Part I. Getting Started: 1. The basics of Bayesian analysis 2. Getting started with WinBUGS Part II. Parameter Estimation: 3. Inferences with binomials 4. Inferences with Gaussians 5. Some examples of data analysis 6. Latent mixture models Part III. Model Selection: 7. Bayesian model comparison 8. Comparing Gaussian means 9. Comparing binomial rates Part IV. Case Studies: 10. Memory retention 11. Signal detection theory 12. Psychophysical functions 13. Extrasensory perception 14. Multinomial processing trees 15. The SIMPLE model of memory 16. The BART model of risk taking 17. The GCM model of categorization 18. Heuristic decision-making 19. Number concept development.

Bayesian Cognitive Modeling: A Practical Course

Bayesian Hypothesis Testing for Psychologists: A Tutorial on the Savage-Dickey Method.

This is an important contribution to a modern area of applied probability that deals with nonstationary Markov chains in continuous time. This area is becoming increasingly useful in engineering, economics, communication theory, active networking, and so forth, where the Markov-chain system is subject to frequent  uctuations with clusters of states such that the chain  uctuates very rapidly among different states of a cluster but changes less rapidly from one cluster to another. The authors use the setting of singular perturbations, which allows them to study both weak and strong interactions among the states of the chain. This leads to simpli cations through the averaging principle, aggregation, and decomposition. The main results include asymptotic expansions of the corresponding probability distributions, occupations measures, limiting normality, and exponential rates. These results give the asymptotic behavior of many controlled stochastic dynamic systems when the perturbation parameter tends to 0. The classical analytical method employs the asymptotic expansions of onedimensional distributions of the Markov chain as solutions to a system of singularly perturbed ordinary differential equations. Indeed, the asymptotic behavior of solutions of such equations is well studied and understood. A more probabilistic approach also used by the authors is based on the tightness of the family of probability measures generated by the singularly perturbed Markov chain with the corresponding weak convergence properties. Both of these methods are illustrated by practical dynamic optimization problems, in particular by hierarchical production planning in a manufacturing system. An important contribution is the last chapter, Chapter 10, which describes numerical methods to solve various control and optimization problems involving Markov chains. Altogether the monograph consists of three parts, with Part I containing necessary, technically rather demanding facts about Markov processes (which in the nonstationary case are de ned through martingales.) Part II derives the mentioned asymptotic expansions, and Part III deals with several applications, including Markov decision processes and optimal control of stochastic dynamic systems. This technically demanding book may be out of reach of many readers of Technometrics. However, the use of Markov processes has become common for numerous real-life complex stochastic systems. To understand the behavior of these systems, the sophisticated mathematical methods described in this book may be indispensable.

Monte Carlo Methods in Bayesian Computation

The integrated likelihood (also called the marginal likelihood or the normalizing constant) is a central quantity in Bayesian model selection and model averaging. It is defined as the integral over the parameter space of the likelihood times the prior density. The Bayes factor for model comparison and Bayesian testing is a ratio of integrated likelihoods, and the model weights in Bayesian model averaging are proportional to the integrated likelihoods. We consider the estimation of the integrated likelihood from posterior simulation output, aiming at a generic method that uses only the likelihoods from the posterior simulation iterations. The key is the harmonic mean identity, which says that the reciprocal of the integrated likelihood is equal to the posterior harmonic mean of the likelihood. The simplest estimator based on the identity is thus the harmonic mean of the likelihoods. While this is an unbiased and simulation-consistent estimator, its reciprocal can have infinite variance and so it is unstable in general.

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Estimating the Integrated Likelihood via Posterior Simulation Using the Harmonic Mean Identity

Model Comparison and the Principle of Parsimony Joachim Vandekerckhove Department of Cognitive Sciences, University of California, Irvine Dora Matzke Department of Psychology, University of Amsterdam Eric-Jan Wagenmakers Department of Psychology, University of Amsterdam Introduction At its core, the study of psychology is concerned with the discovery of plausible explanations for human behavior. For instance, one may observe that “practice makes perfect”: as people become more familiar with a task, they tend to execute it more quickly and with fewer errors. More interesting is the observation that practice tends to improve performance such that most of the beneﬁt is accrued early on, a pattern of diminishing returns that is well described by a power law (Logan, 1988; but see Heathcote, Brown, & Mewhort, 2000). This pattern occurs across so many diﬀerent tasks (e.g., cigar rolling, maze solving, fact retrieval, and a variety of standard psychological tasks) that it is known as the “power law of practice”. Consider, for instance, the lexical decision task, a task in which participants have to decide quickly whether a letter string is an existing word (e.g., sunscreen) or not (e.g., tolphin). When repeatedly presented with the same stimuli, participants show a power law decrease in their mean response latencies; in fact, they show a power law decrease in the entire response time distribution, that is, both the fast responses and the slow responses speed up with practice according to a power law (Logan, 1992). The observation that practice makes perfect is trivial, but the ﬁnding that practice- induced improvement follows a general law is not. Nevertheless, the power law of practice only provides a descriptive summary of the data and does not explain the reasons why practice should result in a power law improvement in performance. In order to go beyond direct observation and statistical summary, it is necessary to bridge the divide between observed performance on the one hand and the pertinent psychological processes on the other. Such bridges are built from a coherent set of assumptions about the underlying cognitive processes—a theory. Ideally, substantive psychological theories are formalized as quantitative models (Busemeyer & Diederich, 2010; Lewandowsky & Farrell, 2010). For example, the power law of practice has been explained by instance theory (Logan, 1992, This work was partially supported by the starting grant “Bayes or Bust” awarded by the European Research Council to EJW, and NSF grant #1230118 from the Methods, Measurements, and Statistics panel to JV.

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Model Comparison and the Principle of Parsimony

In this article, we propose new Monte Carlo methods for computing a single marginal likelihood or several marginal likelihoods for the purpose of Bayesian model comparisons. The methods are motivated by Bayesian variable selection, in which the marginal likelihoods for all subset variable models are required to compute. The proposed estimates use only a single Markov chain Monte Carlo (MCMC) output from the joint posterior distribution and it does not require the specific structure or the form of the MCMC sampling algorithm that is used to generate the MCMC sample to be known. The theoretical properties of the proposed method are examined in detail. The applicability and usefulness of the proposed method are demonstrated via ordinal data probit regression models. A real dataset involving ordinal outcomes is used to further illustrate the proposed methodology.

Ming-Hui Chen

Papers

Computing marginal likelihoods from a single MCMC output