ongoing discussion about verisimilitude to a more general level of abstraction. After some preparatory technical steps, we show that Arrow’s requirements for voting procedures in social choice are also natural desiderata for a general verisimilitude definition that places content and likeness considerations on the same footing. Our main result states that no qualitative unifying procedure of a functional form can simultaneously satisfy the requirements of Unanimity, Independence of irrelevant alternatives and Non-dictatorship at the level of sentence variables. By giving a formal account of the incompatibility of the considerations of content and likeness, our impossibility result makes it possible to systematize the discussion about verisimilitude, and to understand it in more general terms.     

假言推理中的概率效应

近来Oaksford等人的研究表明,概率信息对假言推理的认知过程有着显著的影响,在一定程度上可以用它来解释和预测个体在推理中的行为反应。该文针对这种新的观点,就假言推理中所涉及的前后件概率信息、条件概率信息、连接概率信息、因果概率信息以及额外前提条件等对假言推理的影响分别做了简要的阐述,并介绍了一些相关的实验研究,最后作了总结和展望。     

Preferred Difficulty and Subjective Probability in Different Action Phases

Drawing on Gollwitzer's deliberative–implemental mindset distinction (P. M. Gollwitzer, 1990), it was predicted that people who are deliberating on different actions or goals would be more cautious or more realistic in their expectation of success in subsequent tasks than people who are going to implement a chosen action or goal. Participants were given a choice between different test-materials. They were interrupted before (deliberative) or immediately after decision-making (implemental). They then either had to choose between various levels of difficulty within one type of task (Experiment 1) or they had to predict their own future performance (Experiment 2). The results showed that deliberative participants preferred less difficult tasks and overestimated their probability of success less than implemental participants. In addition, deliberative participants referred more than implemental participants to their past performance when selecting levels of difficulty or predicting future performance; however, the two groups did not differ in actual performance. Taken together, the findings suggest that people are more realistic in a deliberative than in an implemental state of mind. The present studies extend prior research because for the first time they document mindset effects on peoples' estimates concerning their future performance in the achievement domain.