NUMERACY

Numeracy

Numeracy

Introduction

As mathematical modeling increases in popularity in cognitive psychology, it is important for there to be tools to evaluate models in an effort to make modeling as productive as possible. Our aim in this chapter is to introduce some of these tools in the context of the problems they were meant to address. After a brief introduction to building models, the chapter focuses on their testing and selection. Before doing so, however, we raise a few questions about modeling that are meant to serve as a backdrop for the subsequent material.

Why mathematical modeling?

The study of cognition is concerned with uncovering the architecture of the mind. We want answers to questions such as how decisions are made, how verbal and written modes of communication are performed, and how people navigate through the environment. The source of information for addressing these questions are the data collected in experiments. Data are the sole link to the cognitive process of interest, and models of cognition evolve from researchers inferring the characteristics of the processes from these data. Most models are specified verbally when first proposed, and constitute a set of assumptions about the structure and function of the processing system.

A verbal form of a model serves a number of important roles in the research enterprise, such as providing a good conceptual starting point for experimentation. When specified in sufficient detail, a verbal model can stimulate a great deal of research, as its predictions are tested and evaluated. Even when the model is found to be incorrect (which will always be the case), the mere existence of the model will have served an important purpose in advancing our understanding and pushing the field forward. When little is known about the process of interest (i.e. data are scarce), verbal modeling is a sensible approach to studying cognition.

However, simply by virtue of being verbally specified, there is a limit to how much can be learned about the process. When this point is reached, one must turn from a verbal description to a mathematical description to make progress. In this regard, mathematical modeling takes the scientific enterprise a step further to gain new insights into the underlying process and derives quantitative predictions, which are rarely possible with verbal models. In the remainder of this section, we highlight some points to consider when making this transition.

Model as a parametric family of probability distributions

From a statistical standpoint, observed data is a random sample from an unknown population, which represents the underlying cognitive process of interest. Ideally, the goal of modeling is to deduce the population that generated the observed data. A model is in some sense a collection of populations. In statistics, associated with each population is a probability distribution indexed by the model's parameters. Formally, a model is defined as a parametric family of probability distributions.

Let us use f(y | w) to denote the probability density function that gives the probability of observing data y = (y 1,…, ym) given the model's parameter vector w = (w1,…, ...