Iterative Probability Kinematics

Following the pioneer work of Bruno De Finetti [12], conditional probability spaces (allowing for conditioning with events of measure zero) have been studied since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's in [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as Popper functions, and that every Popper function is representable in terms of the standard real values of some infinitesimal measure.Our main goal in this article is to study the constraints on (qualitative and probabilistic) change imposed by an extended version of McGee's result. We focus on an extension capable of allowing for iterated changes of view. Such extension, we argue, seems to be needed in almost all considered applications. Since most of the available axiomatizations stipulate (definitionally) important constraints on iterated change, we propose a non-question-begging framework, Iterative Probability Systems (IPS) and we show that every Popper function can be regarded as a Bayesian IPS. A generalized version of McGee's result is then proved and several of its consequences considered. In particular we note that our proof requires the imposition of Cumulativity, i.e. the principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30].     

Probability Semantics for Quantifier Logic

By supplying propositional calculus with a probability semantics we showed, in our 1996, that finite stochastic problems can be treated by logic-theoretic means equally as well as by the usual set-theoretic ones. In the present paper we continue the investigation to further the use of logical notions in probability theory. It is shown that quantifier logic, when supplied with a probability semantics, is capable of treating stochastic problems involving countably many trials.     

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。     

Conditional Probability and Defeasible Inference

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey–Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of non-monotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).Expectation is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some form of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. The research of both authors was supported in part by a grant from NSF, and, for Parikh, also by support from the research foundation of CUNY.     

概率估计的趋势效应及其对决策的影响机制——基于心理动量的视角

概率是反映风险与不确定性的重要指标, 概率估计具有趋势效应, 会对决策产生影响。文章描述了概率估计趋势效应的两种表现形式, 概率估计变化的趋势性(即不同时间点概率估计变化产生的趋势作用)与单边概率估计的趋势性(即高于或低于某个概率区间范畴的上界或下界的估计表述所产生的趋势作用), 揭示了概率估计趋势效应对于个体判断、决策行为和非理性决策偏差的影响; 基于心理动量的理论视角提出了一个整合模型, 阐述了概率估计趋势效应催生心理动量体验继而引发后续决策行为的内在机理。未来的研究可进一步关注：多方信息来源主体下概率估计的趋势效应; 动态趋势效应与静态概率估计的交互作用; 风险沟通中的概率估计变化趋势。     

有关条件推理中概率效应的实验研究

通过预备实验选取了四种不同条件概率的条件规则和四种不同前后件概率组合的条件规则作为实验材料,以大学生为被试,考察了两种概率因素(条件概率和前后件概率)对条件推理的演绎形式(MP、DA、AC、MT)以及变通形式(四卡问题)的影响。结果表明,两种概率因素对四种条件推理的影响都非常显著,研究进一步证实了人们对四种推理的认可程度主要与范畴前提的概率成正比的结论;但概率因素对四卡问题的解决影响不明显。     

用文字概率衡量不确定性：特征和问题

文字概率是衡量不确定性的方式之一, 即人们使用诸如“也许”、“未必”的词汇来描述特定事件发生的可能性。文字概率不同于数字概率, 主要体现在文字概率的模糊性、非概率运算性和语义特性上。这使得相对于数字概率, 用文字概率衡量不确定性既有优势也有问题, 进而对人们的不确定信息沟通和风险决策造成影响。虽然文字概率与数字概率存在特征上的差异并且人们在日常交流中偏爱文字概率, 但是大部分风险领域的研究却仅局限于数字概率, 今后有必要研究使用文字概率测量的风险决策。在已有文字概率特征研究的基础上, 还可以进一步探究其不同于数字概率的其它特征(文字/数字概率与双系统模型的联系、文字概率的跨文化差异等)及其对风险决策的影响。     

Absolute Probability Functions for Intuitionistic Propositional Logic

Provided here is a characterisation of absolute probability functions for intuitionistic (propositional) logic L, i.e. a set of constraints on the unary functions P from the statements of L to the reals, which insures that (i) if a statement A of L is provable in L, then P(A) = 1 for every P, L's axiomatisation being thus sound in the probabilistic sense, and (ii) if P(A) = 1 for every P, then A is provable in L, L's axiomatisation being thus complete in the probabilistic sense. As there are theorems of classical (propositional) logic that are not intuitionistic ones, there are unary probability functions for intuitionistic logic that are not classical ones. Provided here because of this is a means of singling out the classical probability functions from among the intuitionistic ones.     

On the existence of extremal cones and comparative probability orderings

We study the recently discovered phenomenon [Conder, M. D. E., & Slinko, A. M. (2004). A counterexample to Fishburn's conjecture. Journal of Mathematical Psychology, 48(6), 425-431] of existence of comparative probability orderings on finite sets that violate the Fishburn hypothesis [Fishburn, P. C. (1996). Finite linear qualitative probability. Journal of Mathematical Psychology, 40, 64-77; Fishburn, P. C. (1997). Failure of cancellation conditions for additive linear orders. Journal of Combinatorial Designs, 5, 353-365]—we call such orderings and the discrete cones associated with them extremal. Conder and Slinko constructed an extremal discrete cone on a set of n=7 elements and showed that no extremal cones exist on a set of n?6 elements. In this paper we construct an extremal cone on a finite set of prime cardinality p if p satisfies a certain number theoretical condition. This condition has been computationally checked to hold for 1725 of the 1842 primes between 132 and 16,000, hence for all these primes extremal cones exist.     

The Smith-Walley Interpretation of Subjective Probability: An Appreciation

The right interpretation of subjective probability is implicit in the theories of upper and lower odds, and upper and lower previsions, developed, respectively, by Cedric Smith (1961) and Peter Walley (1991). On this interpretation you are free to assign contingent events the probability 1 (and thus to employ conditionalization as a method of probability revision) without becoming vulnerable to a weak Dutch book. Special Issue Formal Epistemology I. Edited by Branden Fitelson     

On qualitative axiomatizations for probability theory

In the literature, there are many axiomatizations of qualitative probability. They all suffer certain defects: either they are too nonspecific and allow nonunique quantitative interpretations or are overspecific and rule out cases with unique quantitative interpretations. In this paper, it is whown that the class of qualitative probability structures with nonunique quantitative interpretations is not first order axiomatizable and that the class of qualitative probability structures with a unique quantitative interpretation is not a finite, first order extension of the theory of qualitative probability. The idea behind the method of proof is quite general and can be used in other measurement situations.This research was partially supported by the national Science Foundation grant NSF BNS7702911 and by the joint NSF-NIE grant NSF SED 78-22271 to the University of California, Irvine.     

Kaufmann on the Probabilities of Conditionals

Kaufmann has recently argued that the thesis according to which the probability of an indicative conditional equals the conditional probability of the consequent given the antecedent under certain specifiable circumstances deviates from intuition. He presents a method for calculating the probability of a conditional that does seem to give the intuitively correct result under those circumstances. However, the present paper shows that Kaufmann’s method is inconsistent in that it may lead one to assign different probabilities to a single conditional at the same time.     

The Measurement of Qualitative Probability

This essay presents necessary and sufficient conditions for representing a binary relation on a space of bounded random variables with a unique quantitative expectations operator. This result is used to provide a new characterization of qualitative probability. We also show that two distinct probability measures can induce the same qualitative ordering of events, even though they always produce different qualitative expectations relations. Copyright 2000 Academic Press.     

Conditional Probability in the Light of Qualitative Belief Change

We explore ways in which purely qualitative belief change in the AGM tradition throws light on options in the treatment of conditional probability. First, by helping see why it can be useful to go beyond the ratio rule defining conditional from one-place probability. Second, by clarifying what is at stake in different ways of doing that. Third, by suggesting novel forms of conditional probability corresponding to familiar variants of qualitative belief change, and conversely. Likewise, we explain how recent work on the qualitative part of probabilistic inference leads to a very broad class of ‘proto-probability’ functions.     

A PILOT STUDY OF NATURALLY OCCURRING HIGH‐PROBABILITY REQUEST SEQUENCES IN HOSTAGE NEGOTIATIONS

In the current study, the audiotapes from three hostage‐taking situations were analyzed. Hostage negotiator requests to the hostage taker were characterized as either high or low probability. The results suggested that hostage‐taker compliance to a hostage negotiator's low‐probability request was more likely when a series of complied‐with high‐probability requests preceded the low‐probability request. However, two of the three hostage‐taking situations ended violently; therefore, the implications of the high‐probability request sequence for hostage‐taking situations should be assessed in future research.     

How Affect Shapes Risky Choice: Distorted Probability Weighting Versus Probability Neglect

People's choices between prospects with relatively affect‐rich outcomes (e.g., medical side effects) can diverge markedly from their choices between prospects with relatively affect‐poor outcomes (e.g., monetary losses). We investigate the cognitive mechanisms underlying this “affect gap” in risky choice. One possibility is that affect‐rich prospects give rise to more distortion in probability weighting. Another is that they lead to the neglect of probabilities. To pit these two possibilities against each other, we fitted cumulative prospect theory (CPT) to the choices of individual participants, separately for choices between options with affect‐rich outcomes (adverse medical side effects) and options with affect‐poor outcomes (monetary losses); additionally, we tested a simple model of probability neglect, the minimax rule. The results indicated a qualitative difference in cognitive mechanisms between the affect‐rich and affect‐poor problems. Specifically, in affect‐poor problems, the large majority of participants were best described by CPT; in affect‐rich problems, the proportion of participants best described by the minimax rule was substantially higher. The affect gap persisted even when affect‐rich outcomes were supplemented by numerical information, thus providing no support for the thesis that choices in affect‐rich and affect‐poor problems diverge because the information provided in the former is nonnumerical. Our findings suggest that the traditional expectation‐based framework for modeling risky decision making may not readily generalize to affect‐rich choices. Copyright © 2015 John Wiley & Sons, Ltd.     

An account of subjective probability judgment for joint events: Conjunctive and disjunctive

Probability judgment is a vital part of many aspects of everyday life. In the present paper, we present a new theory of the way in which individuals produce probability estimates for joint events: conjunctive and disjunctive. We propose that a majority of individuals produce conjunctive (disjunctive) estimates by making a quasi‐random adjustment, positive or negative, from the less (more) likely component probability with the other component playing no obvious role. In two studies, we produce evidence supporting propositions that follow from our theory. First, the component probabilities do appear to play the distinct roles we propose in determining the joint event probabilities. Second, contrary to probability theory and other accounts of probability judgment, we show that the conjunctive‐less likely probability difference is unrelated to the more likely disjunctive probability difference (in normative theory these quantities are identical). In conclusion, while violating the norms of probability judgment, we argue that estimates produced in the manner we propose will be close enough to the normative values especially given the changing nature of the external environment and the incomplete nature of available information.     

An evidential support accumulation model of subjective probability

A model of cue-based probability judgment is developed within the framework of support theory. Cue diagnosticity is evaluated from experience as represented by error-free frequency counts. When presented with a pattern of cues, the diagnostic implications of each cue are assessed independently and then summed to arrive at an assessment of the support for a hypothesis, with greater weight placed on present than on absent cues. The model can also accommodate adjustment of support in light of the baserate or prior probability of a hypothesis. Support for alternatives packed together in a "residual" hypothesis is discounted; fewer cues are consulted in assessing support for alternatives as support for the focal hypothesis increases. Results of fitting this and several alternative models to data from four new multiple-cue probability learning experiments are reported.     

A cognitive‐emotional account of the shape of the probability weighting function

Research has shown a tendency of decision makers to overweight small probabilities and to underweight moderate and large probabilities. In standard treatments this is graphically modeled by an inverse S‐shaped probability weighting function. We suggest that emotions play a significant role in the shaping of the probability weighting function. In particular, the weighting function is proposed to be some function of objective probability, expected elation, and expected disappointment. The overweighting of small probabilities results from the anticipated elation after having won, given that winning was very unlikely. The underweighting of large probabilities results from anticipated disappointment after having failed to win, given that winning was very likely. Hence, probability is assumed to influence utility. Three experiments investigate these hypotheses. Experiments 1 and 2 show that a convex function relates probability to surprise. Experiment 3 elicits choice data and further supports the proposed hypotheses. The model adds to the understanding of the cognitive and emotional processes underlying the shape of the probability weighting function. Copyright © 2002 John Wiley & Sons, Ltd.