Determining informative priors for cognitive models

The development of cognitive models involves the creative scientific formalization of assumptions, based on theory, observation, and other relevant information. In the Bayesian approach to implementing, testing, and using cognitive models, assumptions can influence both the likelihood function of the model, usually corresponding to assumptions about psychological processes, and the prior distribution over model parameters, usually corresponding to assumptions about the psychological variables that influence those processes. The specification of the prior is unique to the Bayesian context, but often raises concerns that lead to the use of vague or non-informative priors in cognitive modeling. Sometimes the concerns stem from philosophical objections, but more often practical difficulties with how priors should be determined are the stumbling block. We survey several sources of information that can help to specify priors for cognitive models, discuss some of the methods by which this information can be formalized in a prior distribution, and identify a number of benefits of including informative priors in cognitive modeling. Our discussion is based on three illustrative cognitive models, involving memory retention, categorization, and decision making.     

Prior sensitivity in theory testing: An apologia for the Bayes factor

A commonly voiced concern with the Bayes factor is that, unlike many other Bayesian and non-Bayesian quantitative measures of model evaluation, it is highly sensitive to the parameter prior. This paper argues that, when dealing with psychological models that are quantitatively instantiated theories, being sensitive to the prior is an attractive feature of a model evaluation measure. This assertion follows from the observation that in psychological models parameters are not completely unknown, but correspond to psychological variables about which theory often exists. This theory can be formally captured in the prior range and prior distribution of the parameters, indicating which parameter values are allowed, likely, unlikely and forbidden. Because the prior is a vehicle for expressing psychological theory, it should, like the model equation, be considered as an integral part of the model. It is argued that the combined practice of building models using informative priors, and evaluating models using prior sensitive measures advances knowledge.     

Coherent psychometric modelling with Bayesian nonparametrics

In this paper we argue that model selection, as commonly practised in psychometrics, violates certain principles of coherence. On the other hand, we show that Bayesian nonparametrics provides a coherent basis for model selection, through the use of a ‘nonparametric’ prior distribution that has a large support on the space of sampling distributions. We illustrate model selection under the Bayesian nonparametric approach, through the analysis of real questionnaire data. Also, we present ways to use the Bayesian nonparametric framework to define very flexible psychometric models, through the specification of a nonparametric prior distribution that supports all distribution functions for the inverse link, including the standard logistic distribution functions. The Bayesian nonparametric approach provides a coherent method for model selection that can be applied to any statistical model, including psychometric models. Moreover, under a ‘non‐informative’ choice of nonparametric prior, the Bayesian nonparametric approach is easy to apply, and selects the model that maximizes the log likelihood. Thus, under this choice of prior, the approach can be extended to non‐Bayesian settings where the parameters of the competing models are estimated by likelihood maximization, and it can be used with any psychometric software package that routinely reports the model log likelihood.     

Bayesian Fundamentalism or Enlightenment? On the explanatory status and theoretical contributions of Bayesian models of cognition

The prominence of Bayesian modeling of cognition has increased recently largely because of mathematical advances in specifying and deriving predictions from complex probabilistic models. Much of this research aims to demonstrate that cognitive behavior can be explained from rational principles alone, without recourse to psychological or neurological processes and representations. We note commonalities between this rational approach and other movements in psychology - namely, Behaviorism and evolutionary psychology - that set aside mechanistic explanations or make use of optimality assumptions. Through these comparisons, we identify a number of challenges that limit the rational program's potential contribution to psychological theory. Specifically, rational Bayesian models are significantly unconstrained, both because they are uninformed by a wide range of process-level data and because their assumptions about the environment are generally not grounded in empirical measurement. The psychological implications of most Bayesian models are also unclear. Bayesian inference itself is conceptually trivial, but strong assumptions are often embedded in the hypothesis sets and the approximation algorithms used to derive model predictions, without a clear delineation between psychological commitments and implementational details. Comparing multiple Bayesian models of the same task is rare, as is the realization that many Bayesian models recapitulate existing (mechanistic level) theories. Despite the expressive power of current Bayesian models, we argue they must be developed in conjunction with mechanistic considerations to offer substantive explanations of cognition. We lay out several means for such an integration, which take into account the representations on which Bayesian inference operates, as well as the algorithms and heuristics that carry it out. We argue this unification will better facilitate lasting contributions to psychological theory, avoiding the pitfalls that have plagued previous theoretical movements.     

Bayesian nonparametric model selection and model testing

This article examines a Bayesian nonparametric approach to model selection and model testing, which is based on concepts from Bayesian decision theory and information theory. The approach can be used to evaluate the predictive-utility of any model that is either probabilistic or deterministic, with that model analyzed under either the Bayesian or classical-frequentist approach to statistical inference. Conditional on an observed set of data, generated from some unknown true sampling density, the approach identifies the “best” model as the one that predicts a sampling density that explains the most information about the true density. Furthermore, in the approach, the decision is to reject a model when it does not explain enough information about the true density (according to a straightforward calibration of the Kullback-Leibler divergence measure). The posterior estimate of the true density is based on a Bayesian nonparametric prior that can give positive support to the entire space of sampling densities (defined on some sample space). This article also discusses the theoretical and practical advantages of the Bayesian nonparametric approach over all other types of model selection procedures, and over any model testing procedure that depends on interpreting a p-value. Finally, the Bayesian nonparametric approach is illustrated on four real data sets, in the comparison and testing of order-constrained models, cognitive models, models of choice-behavior, and a test of a general psychometric model.     

A Bayesian analysis of human decision-making on bandit problems

The bandit problem is a dynamic decision-making task that is simply described, well-suited to controlled laboratory study, and representative of a broad class of real-world problems. In bandit problems, people must choose between a set of alternatives, each with different unknown reward rates, to maximize the total reward they receive over a fixed number of trials. A key feature of the task is that it challenges people to balance the exploration of unfamiliar choices with the exploitation of familiar ones. We use a Bayesian model of optimal decision-making on the task, in which how people balance exploration with exploitation depends on their assumptions about the distribution of reward rates. We also use Bayesian model selection measures that assess how well people adhere to an optimal decision process, compared to simpler heuristic decision strategies. Using these models, we make inferences about the decision-making of 451 participants who completed a set of bandit problems, and relate various measures of their performance to other psychological variables, including psychometric assessments of cognitive abilities and personality traits. We find clear evidence of individual differences in the way the participants made decisions on the bandit problems, and some interesting correlations with measures of general intelligence.     

A Comparison of ML,WLSMV, and Bayesian Methods for Multilevel Structural Equation Models in Small Samples: A Simulation Study

Multilevel structural equation models are increasingly applied in psychological research. With increasing model complexity, estimation becomes computationally demanding, and small sample sizes pose further challenges on estimation methods relying on asymptotic theory. Recent developments of Bayesian estimation techniques may help to overcome the shortcomings of classical estimation techniques. The use of potentially inaccurate prior information may, however, have detrimental effects, especially in small samples. The present Monte Carlo simulation study compares the statistical performance of classical estimation techniques with Bayesian estimation using different prior specifications for a two-level SEM with either continuous or ordinal indicators. Using two software programs (Mplus and Stan), differential effects of between- and within-level sample sizes on estimation accuracy were investigated. Moreover, it was tested to which extent inaccurate priors may have detrimental effects on parameter estimates in categorical indicator models. For continuous indicators, Bayesian estimation did not show performance advantages over ML. For categorical indicators, Bayesian estimation outperformed WLSMV solely in case of strongly informative accurate priors. Weakly informative inaccurate priors did not deteriorate performance of the Bayesian approach, while strong informative inaccurate priors led to severely biased estimates even with large sample sizes. With diffuse priors, Stan yielded better results than Mplus in terms of parameter estimates.     

A Bayesian psychophysical model for angular variables

Bayesian theories of perception provide a link between observed response distributions and theoretical constructs from Bayesian decision theory. Using Bayesian psychophysics we derive response distributions for two cases, one based on a normal distribution and one on a von Mises distribution for angular variables. Interestingly, where the theoretical response distribution is always unimodal in the case of normal distributions, it can become bimodal in the angular setting in the case when prior and likelihood are about equally strong.     

Three case studies in the Bayesian analysis of cognitive models

Bayesian statistical inference offers a principled and comprehensive approach for relating psychological models to data. This article presents Bayesian analyses of three influential psychological models: multidimensional scaling models of stimulus representation, the generalized context model of category learning, and a signal detection theory model of decision making. In each case, the model is recast as a probabilistic graphical model and is evaluated in relation to a previously considered data set. In each case, it is shown that Bayesian inference is able to provide answers to important theoretical and empirical questions easily and coherently. The generality of the Bayesian approach and its potential for the understanding of models and data in psychology are discussed.     

Bayesian Model Averaging for Propensity Score Analysis

This article considers Bayesian model averaging as a means of addressing uncertainty in the selection of variables in the propensity score equation. We investigate an approximate Bayesian model averaging approach based on the model-averaged propensity score estimates produced by the R package BMA but that ignores uncertainty in the propensity score. We also provide a fully Bayesian model averaging approach via Markov chain Monte Carlo sampling (MCMC) to account for uncertainty in both parameters and models. A detailed study of our approach examines the differences in the causal estimate when incorporating noninformative versus informative priors in the model averaging stage. We examine these approaches under common methods of propensity score implementation. In addition, we evaluate the impact of changing the size of Occam’s window used to narrow down the range of possible models. We also assess the predictive performance of both Bayesian model averaging propensity score approaches and compare it with the case without Bayesian model averaging. Overall, results show that both Bayesian model averaging propensity score approaches recover the treatment effect estimates well and generally provide larger uncertainty estimates, as expected. Both Bayesian model averaging approaches offer slightly better prediction of the propensity score compared with the Bayesian approach with a single propensity score equation. Covariate balance checks for the case study show that both Bayesian model averaging approaches offer good balance. The fully Bayesian model averaging approach also provides posterior probability intervals of the balance indices.     

Identifying expectations about the strength of causal relationships

When we try to identify causal relationships, how strong do we expect that relationship to be? Bayesian models of causal induction rely on assumptions regarding people’s a priori beliefs about causal systems, with recent research focusing on people’s expectations about the strength of causes. These expectations are expressed in terms of prior probability distributions. While proposals about the form of such prior distributions have been made previously, many different distributions are possible, making it difficult to test such proposals exhaustively. In Experiment 1 we used iterated learning—a method in which participants make inferences about data generated based on their own responses in previous trials—to estimate participants’ prior beliefs about the strengths of causes. This method produced estimated prior distributions that were quite different from those previously proposed in the literature. Experiment 2 collected a large set of human judgments on the strength of causal relationships to be used as a benchmark for evaluating different models, using stimuli that cover a wider and more systematic set of contingencies than previous research. Using these judgments, we evaluated the predictions of various Bayesian models. The Bayesian model with priors estimated via iterated learning compared favorably against the others. Experiment 3 estimated participants’ prior beliefs concerning different causal systems, revealing key similarities in their expectations across diverse scenarios.     

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.     

How cognitive modeling can benefit from hierarchical Bayesian models

Hierarchical Bayesian modeling provides a flexible and interpretable way of extending simple models of cognitive processes. To introduce this special issue, we discuss four of the most important potential hierarchical Bayesian contributions. The first involves the development of more complete theories, including accounting for variation coming from sources like individual differences in cognition. The second involves the capability to account for observed behavior in terms of the combination of multiple different cognitive processes. The third involves using a few key psychological variables to explain behavior on a wide range of cognitive tasks. The fourth involves the conceptual unification and integration of disparate cognitive models. For all of these potential contributions, we outline an appropriate general hierarchical Bayesian modeling structure. We also highlight current models that already use the hierarchical Bayesian approach, as well as identifying research areas that could benefit from its adoption.     

A Bayesian view of covariation assessment

When participants assess the relationship between two variables, each with levels of presence and absence, the two most robust phenomena are that: (a) observing the joint presence of the variables has the largest impact on judgment and observing joint absence has the smallest impact, and (b) participants' prior beliefs about the variables' relationship influence judgment. Both phenomena represent departures from the traditional normative model (the phi coefficient or related measures) and have therefore been interpreted as systematic errors. However, both phenomena are consistent with a Bayesian approach to the task. From a Bayesian perspective: (a) joint presence is normatively more informative than joint absence if the presence of variables is rarer than their absence, and (b) failing to incorporate prior beliefs is a normative error. Empirical evidence is reported showing that joint absence is seen as more informative than joint presence when it is clear that absence of the variables, rather than their presence, is rare.     

Constructing informative model priors using hierarchical methods

Despite their negative reputation, informative priors are very useful in inference. Priors that express psychologically meaningful intuitions damp out random fluctuations in the data due to sampling variability, without sacrificing flexibility. This article focuses on how an intuitively satisfying informative prior distribution can be constructed. In particular, it demonstrates how the hierarchical introduction of a parameterized generative account of the set of models under consideration naturally imposes a non-uniform prior distribution over the models, encoding existing intuitions about the models. The hierarchical approach for constructing informative model priors is made concrete using a worked example, the Varying Abstraction Model (VAM), a family of categorization models including and expanding the exemplar and prototype models. It is shown how psychological intuitions about the relative plausibilities of the models in the VAM can be formally captured in an informative prior distribution over these models, by specifying a theoretically informed process for generating the models in the VAM. The smoothing effect of the informative prior in estimation is demonstrated by considering ten previously published data sets from the category learning literature.     

Exemplar models as a mechanism for performing Bayesian inference

Probabilistic models have recently received much attention as accounts of human cognition. However, most research in which probabilistic models have been used has been focused on formulating the abstract problems behind cognitive tasks and their optimal solutions, rather than on mechanisms that could implement these solutions. Exemplar models are a successful class of psychological process models in which an inventory of stored examples is used to solve problems such as identification, categorization, and function learning. We show that exemplar models can be used to perform a sophisticated form of Monte Carlo approximation known as importance sampling and thus provide a way to perform approximate Bayesian inference. Simulations of Bayesian inference in speech perception, generalization along a single dimension, making predictions about everyday events, concept learning, and reconstruction from memory show that exemplar models can often account for human performance with only a few exemplars, for both simple and relatively complex prior distributions. These results suggest that exemplar models provide a possible mechanism for implementing at least some forms of Bayesian inference.     

Philosophy and the practice of Bayesian statistics

A substantial school in the philosophy of science identifies Bayesian inference with inductive inference and even rationality as such, and seems to be strengthened by the rise and practical success of Bayesian statistics. We argue that the most successful forms of Bayesian statistics do not actually support that particular philosophy but rather accord much better with sophisticated forms of hypothetico‐deductivism. We examine the actual role played by prior distributions in Bayesian models, and the crucial aspects of model checking and model revision, which fall outside the scope of Bayesian confirmation theory. We draw on the literature on the consistency of Bayesian updating and also on our experience of applied work in social science. Clarity about these matters should benefit not just philosophy of science, but also statistical practice. At best, the inductivist view has encouraged researchers to fit and compare models without checking them; at worst, theorists have actively discouraged practitioners from performing model checking because it does not fit into their framework.