Note on lexicographic‐order preservation and stochastic dominance

This note concerns two issues left unresolved in our study of lexicographic‐order preservation and stochastic dominance in settings where preferences are represented by utility vectors, ordered lexicographically, and judgements emerge as matrices that premultiply utility vectors in expected utility sums. First, a generalization of the ‘Conjecture Σ’, which implied transitivity of a stochastic dominance relation under non‐vacuous resolution‐level information, is proved. Second, this paper comments on using resolution‐level information in higher as well as in first degree stochastic dominance analysis. Copyright © 1999 John Wiley & Sons, Ltd.     

Probability judgment accuracy for general knowledge. Cross-national differences and assessment methods

In this study we compare the probability judgment accuracy of subjects from the United States and Turkey. Three different response modes were employed — numerical probabilities, pie diagrams, and odds. The questions employed in the study were restricted to two-alternative, general-knowledge items. The observed pattern of differences in the components of probability judgment accuracy paralleled those of studies that have compared Western and Asian subjects. In particular, Turkish subjects exhibited better discrimination but worse calibration than their US counterparts. This result persisted across all three response modes. These findings lend support to previous assertions that observed cross-national differences arise from socioeconomic rather than Asian versus Western cultural differences. However, the consistency of the observed differences across response modes refutes a previous assertion that observed cultural differences are merely the result of response bias.     

The Quantitative/Qualitative Watershed for Rules of Uncertain Inference

We chart the ways in which closure properties of consequence relations for uncertain inference take on different forms according to whether the relations are generated in a quantitative or a qualitative manner. Among the main themes are: the identification of watershed conditions between probabilistically and qualitatively sound rules; failsafe and classicality transforms of qualitatively sound rules; non-Horn conditions satisfied by probabilistic consequence; representation and completeness problems; and threshold-sensitive conditions such as ‘preface’ and ‘lottery’ rules. Special Issue Formal Epistemology I. Edited by Branden Fitelson     

特征概率影响多维和少维类别的分类学习和特征学习

探讨了特征概率对多维和少维类别的分类学习和特征学习的效果及策略的影响。结果表明高特征概率条件下,多维比少维类别的分类学习更容易,而且学到更多的特征知识,多维条件下人们更倾向于整体性加工策略,而少维条件下人们倾向于分析性加工策略。低特征概率条件下,多维比少维类别的分类学习和特征学习都困难,且两种条件下人们都倾向于采取分析性加工策略。     

Waiting for the bus: when base-rates refuse to be neglected

The paper reports the results from 16 versions of a simple probability estimation task, where probability estimates derived from base-rate information have to be modified by case knowledge. In the bus problem [adapted from Falk, R., Lipson, A., & Konold, C. (1994). The ups and downs of the hope function in a fruitless search. In G. Wright & P. Ayton (Eds.), Subjective probability (pp. 353-377). Chichester, UK: Wiley], a passenger waits for a bus that departs before schedule in 10% of the cases, and is more than 10 min delayed in another 10%. What are Fred's chances of catching the bus on a day when he arrives on time and waits for 10 min? Most respondents think his probability is 10%, or 90%, instead of 50%, which is the correct answer. The experiments demonstrate the difficulties people have in replacing the original three-category 1/8/1 partitioning with a normalized, binary partitioning, where the middle category is discarded. In contrast with typical studies of "base-rate neglect", or under-weighing of base-rates, this task demonstrates a reversed base-rate fallacy, where frequentistic information is overextended and case information ignored. Possible explanations for this robust phenomenon are briefly discussed.     

Understanding ‘Unlikely (20% Likelihood)’ or ‘20% Likelihood (Unlikely)’ Outcomes: The Robustness of the Extremity Effect

Calls to communicate uncertainty using mixed, verbal‐numerical formats (‘unlikely [0–33%]’) have stemmed from research comparing mixed with solely verbal communications. Research using the new ‘which outcome’ approach to investigate understanding of verbal probability expressions suggests, however, that mixed formats might convey disadvantages compared with purely numerical communications. When asked to indicate an outcome that is ‘unlikely’, participants have been shown to often indicate outcomes with a value exceeding the maximum value shown, equivalent to a 0% probability —an ‘extremity effect’. Recognising the potential consequences of communication recipients expecting an ‘unlikely’ event to never occur, we extend the ‘which outcome’ work across four experiments, using verbal, numerical, and verbal‐numerical communication formats, as well as a previously unconsidered numerical‐verbal format. We examine how robust the effect is in the context of consequential outcomes and over non‐normal distributions. We also investigate whether participants are aware of the inconsistency in their responses from a traditional ‘how likely’ and ‘which outcome’ task. We replicate and extend previous findings, with preference for extreme outcomes (including above maximum values) observed in both verbal and verbal‐numerical formats. Our results suggest caution in blanket usage of recently recommended verbal‐numerical formats for the communication of uncertainty. Copyright © 2018 John Wiley & Sons, Ltd.     

A process model of the understanding of uncertain conditionals

To build a process model of the understanding of conditionals we extract a common core of three semantics of if-then sentences: (a) the conditional event interpretation in the coherencebased probability logic, (b) the discourse processingtheory of Hans Kamp, and (c) the game-theoretical approach of Jaakko Hintikka. The empirical part reports three experiments in which each participant assessed the probability of 52 if-then sentencesin a truth table task. Each experiment included a second task: An n-back task relating the interpretation of conditionals to working memory, a Bayesian bookbag and poker chip task relating the interpretation of conditionals to probability updating, and a probabilistic modus ponens task relating the interpretation of conditionals to a classical inference task. Data analysis shows that the way in which the conditionals are interpreted correlates with each of the supplementary tasks. The results are discussed within the process model proposed in the introduction.     

A modified beta binomial model with applications to multiple choice and taste tests

A modified beta binomial model is presented for use in analyzing ramdom guessing multiple choice tests and certain forms of taste tests. Detection probabilities for each item are distributed beta across the population subjects. Properties for the observable distribution of correct responses are derived. Two concepts of true score estimates are presented. One, analogous to Duncan's empirical Bayes posterior mean score, is appropriate for assessing the subject's performance on that particular test. The second is more suitable for predicting outcomes on similar tests.This research was made possible by a grant from the Center for Food Policy Research, Graduate School of Business, Columbia University.     

Ambiguous probabilities: when does p=0.3 reflect a possibility,and when does it express a doubt?

Verbal phrases denoting uncertainty are usually held to be more vague than numerical probability statements. They are, however, directionally more precise, in the sense that they are either positive, suggesting the occurrence of a target outcome, or negative, drawing attention to its non‐occurrence. A numerical probability will, in contrast, sometimes be perceived as positive and sometimes as negative. When asked to complete sentences such as ‘The operation has a 30% chance of success, because’ some people will give reasons for success (‘the doctors are expert surgeons’), whereas others will give reasons for failure (‘it is a difficult operation’). It is shown in two experiments that positive reasons are given more often than negative ones, even for p values below 0.5, especially when the probability is higher than expected, and the target outcome is non‐normal, undesirable, and phrased as a negation. We conclude that the directionality of numerical probabilities (as opposed to verbal phrases) is context‐dependent, but biased towards a positive interpretation. Copyright © 2000 John Wiley & Sons, Ltd.