The marginal likelihood plays an important role in many areas of Bayesian statistics such as parameter estimation, model comparison, and model averaging. In most applications, however, the marginal likelihood is not analytically tractable and must be approximated using numerical methods. Here we provide a tutorial on bridge sampling (Bennett, 1976; Meng & Wong, 1996), a reliable and relatively straightforward sampling method that allows researchers to obtain the marginal likelihood for models of varying complexity. First, we introduce bridge sampling and three related sampling methods using the beta-binomial model as a running example. We then apply bridge sampling to estimate the marginal likelihood for the Expectancy Valence (EV) model—a popular model for reinforcement learning. Our results indicate that bridge sampling provides accurate estimates for both a single participant and a hierarchical version of the EV model. We conclude that bridge sampling is an attractive method for mathematical psychologists who typically aim to approximate the marginal likelihood for a limited set of possibly high-dimensional models.

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We investigate a question relevant to the psychology and neuroscience of perceptual decision-making: whether decisions are based on steadily accumulating evidence, or only on the most recent evidence. We report an empirical comparison between two of the most prominent examples of these theoretical positions, the diffusion model and the urgency-gating model, via model-based qualitative and quantitative comparisons. Our findings support the predictions of the diffusion model over the urgency-gating model, and therefore, the notion that evidence accumulates without much decay. Gross qualitative patterns and fine structural details of the data are inconsistent with the notion that decisions are based only on the most recent evidence. More generally, we discuss some strengths and weaknesses of scientific methods that investigate quantitative models by distilling the formal models to qualitative predictions.

Children make faster and more accurate decisions about perceptual information as they get older, but it is unclear how different aspects of the decision-making process change with age. Here, we used hierarchical Bayesian diffusion models to decompose performance in a perceptual task into separate processing components, testing age-related differences in model parameters and links to neural data. We collected behavioural and EEG data from 96 6-to 12-year-old children and 20 adults completing a motion discrimination task. We used a component decomposition technique to identify two response-locked EEG components with ramping activity preceding the response in children and adults: one with activity that was maximal over centro-parietal electrodes and one that was maximal over occipital electrodes. Younger children had lower drift rates (reduced sensitivity), wider boundary separation (increased response caution) and longer non-decision times than older children and adults. Yet, model comparisons suggested that the best model of children's data included age effects only on drift rate and boundary separation (not non-decision time). Next, we extracted the slope of ramping activity in our EEG components and covaried these with drift rate. The slopes of both EEG components related positively to drift rate, but the best model with EEG covariates included only the centro-parietal component. By decomposing performance into distinct components and relating them to neural markers, diffusion models have the potential to identify the reasons why children with developmental conditions perform differently to typically developing children and to uncover processing differences inapparent in the response time and accuracy data alone.

Psychological experiments often yield data that are hierarchically structured. A number of popular shortcut strategies in cognitive modeling do not properly accommodate this structure and can result in biased conclusions. To gauge the severity of these biases, we conducted a simulation study for a two-group experiment. We first considered a modeling strategy that ignores the hierarchical data structure. In line with theoretical results, our simulations showed that Bayesian and frequentist methods that rely on this strategy are biased towards the null hypothesis. Secondly, we considered a modeling strategy that takes a two-step approach by first obtaining participant-level estimates from a hierarchical cognitive model and subsequently using these estimates in a follow-up statistical test. Methods that rely on this strategy are biased towards the alternative hypothesis. Only hierarchical models of the multilevel data lead to correct conclusions. Our results are particularly relevant for the use of hierarchical Bayesian parameter estimates in cognitive modeling.

The marginal likelihood plays an important role in many areas of Bayesian statistics such as parameter estimation, model comparison, and model averaging. In most applications, however, the marginal likelihood is not analytically tractable and must be approximated using numerical methods. Here we provide a tutorial on bridge sampling (Bennett, 1976; Meng & Wong, 1996), a reliable and relatively straightforward sampling method that allows researchers to obtain the marginal likelihood for models of varying complexity. First, we introduce bridge sampling and three related sampling methods using the beta-binomial model as a running example. We then apply bridge sampling to estimate the marginal likelihood for the Expectancy Valence (EV) model---a popular model for reinforcement learning. Our results indicate that bridge sampling provides accurate estimates for both a single participant and a hierarchical version of the EV model. We conclude that bridge sampling is an attractive method for mathematical psychologists who typically aim to approximate the marginal likelihood for a limited set of possibly high-dimensional models.

For decades sequential sampling models have successfully accounted for human and monkey decision-making, relying on the standard assumption that decision makers maintain a pre-set decision standard throughout the decision process. Based on the theoretical argument of reward rate maximization, some authors have recently suggested that decision makers become increasingly impatient as time passes and therefore lower their decision standard. Indeed, a number of studies show that computational models with an impatience component provide a good fit to human and monkey decision behavior. However, many of these studies lack quantitative model comparisons and systematic manipulations of rewards. Moreover, the often-cited evidence from single-cell recordings is not unequivocal and complimentary data from human subjects is largely missing. We conclude that, despite some enthusiastic calls for the abandonment of the standard model, the idea of an impatience component has yet to be fully established; we suggest a number of recently developed tools that will help bring the debate to a conclusive settlement.

An important tool in the advancement of cognitive science are quantitative models that represent different cognitive variables in terms of model parameters. To evaluate such models, their parameters are typically tested for relationships with behavioral and physiological variables that are thought to reflect specific cognitive processes. However, many models do not come equipped with the statistical framework needed to relate model parameters to covariates. Instead, researchers often revert to classifying participants into groups depending on their values on the covariates, and subsequently comparing the estimated model parameters between these groups. Here we develop a comprehensive solution to the covariate problem in the form of a Bayesian regression framework. Our framework can be easily added to existing cognitive models and allows researchers to quantify the evidential support for relationships between covariates and model parameters using Bayes factors. Moreover, we present a simulation study that demonstrates the superiority of the Bayesian regression framework to the conventional classification-based approach.