Subjective probability of disjunctive hypotheses: local-weight models for decomposition of evidential support

When the probability of a single member of a set of mutually exclusive and exhaustive possibilities is judged, its alternatives are evaluated as a composite "residual" hypothesis. Support theory (Rottenstreich & Tversky, 1997; Tversky & Koehler, 1994) implies that the process of packing alternatives together in the residual reduces the perceived evidential support for the set of alternatives and consequently inflates the judged probability of the focal hypothesis. Previous work has investigated the global weights that determine the extent to which the overall evidential support for the alternatives is discounted by this packing operation (Koehler, Brenner, & Tversky, 1997). In the present investigation, we analyze this issue in greater detail, examining the local weights that measure the specific contribution of each component hypothesis included implicitly in the residual. We describe a procedure for estimating local weights and introduce a set of plausible properties that impose systematic ordinal relationships among local weights. Results from four experiments testing these properties are reported, and a local-weight model is developed that accounts for nearly all of the variance in the probability judgments in these empirical tests. Local weights appear to be sensitive both to the individual component with which they are associated and to the residual hypothesis in which the component resides.     

The Δ–∑ hypothesis: How contrast and reinforcement rate combine to generate suboptimal choice

When given a choice between two alternatives, each offering food after the same delay with different but signaled probabilities, pigeons often prefer the low probability alternative. This preference is surprising because pigeons fail to maximize the rate of food intake; they exhibit a suboptimal preference. We advance a new explanation, the Δ–∑ hypothesis, in which the difference in probability of reinforcement within terminal links (Δ) and the overall reinforcement probability rate of each alternative (∑) are the key variables responsible for such suboptimal preference. We tested the Δ–∑ hypothesis in two experiments. In Experiment 1, we manipulated the Δs while maintaining constant all other parameters of the task, in particular the ∑s. We predicted a preference for the alternative with the larger Δ. In Experiment 2, we examined the effect of the overall reinforcement probabilities, the ∑s, while maintaining constant all other parameters of the task, in particular the Δs. We predicted a preference for the larger ∑. The results of both experiments support the Δ–∑ hypothesis.     

Probability judgment and subadditivity: The role of working memory capacity and constraining retrieval

In this research, we examined the role that individual differences in working memory (WM) capacity, the strength of alternatives, and time constraints play in probability judgment and subadditivity. With a laboratory-based learning task, Experiment 1 revealed that the degree to which participants’ probability judgments were subadditive was negatively correlated with a measure of WM capacity, even when variance due to short-term memory capacity was removed. In addition, participants were more subadditive when the viable alternatives were all rather weak. Experiment 2 extended the WM-capacity-subadditivity correlation to a population judgment task and revealed that subadditivity increases when the judgment task is performed under time constraints. Results support a model that assumes that people make probability judgments by comparing the focal hypothesis with relevant alternatives retrieved from long-term memory and that people high in WM span include more alternatives in the comparison process. Time constraints are assumed to truncate the alternative generation process, leading to fewer alternatives being recalled from long-term memory.     

Inconsistent probability estimates of a hypothesis: the role of contrasting support

This paper studies consistency in the judged probability of a target hypothesis in lists of mutually exclusive nonexhaustive hypotheses. Specifically, it controls the role played by the support of displayed competing hypotheses and the relatedness between the target hypothesis and its alternatives. Three experiments are reported. In all experiments, groups of people were presented with a list of mutually exclusive nonexhaustive causes of a person's death. In the first two experiments, they were asked to judge the probability of each cause as that of the person's decease. In the third experiment, people were asked for a frequency estimation task. Target causes were presented in all lists. Several other alternative causes to the target ones differed across the lists. Findings show that the judged probability/frequency of a target cause changes as a function of the support of the displayed competing causes. Specifically, it is higher when its competing displayed causes have low rather than high support. Findings are consistent with the contrastive support hypothesis within the support theory.     

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。     

The power of representation and interpretation: Doubling statistical reasoning performance with icons and frequentist interpretations of ambiguous numbers

Abstract

Six highly familiar melodies were submitted to three transformations: reduction and two rhythmic group transformations. These three transformations offered the opportunity to compare the role of various means of melody recognition: melodic contour, harmonic structure, local surface cues. If melody recognition relies on melodic contour, an original melody would be easier to recognise after rhythmic group transformation than after reduction; the rhythmic group transformation, but not the reduction, preserves the melodic contour. If melody recognition depends on the harmonic structure, an original melody would be easier to recognise after reduction than after a rhythmic group transformation; the reduction, but not the rhythmic group transformation, respects the underlying harmonic structure. The results of two experiments, one with children and one with adults, showed that recognition was better for rhythmic group transformation but only when local surface cues were preserved, a result that could neither be predicted by the melodic contour hypothesis nor by the harmonic structure hypothesis. The results give support to the cue abstraction hypothesis, which suggests that melody recognition relies on the recognition of certain surface cues abstracted while hearing and which are then memorised. Recognition performances and speed of recognition served as dependent variables.     

Testing the Δ-∑ hypothesis in the suboptimal choice task: Same delta with different probabilities of reinforcement

In a concurrent-chain procedure, pigeons choose between 2 initial-link stimuli; one is followed by terminal link stimuli that signal reliably whether food will be delivered after a delay; the other is followed by terminal link stimuli that do not signal whether food will be delivered after the delay. Pigeons prefer the former alternative even when it yields a lower overall probability of food. Recently, we proposed the Delta-Sigma (∆-∑) hypothesis to explain the effect: Preference depends on the difference (∆) between the reinforcement probabilities associated with the terminal link stimuli, and the overall probability of reinforcement (∑) associated with the alternative. The hypothesis predicts that, for constant ∑, animals should prefer alternatives with greater ∆ values regardless of the specific probabilities of reinforcement that determine ∆. In 2 experiments, we tested this prediction by comparing a ∆ = .5 against a ∆ = 0 alternative, with the former obtained with different pairs of reinforcement probabilities across conditions. The results supported the hypothesis when the 2 probabilities defining ∆ were significantly greater than 0, but not when one of them was close to 0. The results challenge our theoretical accounts of suboptimal choice and the variables considered to determine pigeons’ preference.     

The probability of causal conditionals

Conditionals in natural language are central to reasoning and decision making. A theoretical proposal called the Ramsey test implies the conditional probability hypothesis: that the subjective probability of a natural language conditional, P(if p then q), is the conditional subjective probability, P(q/p). We report three experiments on causal indicative conditionals and related counterfactuals that support this hypothesis. We measured the probabilities people assigned to truth table cases, P(pq), P(p notq), P( notpq) and P( notp notq). From these ratings, we computed three independent predictors, P(p), P(q/p) and P(q/ notp), that we then entered into a regression equation with judged P(if p then q) as the dependent variable. In line with the conditional probability hypothesis, P(q/p) was by far the strongest predictor in our experiments. This result is inconsistent with the claim that causal conditionals are the material conditionals of elementary logic. Instead, it supports the Ramsey test hypothesis, implying that common processes underlie the use of conditionals in reasoning and judgments of conditional probability in decision making.     

Probability and conspiratorial thinking

Conspiracy theories as alternative explanations for events and states of affairs enjoy widespread popularity. We test one possible explanation for why people are prone to conspiratorial thinking: We hypothesize that conspiratorial thinking as an explanation for events increases as the probability of those events decreases. In order to test this hypothesis, we have conducted five experiments in which participants were exposed to different information about probabilities of fictional events. The results of all experiments support the hypothesis: The lower the probability of an event, the stronger participants embrace conspiratorial explanations. Conspiratorial thinking, we conclude, potentially represents a cognitive heuristic: A coping mechanism for uncertainty.     

Sensitivity to alternatives in social hypothesis-testing

Past research has documented a hypothesis-testing strategy wherein evidence is sought to the extent that it is probable under the hypothesis. This strategy may yield nondiagnostic information and even biased confirmation of the hypothesis if the simultaneous probability of the evidence under the alternatives is disregarded. The results of three experiments demonstrated that hypothesis-testers were in fact sensitive to the probability of the evidence under the alternatives. In the first experiment, subjects tested a hypothesis under which two kinds of personal features, A-features and B-features, were highly probable. Subjects could test their hypothesis by selecting questions from a list of questions about A-features and B-features. The rerults showed that subjects' questions depended on the probability of the features under the alternative. Specifically, when the hypothesis shared A-features with the alternative, subjects preferred questions about B-features, but when the hypothesis shared B-features with the alternative, subjects preferred questions about A-features. Experiment 2 extended these findings to self-generated questions about a broader range of hypotheses and alternatives. Experiment 3 found that subjects who were provided with a specific alternative asked more diagnostic questions than subjects who were not provided with a specific alternative. Together, these results suggest that the process of generating and evaluating alternatives plays a crucial role in social hypothesis-testing and categorization.     

Readers’ sensitivity to linguistic cues in narratives: How salience influences anaphor resolution

Despite the general assumption that anaphoric inferences are necessary inferences, Levine, Guzmán, and Klin (2000) concluded that the probability of resolving noun phrase anaphors depends both on the degree of accessibility in memory of the antecedent concepts and the extent to which resolution is necessary to create a coherent discourse representation. Four experiments are presented in which the factors that influence readers' standard of coherence are investigated. We examine the hypothesis that readers are more likely to resolve anaphors that are perceived as salient; salience was manipulated both with a syntactic focusing structure (wh- clefts) and with the addition of prenominal adjectival modifiers. The results of a probe recognition time task provide support for the hypothesis that a variety of linguistic cues serve as mental processing instructions (Givón, 1992), which instruct readers as to how much attention to devote to processing.     

线索-反应相容性效应和线索有效概率的影响

运用前置线索技术,考察了线索－反应的相容性效应,研究了不同线索与反应的关系和线索有效概率对视觉剌激进行命名反应的时间的影响。结果表明：线索－反应的相容性存在,且符合维度重合理论,支持了相容性是信息加工过程的基本特征;线索客观有效概率影响了线索－反应的相容性效应的强度,线索客观有效概率越高,线索－反应的相容性效应越强。     

Cue set size effects: Sampling activated associates or cross-target interference?

Previous findings indicate that test cues linked to more associates (more knowledge) produce lower levels of recall than cues with fewer associates. One hypothesis attributes this effect to cross-target interference arising during retrieval on the assumption that cues with more associates are more likely to be indirectly connected to studied words other than the target. Another attributes the effect to sampling associates of the cue on the assumption that the probability of sampling the target declines as more associates are activated. Findings from four experiments showed that recall varied with cue set size, and, more importantly, that cue set size affected recall independently of the interference produced by cross-target connections. These results were interpreted as supporting a model that attributes cue set size effects to sampling processes associated with the intersection of the test cue and its associates with the target and its associates.     

Verbal probabilities: a question of frame?

Verbal expressions of probability and uncertainty are of two kinds: positive (‘probable’, ‘possible’) and negative (‘not certain’, ‘doubtful’). Choice of term has implications for predictions and decisions. The present studies show that positive phrases are rated to be more optimistic (when the target outcome is positive), and more correct, when the target outcome actually occurs, even in cases where positive and negative phrases are perceived to convey the same probabilities (Experiments 1 and 2). Selection of phrase can be determined by linguistic frame. Positive quantifiers (‘some’, ‘several’) support positive probability phrases, whereas negative quantifiers (‘not all’) suggest negative phrases (Experiment 3). Positive frames induced by numeric frequencies (e.g. the number of students to be admitted) imply positive probability phrases, whereas negative frames (e.g. the number of students to be rejected) call for negative probability phrases (Experiment 4). It is concluded that choice of verbal phrase is based not only on level of probability, but also on situational and linguistic cues. Copyright © 2002 John Wiley & Sons, Ltd.     

Why versus how often: Causal reasoning and the incidence of judgmental bias

When faced with the task of making a prediction or estimating a likelihood, it is argued that people often reason about the presence or absence and relative strength of possible causal mechanisms for the production of relevant outcomes. In so doing people rely on “causal cues” or properties of an inferential problem which indicate the nature of the particular causal processes which give rise to specific outcomes. It is hypothesized that causal cues, precisely because they focus attention and thought on specific causal mechanisms, can obscure the relevance of mathematical laws of probability and lead to statistically biased judgment. Two experiments were conducted. Their results support the hypothesis, showing that the incidence of the conjunction fallacy and the base rate fallacy depend on task-specific cues for causal reasoning.     

“Subordination” and Nonverbal Sensitivity: A Study and Synthesis of Findings Based on Trait Measures

We conducted a primary study and a meta-analysis on the relation of trait subordination measures to trait measures of sensitivity to nonverbal cues, in order to test the hypothesis that more subordinate individuals have enhanced ability to decode nonverbal cues. In the primary study, subordination measures included socioeconomic background, two dominance scales, a capacity for status scale, a control by powerful others scale, self-ratings of leadership and social status in high school, and for women, measures of sex role values. Sensitivity to nonverbal cues was measured using three psychometric tests, Results showed little overall support for the subordination hypothesis, and some results were significantly opposite to the prediction. The meta-analysis showed that dominance, capacity for status, socioeconomic status, and women's sex role values all had relations to nonverbal sensitivity that were opposite to the predictions of the subordination hypothesis. Future prospects for the subordination hypothesis with respect to nonverbal sensitivity are discussed.     

Multimodal communication and spatial binding in pied currawongs (<Emphasis Type="Italic">Strepera graculina</Emphasis>)

To make accurate assessments about their environment, animals must integrate a variety of sensory cues into a single unified percept. The effects of redundant multimodal signaling may be equivalent to the responses elicited by each individual cue, or enhanced when cues are combined. Binding of two seemingly coupled cues can persist despite small spatial and temporal discrepancies in signal presentation, a phenomenon termed the ventriloquist effect. Our study had two aims: first, to test the cognitive ability of a territorial, forest-dwelling bird to bind two spatially disparate cues; and second, to define the processing of the acoustic and visual cues as having either equivalent or enhanced effects when presented together. We broadcasted pied currawong (Strepera graculina) vocalizations alone or in the presence of a model currawong situated either adjacent to, or far away from a speaker, to free-living currawongs. The number of locomotive events and the average standard deviation in the distance from the speaker maintained by the focal currawong were greater in response to "far" than "close" treatments. Additionally, the average standard deviation of the distance to speaker for the uni-modal, speaker only treatment was similar to "far" responses. These findings support our hypothesis that currawongs cognitively bind two stimuli in close spatial proximity. In nature, this would result in an enhanced level of response toward territorial intruders. Our study was novel in its attempt to assess cognitive processes involved in the integration of spatially disparate bimodal signaling events in free-living birds.     

Implicit attentional orienting in a target detection task with central cues

Studies using Posner’s spatial cueing paradigm have demonstrated that participants can allocate their attention to specific target locations based on the predictiveness of preceding cues. Four experiments were conducted to investigate attentional orienting processes operating in a high probability condition (cues 75% predictive) as compared to a low probability condition (cues 50% predictive) using various types of centrally-presented cues. Spatially-informative cues (arrows and circles with gaps) resulted in cueing effects (CEs) for both probability conditions, with significantly larger CEs in the high probability conditions than the low probability conditions. Participants in the high probability conditions reported little or no awareness of cue–target probabilities after task completion. Our results provide support for an implicit learning account of the proportion valid effect under experimental conditions involving spatially-informative central cues and relatively short stimulus onset asynchronies (SOAs).     

Hypothesis generation,probability judgment,and individual differences in working memory capacity

This research examined the role of working memory (WM) in probability judgment and hypothesis generation using a simulated task that involved estimating the likelihood that particular menu items would be ordered by customers at a dinner. Five main findings were observed. First, judgments of the likelihood of individual items were made relative to alternatives retrieved from long-term memory. Second, the number of alternatives retrieved was positively correlated with a measure of WM-capacity (the operation-span task). Third, participants' probability judgments were subadditive (summing to well over 100%). Fourth, the degree to which participants' judgments were subadditive was affected by the number and strength of the alternatives retrieved from long-term memory. Fifth, the degree to which participants were subadditive was negatively correlated with WM-capacity. The results suggest that individual differences in WM-capacity are fundamental to hypothesis generation and probability judgment.