Types and states: Mixture and hidden Markov models for cognitive science
Ingmar Visser, Maarten Speekenbrink · eScholarship (California Digital Library) · 2014
Types and states: Mixture and hidden Markov models for cognitive science Ingmar Visser ([email protected]) Department of Developmental Psychology, University of Amsterdam Weesperplein 4, 1018 XA Amsterdam, The Netherlands Maarten Speekenbrink ([email protected]) Cognitive, Perceptual, and Brain Sciences, University College London Gower Street, London WC1E 6BT, United Kingdom tively rare. This is unfortunate, as MMs and HMMs are ide- ally suited to test and explore important theoretical ideas in cognitive science. The objective of this tutorial is to provide researchers in cognitive science with an accessible introduc- tion to MMs and HMMs and provide them with the necessary skills to apply them in their own research. Keywords: Mixture model, hidden Markov model, latent class analysis, categories, types, states, transitions Objectives and scope There are many situations in which one may encounter dis- tinct types of entities, such as different animal species, and different states in which these entities may exist, for example motivational states like hunger. Cognition is sometimes also best understood in terms of discrete types and states. For ex- ample, forms of cognitive development can be characterised as the acquisition of increasingly complex rules which con- stitute different types of reasoning and associated response patterns in reasoning tasks (Jansen, Raij-makers, & Visser, 2007). And rather than a gradually shifting trade-off, people may switch rapidly between distinct decision-making modes favouring either speed or accuracy (Dutilh, Wagenmakers, Visser, & Maas, 2011). The idea that cognitive processes are guided by qualitatively different strategies underlies a wide range of theories concerned with topics such as word recognition, cognitive development, categorization, and de- cision making, to name but a few (for an overview, see e.g. Scheibehenne, Rieskamp, & Wagenmakers, 2013). As the identity of types and states is generally not directly observable, appropriate statistical techniques are required to identify them. This tutorial will focus on mixture models (MMs) and hidden Markov models (HMMs), which are the basis of such techniques. In the context of MMs, a type or state (e.g., a cognitive strategy) is formalized as a probability distribution over observables. Because a dataset may con- tain different types, the overall distribution is a mixture of such individual component distributions. As the component distributions need not be of the same parametric family (e.g., Gaussian distributions can be mixed with other distributions), MMs allow for considerable flexibility in the definition of types and states. HMMs are a natural extension of MMs, al- lowing switches between states over time. For example, these models are useful when people can switch between cognitive strategies during a task. In addition to identifying the dif- ferent states, HMMs allow one to also focus on the process underlying state transitions. While mixture models (MMs) and hidden Markov models (HMMs) are widely used in fields such as computational biol- ogy (e.g., for DNA sequence analysis) and machine learning (e.g., for speech recognition and estimation of topic models), their use in the analysis of cognition and behaviour is rela- Outline of the tutorial The tutorial is divided into two parts. The first part introduces the theory behind MMs and HMMs. The second part will be more practical, using a number of examples to show (a) how to apply MMs and HMMs with user-friendly and freely available software, (b) how to interpret these models, and (c) how the models can reveal aspects in the data which remain hidden with more traditional analyses. The first part of the tutorial will be delivered as a classroom style lecture. The second part will use a more hands-on approach with practi- cal computer-based examples and exercises. The audience is encouraged to bring a laptop. All necessary software and material will be made available in advance. Part I: Mixture models Introduction to mixture models. This part will introduce the basic structure of mixture models and the use of graphical and other techniques to determine whether MMs might be applicable. Estimation and inference This part will provide an intu- itive treatment of maximum likelihood estimation and intro- duce numerical optimization and Expectation-Maximization (EM), the two main methods for this type of estimation of MMs and HMMs. Practical issues such as starting values and local maxima will also be discussed. We then discuss meth- ods for model selection and how to determine the number of components (i.e., types, or latent classes). We will also dis- cuss methods to test parameters for significance and the use of posterior probabilities to determine the component to which a data point belongs. Introduction to depmixS4 This part will introduce depmixS4 (Visser & Speekenbrink, 2010), a flexible package to estimate MMs and HMMs in the R environment for sta- tistical computing (R Development Core Team, 2010). The examples in the remainder of the tutorial will use this pack- age.