Most students and researchers are familiar with linear statistical models such as ANOVA and linear regression. The advantage of linear models is that they are flexible and can be used for inference across many disciplines. They are, however, often poor models of cognitive and psychological processes. For example, researchers may be interested in assessing the roles of storage and retrieval processes in a memory task. The relationship between storage and retrieval is surely not linear.

This course is about a different class of models for psychology. Two main paths of cognitive modeling have evolved in mathematical psychology, depending on how we deal with the 'black box.' One path of modeling is concerned with uncovering the structure within the black box; it aims to provide detailed, substantive, and formal accounts of specific mental processes. The other path of modeling focuses on capturing the properties of the black box by the mathematical model; it aims to provide the representation that might characterize a large family of processing models. This course is designed to the introduction of important general concepts in modeling.

One of the goals is to teach you a unified principle for all statistics:
likelihood. We will show you how to write down likelihoods of various models and how to use computational techniques to maximize likelihood. We will also mention issues on model selection based on nested likelihood and others. Moreover, since simulation can help developing insight about how models account for phenomena, we will use simulations in this regard from time to time.

To summarize, in this course we will introduce some mathematical modeling approaches in psychology. We first review some basic concepts of probability and random variables. We then introduce the concept of maximum likelihood, a model- fitting approach commonly used in mathematical psychology. In the second part of the course we illustrate the use of mathematical methods with examples from psychophysics. Several applications of mathematical modeling also will be introduced. Topics include signal detection theory, threshold models, multinomial processing tree models, etc.

We will use R, a free software environment for statistical computing and graphics that can be downloaded from the web page http://www.r-project.org/, for some of the homework problems.