An overview of Bayesian reasoning in the analysis of delay‐discounting data

Statistical inference (including interval estimation and model selection) is increasingly used in the analysis of behavioral data. As with many other fields, statistical approaches for these analyses traditionally use classical (i.e., frequentist) methods. Interpreting classical intervals and p‐values correctly can be burdensome and counterintuitive. By contrast, Bayesian methods treat data, parameters, and hypotheses as random quantities and use rules of conditional probability to produce direct probabilistic statements about models and parameters given observed study data. In this work, we reanalyze two data sets using Bayesian procedures. We precede the analyses with an overview of the Bayesian paradigm. The first study reanalyzes data from a recent study of controls, heavy smokers, and individuals with alcohol and/or cocaine substance use disorder, and focuses on Bayesian hypothesis testing for covariates and interval estimation for discounting rates among various substance use disorder profiles. The second example analyzes hypothetical environmental delay‐discounting data. This example focuses on using historical data to establish prior distributions for parameters while allowing subjective expert opinion to govern the prior distribution on model preference. We review the subjective nature of specifying Bayesian prior distributions but also review established methods to standardize the generation of priors and remove subjective influence while still taking advantage of the interpretive advantages of Bayesian analyses. We present the Bayesian approach as an alternative paradigm for statistical inference and discuss its strengths and weaknesses.     

Age‐related differences in delay discounting: Immediate reward,reward magnitude,and social influence

Delay discounting refers to the tendency of individuals to subjectively devalue rewards that are to be received in the future, with high rates of delay discounting being associated with a variety of maladaptive life outcomes (e.g., unhealthy dietary and exercise behaviors). The current study explored the psychological and social processes involved in adult age‐related differences in delay discounting of monetary rewards. Younger adults exhibited higher levels of delay discounting than older adults. This increased level of patience in older adults was found whether smaller‐sooner rewards were to be received immediately or in the future. However, there was an interaction with reward magnitude, whereby younger adults exhibited higher levels of delay discounting for smaller reward magnitudes but not larger reward magnitudes. Social influence on delay discounting was investigated by having participants complete three phases of the delay‐discounting task: an individual precollaboration phase, a collaboration phase in age‐group‐matched dyads, and an individual postcollaboration phase. A convergence effect was observed in that dyad members' postcollaboration choices were significantly more similar compared to their baseline choices during the precollaboration phase. Moreover, levels of convergence were comparable between younger and older adults, suggesting age invariance in social influence on delay discounting. The current results demonstrate a degree of malleability in delay discounting that extends into older adulthood, making interventions targeting the construct a promising avenue for future research.     

Termination of logic programs with imperfect information: applications and query procedure

A general logic programming framework allowing for the combination of several adjoint lattices of truth-values is presented. The language is sorted, enabling the combination of several reasoning forms in the same knowledge base. The contribution of the paper is two-fold: on the one hand, sufficient conditions guaranteeing termination of all queries for the fix-point semantics for a wide class of sorted multi-adjoint logic programs are presented and related to some well-known probability-based formalisms; in addition, we specify a general non-deterministic tabulation goal-oriented query procedure for sorted multi-adjoint logic programs over complete lattices. We prove its soundness and completeness as well as independence of the selection ordering. We apply the termination results to probabilistic and fuzzy logic programming languages, enabling the use of the tabulation proof procedure for query answering.     

Internal multidimensional unfolding about a single-ideal—A probabilistic solution

A solution is presented for an internal multidimensional unfolding problem in which all the judgments of a rectangular proximity matrix are a function of a single-ideal object. The solution is obtained by showing that when real and ideal objects are represented by normal distributions in a multidimensional Euclidean space, a vector of distances among a single-ideal and multiple real objects follows a multivariate quadratic form in normal variables distribution. An approximation to the vector's probability density function (PDF) is developed which allows maximum likelihood (ML) solutions to be estimated. Under dependent sampling, the likelihood function contains information about the parametric distances among real object pairs, permitting the estimation of single-ideal solutions and leading to more robust multiple-ideal solutions. Tests for single- vs. multiple-ideal solutions and dependent vs. independent sampling are given. Properties of the proposed model and parameter recovery are explored. Empirical illustrations are also provided.     

DISCOUNTING OF DELAYED FOOD REWARDS IN PIGEONS AND RATS: IS THERE A MAGNITUDE EFFECT?

Temporal discounting refers to the decrease in the present, subjective value of a reward as the time to its receipt increases. Results from humans have shown that a hyperbola-like function describes the form of the discounting function when choices involve hypothetical monetary rewards. In addition, magnitude effects have been reported in which smaller reward amounts are discounted more steeply than larger amounts. The present research examines the cross-species generality of these findings using real rewards, namely food pellets, with both pigeons and rats. As with humans, an adjusting amount procedure was used to estimate the amount of immediate reward judged equal in value to a delayed reward. Different amounts of delayed food rewards (ranging from 5 to 32 pellets in pigeons and from 5 to 20 pellets in rats) were studied at delays varying from 1 s to 32 s. A simple hyperbola, similar to the hyperbola-like mathematical function that describes the discounting of hypothetical monetary rewards in humans, described the discounting of food rewards in both pigeons and rats. These results extend the generality of the mathematical model of discounting. Rates of discounting delayed food rewards were higher for pigeons than for rats. Unlike humans, however, neither pigeons nor rats showed a reliable magnitude effect: Rate of discounting did not vary systematically as a function of the amount of the delayed reward.     

Effects of methylphenidate and morphine on delay-discount functions obtained within sessions

Four rats responded under a "self-control" procedure designed to obtain delay-discount functions within sessions. Each session consisted of seven blocks, with seven trials within each block. Each block consisted of two initial forced-choice trials followed by five free-choice trials. On choice trials, the rats could press either of two retractable levers. A press on one lever was followed by presentation of a smaller reinforcer (a single dipper presentation of a sucrose solution); a press on the other lever was followed by presentation of a larger reinforcer (four consecutive dipper presentations). The delay associated with the smaller reinforcer always was 0 s, whereas the signaled delay associated with the larger reinforcer increased across blocks (from 0 to 50 s). Under these conditions, the percentage of choices of the larger reinforcer decreased across blocks, and relatively reliable delay-discount functions were obtained within sessions. Doses of methylphenidate (1.0 to 17.0 mg/kg) and morphine (0.3 to 17.0 mg/kg) were then administered prior to selected sessions. Typically, intermediate doses of methylphenidate shifted the discount functions to the right (increased choices of the larger reinforcer). For 2 of the rats, this effect was pronounced; for the other 2 rats, this effect occurred after the range of delays for the larger reinforcer was decreased (0 to 20 s). On the other hand, in most cases morphine produced a slight leftward shift in the discount function (decreased choices of the larger reinforcer). The present procedure appears to be a useful and efficient method to characterize drug effects on an entire delay-discount function. As with many procedures used to study self-control choices, however, sources of control other than reinforcement delay and amount may have been operating in the present study, and these sources must be considered when interpreting drug effects.     

Bootstrapping the lexicon: a computational model of infant speech segmentation

Prelinguistic infants must find a way to isolate meaningful chunks from the continuous streams of speech that they hear. BootLex, a new model which uses distributional cues to build a lexicon, demonstrates how much can be accomplished using this single source of information. This conceptually simple probabilistic algorithm achieves significant segmentation results on various kinds of language corpora - English, Japanese, and Spanish; child- and adult-directed speech, and written texts; and several variations in coding structure - and reveals which statistical characteristics of the input have an influence on segmentation performance. BootLex is then compared, quantitatively and qualitatively, with three other groups of computational models of the same infant segmentation process, paying particular attention to functional characteristics of the models and their similarity to human cognition. Commonalities and contrasts among the models are discussed, as well as their implications both for theories of the cognitive problem of segmentation itself, and for the general enterprise of computational cognitive modeling.     

Delay discounting as emotional processing: An electrophysiological study

Both theoretical models and functional imaging studies implicate the involvement of emotions within the delay discounting process. However, defining this role has been difficult to establish with neuroimaging techniques given the automaticity of emotional responses. To address this, the current study examined electrophysiological correlates involved in the detection and evaluation of immediate and delayed monetary outcomes. Our results showed that modulation of both early and later ERP components previously associated with affective stimuli processing are sensitive to the signalling of delayed rewards. Together with behavioural reaction times that favoured immediacy, we demonstrated, for the first time, that time delays modify the incentive value of monetary rewards via mechanisms of emotional bias and selective visual attention. Furthermore, our data are consistent with the hypothesis that delayed and thus intangible rewards are perceived less saliently, and rely on emotion as a common currency within decision making. This study provides a new approach to delay discounting and highlights a potential novel route through which delay discounting may be investigated.     

任务框架及利他人格对社会折扣的影响

与自我存在一定社会距离的他人获得收益或遭受损失都会给自我带来一定的效用,这种效用随着与他人间社会距离的增加而减少,该现象被称为社会折扣。本研究采用情境式选择滴定程序,探索并比较了“为他人得益而放弃自我得益”(NG-G)、“为他人不失而放弃自我得益”(NG-NL)、“为他人得益而承担自我损失”(L-G)以及“为他人不失而承担自我损失”(L-NL)4种任务框架下的社会折扣程度。结果发现：(1)4种框架下,指数模型均比双曲模型更好地描绘了被试的社会折扣规律;(2)任务框架影响社会折扣程度,“为他人不失”框架下(含NG-NL、L-NL)的折扣程度比“为他人得益”框架下(含NG-G、L-G)低;(3)任务框架与社会距离的交互作用影响社会折扣程度。随着社会距离增加, NG-G与NG-NL框架下折扣程度的差异逐渐变小, L-G与L-NL框架下折扣程度的差异也逐渐变小;(4)社会距离与利他人格的交互作用显著,高低利他人格者的社会折扣程度只在社会距离较远时存在显著差异。上述结果表明,个体不仅存在自我损失规避倾向,也存在不同程度的他人损失规避倾向。