Probability logic and combining evidence

In Hailperin 1996 Hailperin, T. 1996. Sentential Probability Logic, Bethlehem and London: Lehigh University Press and Associated University Presses.  [Google Scholar], in addition to its formal development of Probability Logic, there are many sections devoted to historical origins, illustrative examples, and discussion of related work by other authors. Here selected portions of its formal treatment are summarized and then used as a basis for a probability logic treatment of combining evidence.     

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。     

Probability kinematics

Probability kinematics is studied in detail within the framework of elementary probability theory. The merits and demerits of Jeffrey's and Field's models are discussed. In particular, the principle of maximum relative entropy and other principles are used in an epistemic justification of generalized conditionals. A representation of conditionals in terms of Bayesian conditionals is worked out in the framework of external kinematics.     

动机的概率意义——概率归并定理与Monty Hall问题的非贝叶斯解

本文探讨Monty Hall问题认知困惑的形成因素。通过对该问题数学结构的分析,导出一个与该问题的解密切相关的定理——概率归并定理;并依据该定理给出了Monty Hall问题的一种更直观的数学解。     

Editor's note: Special issue on Combining Probability and Logic to Solve Philosophical Problems

Editor's note: Special Issue on Combining Probability and Logic to Solve Philosophical Problems (Progic 2013).     

概率与事物表征

概率表达着世界的可能性。近年来研究发现, 人们倾向于在更抽象的和更核心的水平上表征低概率事件, 而倾向于在更具体和更次要的水平上表征高概率事件。相应的, 高水平表征事物会导致人们的概率估计偏低, 而低水平表征事物会导致概率估计偏高。此外, 习得的刺激间联系也会影响人们的概率估计。未来研究应注意探讨概率影响事物表征的内在原因、概率估计的动态性与多重概率估计、以及其更深层次的内部机制等问题。     

Probability matching in visual selective attention

Jonides (1980, p. 111) suggested a model to account for shifts of spatial attention in visual search tasks with less than 100% cue validity. This two-stage probability matching model was subsequently tested by Jonides (1983). Reaction time distribution characteristics indeed provided some support for the model. In the present paper the relevant data presented in Jonides (1980, 1981, 1983) are further analysed. The results of this analysis strongly support the probability matching hypothesis. Probability matching is apparently a very important strategic characteristic of visual spatial attention that deserves much more study.     

Teachers' Understandings of Probability

Probability is an important idea with a remarkably wide range of applications. However, psychological and instructional studies conducted in the last two decades have consistently documented poor understanding of probability among different populations across different settings. The purpose of this study is to develop a theoretical framework for describing teachers' understandings of probability. To this end, we conducted an 8-day seminar with eight high school statistics teachers in the summer of 2001. The data we collected include videotaped sessions and interviews, teachers' written work, and researchers' field notes. Our analysis of the data revealed that there was a complex mix of conceptions and understandings of probability, both within and across the teachers, which were situationally triggered, often incoherent when the teachers tried to reflect on them, and which did not support their attempts to develop coherent pedagogical strategies regarding probability and statistical inference.     

Probability Transformations In The Study Of Behavior Toward Risk

Probability transformation functions were introduced into modelsof behavior toward risk to allow them to accommodate violations of the expected utility hypothesis.This paper examines the shape of the probability transformation function, its interpretation asoptimism or pessimism, and how the ranking of outcomes becomes important when probability transformationsare used. It also explores two behavioral implications: the overweighting of unlikely, extremeoutcomes, and intertia around certainty. Finally, the rationality of transforming the probabilitydistribution is discussed.     

Testing the Reliability of Paper-Pencil Versions of the fill-in-the-blank and Multiple-Choice Methods of Measuring Probability Discounting for Seven Different Outcomes

The Psychological Record - Probability discounting occurs when the subjective value of an outcome changes because its delivery is uncertain. The present study measured the reliability of rates of...     

Probability as a Theory Dependent Concept

It is argued that probability should be defined implicitly by the distributions of possible measurement values characteristic of a theory. These distributions are tested by, but not defined in terms of, relative frequencies of occurrences of events of a specified kind. The adoption of an a priori probability in an empirical investigation constitutes part of the formulation of a theory. In particular, an assumption of equiprobability in a given situation is merely one hypothesis inter alia, which can be tested, like any other assumption. Probability in relation to some theories – for example quantum mechanics – need not satisfy the Kolmogorov axioms. To illustrate how two theories about the same system can generate quite different probability concepts, and not just different probabilistic predictions, a team game for three players is described. If only classical methods are allowed, a 75% success rate at best can be achieved. Nevertheless, a quantum strategy exists that gives a 100% probability of winning. This revised version was published online in June 2006 with corrections to the Cover Date.     

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 matching in choice under uncertainty: Intuition versus deliberation

Gaissmaier and Schooler (2008) [Gaissmaier, W., & Schooler, L. J. (2008). The smart potential behind probability matching. Cognition, 109, 416-422] argue that probability matching, which has traditionally been viewed as a decision making error, may instead reflect an adaptive response to environments in which outcomes potentially follow predictable patterns. In choices involving monetary stakes, we find that probability matching persists even when it is not possible to identify or exploit outcome patterns and that many “probability matchers” rate an alternative strategy (maximizing) as superior when it is described to them. Probability matching appears to reflect a mistaken intuition that can be, but often is not, overridden by deliberate consideration of alternative choice strategies.     

Probability effects on stimulus evaluation and response processes.

This study investigated the effects of probability information on response preparation and stimulus evaluation. Eight subjects responded with one hand to the target letter H and with the other to the target letter S. The target letter was surrounded by noise letters that were either the same as or different from the target letter. In 2 conditions, the targets were preceded by a warning stimulus unrelated to the target letter. In 2 other conditions, a warning letter predicted that the same letter or the opposite letter would appear as the imperative stimulus with .80 probability. Correct reaction times were faster and error rates were lower when imperative stimuli confirmed the predictions of the warning stimulus. Probability information affected (a) the preparation of motor responses during the foreperiod, (b) the development of expectancies for a particular target letter, and (c) a process sensitive to the identities of letter stimuli but not to their locations.     

风险决策的概率权重偏差：心理机制与优化策略

概率权重偏差指人对事件发生的主观概率估计与客观概率的差异。它影响投资、投保、医患沟通等方面。“重结果轻概率”的非补偿性策略和参照点诱发的情绪波动会引发概率权重偏差; 改变“概率”的描述形式、“结果”的情绪体验、“损益”的参照点、风险的心理距离等可调整权重偏差、优化决策。未来需深究权重偏差的适用情境、机制关联及偏差辨别等问题。     

Use of the Suicide Probability Scale (SPS) with Adolescents

This study examined the utility of the Suicide Probability Scale (SPS) by comparing results obtained with a sample of 217 normal adolescents (M = 16.2 years) to the findings obtained in Cull and Gill's (1988 ) standardization effort. The present sample scored significantly higher than Cull and Gill's normative sample in SPS item, subscale, and total scores. In addition, the SPS generally failed to discriminate between the present, adolescent sample and Cull and Gill's inpatient psychiatric sample. Moreover, item-subscale correlations obtained for the present sample often differed from those reported by Cull and Gill, and factor analysis of SPS item scores failed to replicate Cull and Gill's four-factor solution. Findings suggest the need for caution when using the SPS to assess adolescent suicide potential and underscore the need for additional research regarding the instrument's efficacy in distinguishing between normal and suicidal adolescents.     

Banking on a bad bet. Probability matching in risky choice is linked to expectation generation

Probability matching is the tendency to match choice probabilities to outcome probabilities in a binary prediction task. This tendency is a long-standing puzzle in the study of decision making under risk and uncertainty, because always predicting the more probable outcome across a series of trials (maximizing) would yield greater predictive accuracy and payoffs. In three experiments, we tied the predominance of probability matching over maximizing to a generally adaptive cognitive operation that generates expectations regarding the aggregate outcomes of an upcoming sequence of events. Under conditions designed to diminish the generation or perceived applicability of such expectations, we found that the frequency of probability-matching behavior dropped substantially and maximizing became the norm.     

Probability logic,logical probability,and inductive support

This paper seeks to defend the following conclusions: The program advanced by Carnap and other necessarians for probability logic has little to recommend it except for one important point. Credal probability judgments ought to be adapted to changes in evidence or states of full belief in a principled manner in conformity with the inquirer’s confirmational commitments—except when the inquirer has good reason to modify his or her confirmational commitment. Probability logic ought to spell out the constraints on rationally coherent confirmational commitments. In the case where credal judgments are numerically determinate confirmational commitments correspond to Carnap’s credibility functions mathematically represented by so—called confirmation functions. Serious investigation of the conditions under which confirmational commitments should be changed ought to be a prime target for critical reflection. The necessarians were mistaken in thinking that confirmational commitments are immune to legitimate modification altogether. But their personalist or subjectivist critics went too far in suggesting that we might dispense with confirmational commitments. There is room for serious reflection on conditions under which changes in confirmational commitments may be brought under critical control. Undertaking such reflection need not become embroiled in the anti inductivism that has characterized the work of Popper, Carnap and Jeffrey and narrowed the focus of students of logical and methodological issues pertaining to inquiry.     

Probability matching in risky choice: The interplay of feedback and strategy availability

Probability matching in sequential decision making is a striking violation of rational choice that has been observed in hundreds of experiments. Recent studies have demonstrated that matching persists even in described tasks in which all the information required for identifying a superior alternative strategy—maximizing—is present before the first choice is made. These studies have also indicated that maximizing increases when (1)?the asymmetry in the availability of matching and maximizing strategies is reduced and (2)?normatively irrelevant outcome feedback is provided. In the two experiments reported here, we examined the joint influences of these factors, revealing that strategy availability and outcome feedback operate on different time courses. Both behavioral and modeling results showed that while availability of the maximizing strategy increases the choice of maximizing early during the task, feedback appears to act more slowly to erode misconceptions about the task and to reinforce optimal responding. The results illuminate the interplay between “top-down” identification of choice strategies and “bottom-up” discovery of those strategies via feedback.     

概率贴现和时间贴现在人脑中的分离

本研究运用ERP技术来检验概率贴现和时间贴现是否反映了相同的神经机制过程。在本研究中,我们在一个刺激中同时操纵概率贴现（风险vs.安全）和时间贴现（1个月后vs.今天）水平。通过对两个过程上的ERN以及全时间段的ERP分析来确定它们是否反映了相同的认知过程。结果发现,这两者在ERN上存在差异,同时这两个效应在ERP的时程和出现的电极点上都存在差异。这些差异表明这两者可能包含着不同的神经机制过程。