Types and states: Mixture and hidden Markov models for cognitive science

There are many situations in which one may encounter distinct 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 example, forms of cognitive development can be characterised as the acquisition of increasingly complex rules which constitute 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 decision 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 contain 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, allowing 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 different 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 biology (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 relatively rare. This is unfortunate, as MMs and HMMs are ideally 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 introduction to MMs and HMMs and provide them with the necessary skills to apply them in their own research.