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8680 results in Cognition

Page 169 of 434

  • Preface

  • Simon Farrell, University of Western Australia, Perth, Stephan Lewandowsky, University of Bristol
  • Book:
    Computational Modeling of Cognition and Behavior
    Published online:
    05 February 2018
    Print publication:
    22 February 2018, pp xxi-xxii

  • Summary

    This book presents an integrated approach to the application of computational and mathematical models in psychology. Computational models have been extensively applied to better understand many domains of human behavior, such as perception, memory, reasoning, decision-making, communicating, and deciding. Modeling is often applied in these areas to different purposes – measurement, prediction, and model testing. Our major goal here is to provide a unified view on the interface between theories, simulations, and data, with a view to answering the central question: how can we learn from models of behavior?

    We cover several topics. Part I of the book explains what a computational model is and gives a general overview of models that have been applied to understanding human behavior. We also examine the process of converting theoretical statements into simulation code and give an overview of the various concepts required to understand modeling. Part II examines one use of models: parameter estimation. By fitting models to data, inferences can be made from the resulting parameter estimates, and statements made about the psychological mechanism(s) or representations that generated those data. We cover maximum likelihood estimation and Bayesian estimation, including estimation across multiple participants and hierarchical estimation. Part III explores how inferences can be made from models by using model comparison. We consider under what conditions statements of sufficiency and necessity can be made from data, and how model complexity can be conceptualized and quantified. Part III examines several approaches to accounting for complexity in model comparison, including information criteria and Bayes Factors. Part IV considers the role of computational modeling in advancing psychological theory. We explore use of models as adjuncts to human reasoning, and the interaction between human and artificial intelligence to guide theorizing and generation of conceptual insights. We also consider the use of models as tools to arrive at shared understanding between researchers (i.e. the use of models as common terms of reference), and practices for communicating and sharing models. We finish by giving an overview of the application of models in several popular areas: neural network models, models of choice response time, and the application of models to understand neural data.

    To accomplish all this, we use a freely available computer language, called R, which was initially developed for statistical data analysis but has broad applicability and is now used by many modellers.

  • 6 - Bayesian Parameter Estimation

  • from Part II - Parameter Estimation
  • Simon Farrell, University of Western Australia, Perth, Stephan Lewandowsky, University of Bristol
  • Book:
    Computational Modeling of Cognition and Behavior
    Published online:
    05 February 2018
    Print publication:
    22 February 2018, pp 126-145

  • Summary

    The goal of this chapter is to give the reader a thorough understanding of the principles of Bayesian Parameter Estimation and its application using analytic and numerical methods. Readers interested in the broader background of Bayesian statistics may wish to consult the books by Kruschke (2011), Gelman et al. (2004), or Jaynes (2003), to mention but a few that we find particularly helpful.

    What Is Bayesian Inference?

    In Chapter 4 we introduced the likelihood function as a means of identifying the most likely value of a parameter in light of the observed data. We also cautioned against confusing the likelihood with a probability. Likelihoods permit relative comparisons between different parameter values – thus allowing us to maximize likelihoods in order to obtain parameter estimates – but they are not suited for estimating absolute probabilities.

    Some of the most intuitively obvious questions one might ask of parameter estimates therefore require an approach other than maximum likelihood estimation. Suppose we estimate a parameter M from a given data set that we believe represents the capacity of working memory; that is, how many items people can hold in mind at the same time in the face of distraction (e.g., Kane et al., 2005).We need not worry about how exactly this was done, but let's suppose the best estimate turns out to be 3.2. That punctate information by itself tells us relatively little because we know that however cleverly we designed our experiment, there would be some measurement error associated with our single estimate. What we really want to know is the likely range of the “true” parameter value that we can infer from our measurement. Ideally, we want to have information about the probability distribution of that parameter so we can draw more educated conclusions. This requires the use of Bayesian parameter estimation, and we devote the next four chapters to an exploration of Bayesian concepts. We begin by introducing some additional basic facts about conditional probabilities.

    Analytic Methods for Obtaining Posteriors Our first example of Bayesian inference involves multiple flips of a slightly biased coin. Our example actually involves a natural process but we will not reveal that process until later. Our goal is to identify exactly how biased the “coin” is, and we apply Bayes Theorem to estimate the parameter that characterizes the coin's behavior.

  • 4 - Maximum Likelihood Parameter Estimation

  • from Part II - Parameter Estimation
  • Simon Farrell, University of Western Australia, Perth, Stephan Lewandowsky, University of Bristol
  • Book:
    Computational Modeling of Cognition and Behavior
    Published online:
    05 February 2018
    Print publication:
    22 February 2018, pp 72-104

  • Summary

    In the previous chapters, we encountered one of the key issues in computational modeling: a full, quantitative specification of a model involves not just a description of the model (in the form of algorithms or equations), but also a specification of the parameters of the model and their values. Although in some cases we can use known parameter values (e.g., those determined from previous applications of the model; see Oberauer and Lewandowsky, 2008), in most cases we must estimate those parameters from the data. Chapter 3 described the basics of parameter estimation by minimizing the discrepancy between the data and the model's predictions. Chapter 4 deals with a popular and more principled alternative approach to parameter estimation called maximum likelihood estimation.

    Unlike the techniques discussed in the previous chapter, maximum likelihood estimation is deeply rooted in statistical theory. Maximum likelihood estimators have known properties that are not possessed by estimates obtained via minimizing RMSD (except under specific situations detailed later); for example, maximum likelihood estimates are guaranteed to become more accurate on average with increasing sample size. Additionally, likelihood can be used to make statements about the relative weight of evidence for a particular hypothesis, either about the value of a particular parameter or about a model as a whole. This lays the groundwork for the material in upcoming chapters: likelihood plays a key role in Bayesian parameter estimation, and we will later use the idea of likelihood as the strength of evidence to explore a principled and rigorous technique for evaluating scientific models.

    Basics of Probabilities

    The term “likelihood” in common parlance is used interchangeably with probability; we might consider the likelihood of it raining tomorrow (which varies considerably between the two authors, who at the time of writing live in Australia and the UK), or the likelihood that an individual randomly selected from the population will live past the age of 80. By contrast, when considering statistical or computational modeling, the term likelihood takes on a very strict meaning which is subtly – but fundamentally – different from that of probability.

    The best way to define likelihood, and to distinguish it from probability.

    is to start with the concept of probability itself. We all have some intuitive notion of what a probability is, and these intuitions probably make some connection with the formal definitions we will introduce here.

  • 13 - Neural Network Models

  • from Part IV - Models in Psychology
  • Simon Farrell, University of Western Australia, Perth, Stephan Lewandowsky, University of Bristol
  • Book:
    Computational Modeling of Cognition and Behavior
    Published online:
    05 February 2018
    Print publication:
    22 February 2018, pp 334-368

  • Summary

    The ability to learn is a fundamental one. Many models of cognition incorporate some form of learning, as it is rarely the case that performance on a task is unaffected by experience. This chapter presents a family of models that provide a simple and profound architecture for understanding memory and how we learn about regularities in our world.

    Neural network models, or connectionist models, model human behavior using structures and representations that loosely capture our understanding of how the brain functions. Although some models do attempt to capture fine-grained details of the operation of neurons, the framework we discuss here abstracts some general operating principles from the brain and uses these to account for performance on learning and memory tasks. Most connectionist models have the following features in common:

  • The modeling architecture is composed of units (“neurons”) connected by weights;

  • Each unit has its own firing rate;

  • The pattern of activation (i.e., firing rates) across an ensemble of units can be treated as a representation;

  • The activity of a unit is determined by the incoming weights, and the activity of the units that are passing activation through those weights;

  • Learning is accomplished by modifying the connection strengths – the weights – between units.

    • Don't worry if you found that description very abstract – it is! Like many things, neural networks are better understood by example than by description. We next go through a simple and popular example of a connectionist model to put some flesh on this abstract description.

    Hebbian Models

    The Hebbian Associator

    Associative memory refers to our ability to remember pairings between items or objects. One example of an associative memory task is the cued recall task, where a person is shown pairings of stimuli and responses, and must produce the appropriate response on seeing a stimulus. A real-world example of a cued recall task is remembering someone's name when you see their face at a conference. Here, we will assume that the stimuli and responses are both words; so, for example, if the word trench has been paired with the word scissors, then the goal is to say scissors when the word trench is later seen again.

  • 3 - Basic Parameter Estimation Techniques

  • from Part II - Parameter Estimation
  • Simon Farrell, University of Western Australia, Perth, Stephan Lewandowsky, University of Bristol
  • Book:
    Computational Modeling of Cognition and Behavior
    Published online:
    05 February 2018
    Print publication:
    22 February 2018, pp 47-71

  • Summary

    We know from the preceding chapter that the goal of parameter estimation is to find those parameter values that maximize the agreement between the model's predictions and the data. The extent of that agreement then tells us something (though not everything!) about the utility of the model. Moreover, interpretation of those parameter values can often shed a light on the underlying processes. In this chapter we provide the basic set of tools necessary to achieve these goals.

    Discrepancy Function

    Although we wish to maximise the similarity between the model predictions and the data, most parameter estimation procedures reframe this intention by instead minimizing the discrepancy between predictions and data.

    Minimization requires a continuous discrepancy function that condenses the discrepancy between predictions and data into a single number. That discrepancy function is minimized by gradual and iterative adjustment of the parameters. The discrepancy function is also variously known as objective function, cost function, or error function, and we will consider a few such functions along the way.

    To illustrate, consider Figure 3.1, which presents data from one condition in a forgetting experiment reported by Carpenter et al. (2008). In the experiment, participants studied a set of 60 obscure facts (e.g., “greyhounds have the best eyesight of any dog”), and their memory for those facts was tested after 5 minutes, and again 1, 2, 7, 14, 42 days later. A different subset of items was tested at each retention interval. The figure also shows the best-fitting predictions of a model of forgetting based on a power function. We first encountered the power function in Chapter 1 in connection with practice effects. As we foreshadowed there, the power function is also a good way to characterize forgetting. Carpenter et al. (2008) used this form of the power function:

    Fitting Models to Data: Parameter Estimation Techniques

    How do we minimize the discrepancy function? A number of competing approaches exist, and we will discuss them throughout the remainder of the book. The first two approaches are known as least-squares and maximum likelihood estimation, respectively, and this chapter and the next one is devoted to presenting them. The third approach, which involves application of Bayesian statistics, will be discussed later in Chapters 6 through 9. Although the mechanics of least-squares and maximum likelihood estimation are quite similar, their underlying motivation and properties differ considerably.

  • 11 - Bayesian Model Comparison Using Bayes Factors

  • from Part III - Model Comparison
  • Simon Farrell, University of Western Australia, Perth, Stephan Lewandowsky, University of Bristol
  • Book:
    Computational Modeling of Cognition and Behavior
    Published online:
    05 February 2018
    Print publication:
    22 February 2018, pp 273-308

  • Summary

    The previous chapter explored the issue of model complexity, and focused on the fact that a model may give a good fit to a set of data simply by virtue of being more flexible. We also discussed several methods by which we can correct for complexity to obtain an unbiased measure of the fit of a model, in particular the AIC as a corrected estimator of the distance between the data and the “true” model. We pick up on both of these themes in this chapter, in which we discuss the Bayesian approach to model comparison, and how the Bayesian approach naturally accounts for model complexity.

    We begin by presenting the core component of Bayesian model comparison – the marginal likelihood – and discuss how the relative fit of two models can be expressed in terms of Bayes factors. We then survey several methods for calculating the marginal likelihood, before discussing the particular role of the prior distributions when performing Bayesian model comparison.

    Marginal Likelihoods and Bayes Factors

    To understand Bayes factors, it is useful to first remind ourselves of Bayes theorem, as it applies to Bayesian parameter estimation (Equation 6.6). For convenience, we restate the theorem here:

    For much of our discussion thus far, we focussed on the proportional relationship P(θ|y) ∝ P(y| θ)P(θ), and dropped P(y) as it can be treated as a normalising constant (Equation 7.1). It turns out that P(y) – the marginal likelihood – plays a critical role in Bayesian model comparison. Indeed, it is also called the evidence because it quantifies the evidence the data y provide for the model.

    To understand why, we need to remind ourselves that all of the components in Equation 11.1 are conditional on a particular model M; that is, the equation should really be presented as:

    Accordingly, the evidence P(y|M) tells us about the probability of obtaining data y under model M, and thus how consistent the data are with the model.

    It might not seem obvious how we can calculate P(y|M), but all the information we need is already used in Equation 11.2. The evidence is obtained by calculating the marginal likelihood of the data given the model:

    Effectively, what we are doing is considering how likely the data are for each point in the parameter space, and then averaging the resulting values.

  • 15 - Models in Neuroscience

  • from Part IV - Models in Psychology
  • Simon Farrell, University of Western Australia, Perth, Stephan Lewandowsky, University of Bristol
  • Book:
    Computational Modeling of Cognition and Behavior
    Published online:
    05 February 2018
    Print publication:
    22 February 2018, pp 395-423

  • Summary

    The emphasis in the preceding chapters has generally been on the modeling of behavioral data such as response probability and response time. As outlined in Chapter 1, models give us a number of advantages over purely verbal reasoning about behavioral data. The current chapter shows how these advantages extend to neurophysiological data, by applying models of cognition to data from techniques such as electroencephalophy (EEG) and event-related potentials (ERPs); magnetoencephalography (MEG); functional magnetic resonance imaging (fMRI); diffusion tensor imaging (FRI); and single-cell recordings.

    Historically, many cognitive modelers and mathematical psychologists (and cognitive psychologists more generally) have been critical of the contributions of neurophysiology and neuroscience to theorizing about cognitive processes (e.g., Coltheart, 2006; Lewandowsky and Coltheart, 2012; Page, 2006). These concerns go beyond the methodology of neuroimaging (e.g., Bennett et al., 2009; Vul et al., 2009), and more generally question whether we can learn anything of theoretical import from neuroimaging. One criticism is that some techniques – particularly fMRI – have traditionally been used to infer where cognitive cognitive processing is performed, but this does not necessarily tell us anything about how it was performed (Page, 2006). Another is that neuroscience has been overly concerned with developing taxonomies of processing (Lewandowsky and Coltheart, 2012). In the case of category learning, Newell (2012) argued that evidence for the idea that different memory systems can be used to categorize objects mostly comes from neuroimaging data, and that the behavioral data (e.g., behavioral dissociations) are less clear cut. In the case of recognition memory, neuroscientific data have been argued to be irrelevant to the distinction between recollection and familiarity, or uninformative without a precise specification of the causal (process) mechanisms (Kalish and Dunn, 2012). The general question underlying these different criticisms is whether neural data are able to discriminate between different theoretical explanations for a behavioral phenomenon (Coltheart, 2006).

    Regardless of whether those criticisms turn out to be valid, neuroscience has seen a number of changes over the past decade that have addressed at least some of those concerns. One change has been the use of more fine-grained and dynamic methods for analyzing data, such as functional connectivity (e.g., Van Den Heuvel and Pol, 2010) and multi-voxel pattern analysis (e.g., Norman et al., 2006).

  • 2 - From Words to Models

  • from Part I - Introduction to Modeling
  • Simon Farrell, University of Western Australia, Perth, Stephan Lewandowsky, University of Bristol
  • Book:
    Computational Modeling of Cognition and Behavior
    Published online:
    05 February 2018
    Print publication:
    22 February 2018, pp 24-44

  • Summary

    This chapter introduces the basic terms required to enter the world of modeling while at the same time providing an overview of the entire enterprise. For illustrative purposes, our discussion focuses on a simple model that was developed 50 years ago to describe how people make decisions in a speeded-choice task.

    Response Times in Speeded-Choice Tasks

    You are in the cognitive laboratory to participate in an experiment. A tight cluster of 300 lines at various orientations is projected onto the screen in front of you. Are they predominantly tilted to the left or to the right? The experimenter has instructed you to respond as quickly as possible by pressing one of two keys: “z” for “left” and “/” for “right.” There are many such trials and in addition to being speedy, you are also asked to be as accurate as possible. Because the orientations of individual lines within each stimulus cluster are drawn from a distribution with considerable variance, the task is quite difficult.

    The procedure just summarized was from an experiment by Smith and Vickers (1988) and is representative of a “choice reaction time” task. Although the task sounds simple, the data from such experiments are strikingly rich and can provide a broad window into human cognition. There are two classes of responses (correct and incorrect), and each class is characterized by an entire distribution of response times across the numerous trials of each type. A complete account of human performance in this quintessential decision-making task would thus describe response accuracy and latency, and the relationship between the two, as a function of various experimental manipulations. For example, the mean orientation of the lines might be changed or participants might be instructed to emphasize speed over accuracy, or vice versa.

    There currently exist a number of sophisticated models that can describe performance in choice reaction time tasks (Brown and Heathcote, 2008; Ratcliff, 1978;Wagenmakers et al., 2007), and we will explain some of those in detail later in Chapter 14. For present purposes, we step back in time by approximately 50 years and illustrate the theoretical challenges associated with modeling choice tasks by building a model from scratch.

    We begin with the assumption that when a stimulus is presented, not all information is available to the decision maker instantaneously. Instead, people gradually build up the evidence required to make a decision.

  • 1 - Introduction

  • from Part I - Introduction to Modeling
  • Simon Farrell, University of Western Australia, Perth, Stephan Lewandowsky, University of Bristol
  • Book:
    Computational Modeling of Cognition and Behavior
    Published online:
    05 February 2018
    Print publication:
    22 February 2018, pp 3-23

  • Summary

    This introductory chapter pursues three principal goals. First, we show that computational modeling is essential to ensure progress in cognitive science. Second, we provide an introduction to the abstract idea of modeling and its many and varied applications. Third, we survey some of the issues involved in the interpretation of model output, including in particular how models can help constrain scientists’ own thinking.

    Models and Theories in Science

    Cognitive scientists seek to understand how the mind works. That is, we want to describe and predict people's behavior, and we ultimately wish to explain it, in the same way that physicists predict the motion of an apple that is dislodged from its tree (and can accurately describe its downward path) and explain its trajectory (by appealing to gravity). For example, if you forget someone's name when you are distracted seconds after being introduced to her, we would like to know what cognitive process is responsible for this failure. Was it lack of attention? Forgetting over time? Can we know ahead of time whether or not you will remember that person's name?

    The central thesis of this book is that to answer questions such as these, cognitive scientists must rely on quantitative mathematical models, just like physicists who research gravity. We suggest that to expand our knowledge of the human mind, consideration of the data and verbal theorizing are insufficient on their own. This thesis is best illustrated by considering something that is (just a little) simpler and more readily understood than the mind. Have a look at the data shown in Figure 1.1, which represent the position of planets in the night sky over time.

    How might one describe this peculiar pattern of motion? How would you explain it? The strange loops in the otherwise consistently curvilinear paths describe the famous “retrograde motion” of the planets – that is, their propensity to suddenly reverse direction (viewed against the fixed background of stars) for some time before resuming their initial path. What explains retrograde motion? It took more than a thousand years for a satisfactory answer to that question to become available, when Copernicus replaced the geocentric Ptolemaic system with a heliocentric model.

Page 169 of 434