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We introduce the expectation of a random variable on a probability space, which is its best guess in the least-squares sense. The variance is an operator quantifying the dispersion of its distribution. We recall the expression of the expectation and variance of a weighted sum of variables. The moment generating function is a characterization of a distribution which is powerful for finding the distribution of a linear combination of variables. Those concepts are then extended to pairs of variables, for which covariance and correlation can be defined. The Central Limit Theorem gives another interpretation to the expectation, which is also the asymptotic value taken by the average of variables having the same distribution. Finally, we introduce a special class of random variables called Radon–Nikodym derivatives, which are nonnegative and display unit expectation. This family of variables can be used to build new probability measures starting from a reference probability space. Switching probability measures triggers a modification of the distribution of the random variables at hand. Those concepts are illustrated on various examples including coins, dice, and stock price models.
Our paper reconsiders the triadic design proposed by Cox (Games and Economic Behavior 46:260–281, 2004) to identify trust and reciprocity in investment games. Specifically, we extend the design in two directions. First, we collect information on investors’ choices by using both the direct-response (as does Cox) and strategy methods. Using the latter, we are able to condition reciprocity on initial inequality, which is endogenous when investigating reciprocity. We demonstrate that the triadic design provides evidence for reciprocity once that initial inequality is considered. Second, we elicit expectations and test their coherence with the triadic outcomes. By examining the relationship between trust actions and expected gains, we analyze whether investors’ expectations are consistent with their behavior. Finally, we test for the existence of an emotional bias, i.e., whether expectation mismatches induce trustees to change actual choices from the planned ones.
Wildlife health surveillance is a rapidly evolving field. The goal of this commentary is to share the authors perspectives on the evolving expectations of wildlife health surveillance. We describe the basis for developing our opinions using multiple information sources including a narrative literature review, convenience samples of websites and conversations with experts. With increasing prominence of wildlife health, expectations for surveillance have increased. Situational awareness and threat or vulnerability detection were expected outputs. Action expectation themes included knowledge mobilization, reliable action thresholds and evidence-based decision making. Information expectations were broad and included the need for information on social and ecological risk drivers and impacts and evaluation of surveillance systems. Surveillance systems developers should consider: (1) What methods can equivalently and reliably manage the biases, uncertainties and ambiguities of wildlife health information; (2) How surveillance and intelligence systems support acceptable, ethical, efficient and effective actions that do not generate unintended consequences; and (3) How to generate evidence to show that surveillance and intelligence systems lead to decisions affecting vulnerability or resilience to endemic health threats, emerging diseases, climate change and other conservation threats.
Chapter 2 identifies and describes several methodological considerations in participatory research with adolescents, for example the issue of power sharing. Language is explored as a tool but also a potential barrier for engagement over time. Adolescents can have different roles and levels of involvement. The research setting and the characteristics of the research team are also explored.
The current chapter addresses one of the most controversial concepts in psychology due to the challenge it poses to study from a methodological perspective: the mind. The chapter’s objective is to inform the reader that when accessing digital services, users come with preformed expectations shaped into mental or neural representations, a result of their previous experiences with the service or from what they have learned through social interactions with others. In the decision-making process, where users determine whether to use a digital service or others, they base their choice on a cognitive unit comprised of different mental representations that link a behavior to a reward. The behavior that is decided upon (which would include the use of the digital service) generates a set of expectations, which will be compared to the actual experience. The outcome of this expectation-reality comparison will elicit an emotional response that will become associated with the digital service, altering its value for the user. The congruence between expectation and reality is a fundamental requirement for a cognitively ergonomic design of digital services. Therefore, digital behavior designers must ensure that user expectations align with the reality of interacting with the digital tool. Designing user expectations should be one of the focal points in the design of digital behaviors, as a mismatch between expectation and reality can produce aversive emotions that may lead to the abandonment of the service.
This Element examines the influence of expectation and attention on conscious perception. It explores the debate on whether attention is necessary for conscious perception by presenting empirical evidence from studies on inattentional blindness, change blindness, and the attentional blink. While the evidence strongly suggests that attention is necessary for conscious perception, other research has shown that expectation can shape perception, sometimes leading to illusory experiences where predicted stimuli are perceived despite their absence. This phenomenon, termed 'expectation awareness', suggests that attention may not be necessary for all conscious experiences. These findings are explored within the predictive processing framework, where the brain is characterized as a prediction engine, continuously updating its internal models to minimize prediction errors. Integrating findings from psychology, neuroscience, and cognitive science, this Element provides a predictive processing model of how attention and expectation construct perceptual reality. It also discusses clinical and theoretical implications and suggests future research.
The introduction introduces the concept of imagination used in the book, explains the relation of art and of faith to this concept, and discusses the approach and method of the book, highlighting its understanding of theology and theology’s relationship to phenomenology and to other disciplines. The introduction concludes with an overview of the plan of the book.
The adoption of fungus-resistant grapevines may be a key strategy for substantially reducing fungicide use in pesticide-intensive viticulture. In a representative survey conducted among 436 grapevine growers in Switzerland, we elicited growers’ expected share of land devoted to fungus-resistant varieties in ten years. More specifically, using regression analyses, we explore the main predictors behind the stated adoption intentions. We find that one-third of new plantings in the next decade will be fungus-resistant varieties. As a result, the expected share of land devoted to fungus-resistant varieties in ten years is 27.4% (compared to 10.2% in 2022), thus increasing by 169%. Farmer- and farm characteristics explain most of the adoption dynamics, especially growers’ beneficial health perceptions about fungus-resistant varieties, which correlate positively with their expected land share devoted to these varieties. Moreover, non-organic grapevine growers are particularly likely to increase their land devoted to these varieties. These findings have important implications for agricultural policy and industry in Europe and elsewhere, facilitating the expected plantation increase using a policy mix tailored to farmer- and farm-level characteristics.
Edited by
Alik Ismail-Zadeh, Karlsruhe Institute of Technology, Germany,Fabio Castelli, Università degli Studi, Florence,Dylan Jones, University of Toronto,Sabrina Sanchez, Max Planck Institute for Solar System Research, Germany
Abstract: This chapter presents a third-order predictive modelling methodology which aims at obtaining best-estimate results with reduced uncertainties (acronym: 3rd-BERRU-PM) for applications to large-scale models comprising many parameters. The building blocks of the 3rd-BERRU-PM methodology include quantification of third-order moments of the response distribution in the parameter space using third-order adjoint sensitivity analysis (which overcomes the curse of dimensionality), assimilation of experimental data, model calibration, and posterior prediction of best-estimate model responses and parameters with reduced best-estimate variances/covariances for the predicted responses and parameters. Applications of these concepts to an inverse radiation transmission problem, to an oscillatory dynamical model, and to a large-scale computational model involving 21,976 uncertain parameters, respectively, are also presented, thus illustrating the actual computation and impacts of the first-, second-, and third-order response sensitivities to parameters on the expectation, variance, and skewness of the respective model responses.
A review of basic probability theory – probability density, expectation, mean, variance/covariance, median, median absolute deviation, quantiles, skewness/kurtosis and correlation – is first given. Exploratory data analysis methods (histograms, quantile-quantile plots and boxplots) are then introduced. Finally, topics including Mahalanobis distance, Bayes theorem, classification, clustering and information theory are covered.
Although new information technologies and social networks make a wide variety of opinions and advice easily accessible, one can never be sure to get support on a focal judgment task. Nevertheless, participants in traditional advice taking studies are by default informed in advance about the opportunity to revise their judgment in the light of advice. The expectation of advice, however, may affect the weight assigned to it. The present research therefore investigates whether the advice taking process depends on the expectation of advice in the judge-advisor system (JAS). Five preregistered experiments (total N = 2019) compared low and high levels of advice expectation. While there was no evidence for expectation effects in three experiments with block-wise structure, we obtained support for a positive influence of advice expectation on advice weighting in two experiments implementing sequential advice taking. The paradigmatic disclosure of the full procedure to participants thus constitutes an important boundary condition for the ecological study of advice taking behavior. The results suggest that the conventional JAS procedure fails to capture a class of judgment processes where advice is unexpected and therefore weighted less.
We argue that people choosing prosocial distribution of goods (e.g., in dictator games) make this choice because they do not want to disappoint their partner rather than because of a direct preference for the chosen prosocial distribution. The chosen distribution is a means to fulfil one’s partner’s expectations. We review the economic experiments that corroborate this hypothesis and the experiments that deny that beliefs about others’ expectations motivate prosocial choice. We then formulate hypotheses about what types of expectation motivate someone to do what is expected: these are justifiable hopeful expectations that are clearly about his own choices. We experimentally investigate how people modulate their prosociality when they face low or unreasonably high expectations. In a version of a dictator game, we provide dictators with the opportunity to modulate their transfer as a function of their partner’s expectations. We observe that a significant portion of the population is willing to fulfil their partner’s expectation provided that this expectation expresses a reasonable hope. We conclude that people are averse to disappointing and we discuss what models of social preferences can account for the role of expectations in determining prosocial choice, with a special attention to models of guilt aversion and social esteem.
How accurate are laypeople’s intuitions about probability distributions of events? The economic and psychological literatures provide opposing answers. A classical economic view assumes that ordinary decision makers consult perfect expectations, while recent psychological research has emphasized biases in perceptions. In this work, we test laypeople’s intuitions about probability distributions. To establish a ground truth against which accuracy can be assessed, we control the information seen by each subject to establish unambiguous normative answers. We find that laypeople’s statistical intuitions can be highly accurate, and depend strongly upon the elicitation method used. In particular, we find that eliciting an entire distribution from a respondent using a graphical interface, and then computing simple statistics (such as means, fractiles, and confidence intervals) on this distribution, leads to greater accuracy, on both the individual and aggregate level, than the standard method of asking about the same statistics directly.
Maximizing is characterized by aspirations for the highest standards. The current study explored the relationship between maximizing and risk-taking tendencies in decisions subject to risk. We propose that people first refer to expectation (i.e., the overall utility expected from an alternative) when taking risky decisions. If expectation clearly identifies the best option, maximizing will not be correlated with risk-taking tendencies. If not, people refer to maximizing to reach a decision. Maximizing will be positively associated with risk-taking tendencies because the “upper bound” of risky options helps achieve the goal of seeking the best. Four studies showed that risk-taking tendencies increased with maximizing when the options had similar expectations (Studies 1 to 3). When expectations between options were clearly different (vs. similar), the positive relationship between maximizing and risk-taking tendencies was reduced (Study 4). These findings provide an insight into how maximizing is related to risk seeking.
Investors, like any decision maker, feel regret when they compare the outcome of an investment with what the outcome would have been had they invested differently. We argue and show that this counterfactual comparison process is most likely to take place when the decision maker’s expectations are violated. Across five scenario experiments we found that decision makers were influenced only by forgone investment outcomes when the realized investment fell short of the expected result. However, when their investments exceeded prior expectations, the effect of foregone investment on regret disappeared. In addition, Experiment 4 found that individual differences in the need to maximize further moderated the effects of their expectations, such that maximizers always take into account the forgone investment. The final experiment found that when probed to make counterfactual comparisons, also investments that exceed expectations may lead to regret. Together these experiments reveal insights into the comparative processes leading to decision regret.
We introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials.
Ancient Greek does not have a term equivalent to ‘please’ and the bare imperative is used for requests. The understanding of Ancient Greek can benefit from a comparison with some modern languages where the equivalent to ‘please’ is either more restricted in use (Modern Greek) or absent (Danish): these two languages have different strategies (diminutives for Modern Greek, particles for Danish) in the case of routine interactions where the aim is to signal to the interlocutor that the request is expected in the context.
The aim of the chapter is to evaluate the role of the particle δή in such a function of de-dramatization and trivialization of the potentially threatening speech act, through a corpus study in dialogical texts of the classical period (comedies of Aristophanes, philosophical dialogues of Plato). The study shows that this post-positive particle can function as a positive politeness marker (in Brown and Levinson’s sense), to signal a weakly threatening or expected request in the context.
Borderline personality disorder (BPD) is a severe mental disorder, comprised of heterogeneous psychological and neurobiological pathologies. Here, we propose a predictive processing (PP) account of BPD to integrate these seemingly unrelated pathologies. In particular, we argue that the experience of childhood maltreatment, which is highly prevalent in BPD, leaves a developmental legacy with two facets: first, a coarse-grained, alexithymic model of self and others – leading to a rigidity and inflexibility concerning beliefs about self and others. Second, this developmental legacy leads to a loss of confidence or precision afforded beliefs about the consequences of social behavior. This results in an over reliance on sensory evidence and social feedback, with concomitant lability, impulsivity and hypersensitivity. In terms of PP, people with BPD show a distorted belief updating in response to new information with two opposing manifestations: rapid changes in beliefs and a lack of belief updating despite disconfirmatory evidence. This account of distorted information processing has the potential to explain both the instability (of affect, self-image, and interpersonal relationships) and the rigidity (of beliefs about self and others) which is typical of BPD. At the neurobiological level, we propose that enhanced levels of dopamine are associated with the increased integration of negative social feedback, and we also discuss the hypothesis of an impaired inhibitory control of the prefrontal cortex in the processing of negative social information. Our account may provide a new understanding not only of the clinical aspects of BPD, but also a unifying theory of the corresponding neurobiological pathologies. We conclude by outlining some directions for future research on the behavioral, neurobiological, and computational underpinnings of this model, and point to some clinical implications of it.
Although commonly treated as two separate areas of study in primate cognition, inferential reasoning and problem solving share two key features. They involve going “beyond the information given,” and they compete with associative accounts to explain observable behavior. Despite these commonalities, the study of inferential reasoning and problem solving differ in non-trivial ways from both a methodological and conceptual perspective. They use different setups and use different concepts to investigate how individuals innovate when faced with novel challenges. However, these differences, I will argue, are far less substantial than their commonalities, especially when contrasted with competing frameworks such as associative or perceptual-based accounts of behavior. In this chapter, I will review some of the most relevant empirical studies in primates on inferential reasoning and problem solving. In general, studies on inferential reasoning entail choosing from two or more alternatives to locate a hidden food item (e.g., object permanence) whereas problem-solving studies require individuals to overcome some obstacle that is blocking their access to a visible food item (e.g., tool use). I will then attempt to synthesize this information to extract the key theoretical constructs, paying particular attention to the commonalities and differences between them. Finally, I will contrast the “inferential” approach to other competing approaches (associative, perceptual) in an attempt to strengthen the ties between inferential reasoning and problem solving and propose ways to foster progress in the coming years.