Mathematical Modeling of Cognition

Fall 2017

Wed. 10-12:30, Muen E317

Matt Jones
mcj@colorado.edu
Muenzinger D344
Office hours: MW afternoons, TR mornings, some F afternoons. Drop in or email to set a time to meet.

Class format

Class time will be divided between tutorial-style lecture and group discussion of modeling articles. Tutorials will aim to convey the basic mathematical ideas, and articles will allow us to see how they are applied. Because this is a small and advanced group, even the tutorials should be significantly interactive; it is important that everyone keeps up and asks frequent questions. The goals of the article discussions will to understand the details of how each model was implemented, fitted, and tested; to consider alternative modeling approaches; and to critically analyze what explanatory work the model contributes to the research.

Requirements

Exercises. Homework exercises will range from running code I provide and exploring minor modifications, to analyzing more complex properties of models or coding extensions. Students can choose how ambitious to be with these exercises depending on their level of experience.

Modeling project. Each student will develop and implement a model based on one (or more) of the frameworks we study. Ideally, you will apply the model to data of your own, possibly to some aspect of the data that you would not have otherwise considered. If this is not feasible, I can provide data for you to work with. A primary goal is for the project to be useful to your own research, rather than being a dead-end exercise. As with the class as a whole, the idea is to extend the range of tools you use to address your research questions. In keeping with this goal, writeups need not include extensive background theory or data-collection methods. Just focus on how the model was conceived, implemented, and evaluated. Conclusions should describe what the model tells you about your data (and the underlying psychological processes), and in particular how the model tells you more than you could learn from basic statistics or qualitative analysis.

Project timeline: Everyone should send me initial ideas on their projects by the end of September. If you’re unsure or would like suggestions, email me and we’ll plan a time to meet (also by the end of September). Everyone should have a concrete plan for their project by the end of October. The writeup will be due December 15.

Software

All code used in class will be written in Matlab. See CU’s overview page and installation instructions for students or faculty (you can access only one or the other).

Grading

Class participation: 25%
Weekly exercises: 25%
Modeling project: 50%

Topics

A prioritization of modeling domains and frameworks will be decided by the class in the first week. Some options:

  1. probabilistic models and bayesian inference
  2. reinforcement learning
  3. game theory, utility theory, behavioral economics
  4. concepts, categories, and similarity
  5. tools from machine learning, e.g. kernel methods, support vector machines
  6. mathematical properties of idealized neural networks (e.g., Hopfield networks, Boltzmann machines)
  7. response time and diffusion models
  8. perceptual scaling
  9. analogy and structured relational representations
  10. quantum cognition
Along the way, we’ll cover various aspects of how models are used, again prioritized by student interest.

Schedule

This will be updated as we go.

8/30: Introduction - Reinforcement learning and Rescorla-Wagner

  • outline and exercises
  • code
  • homework solutions and code

    • 9/6: Full Reinforcement Learning

  • outline and exercises
  • code
  • Reading: Sutton, R. S., & Barto, A. G. (1998). Reinforcement learning: An introduction. MIT Press. Ch 1-3.
  • homework solutions

    • 9/13: Bayesian Inference

  • outline and exercises
  • code
  • Reading: Griffiths, T. L., & Yuille, A. (2006). A primer on probabilistic inference. Trends in Cognitive Sciences. Supplement to special issue on Probabilistic Models of Cognition (volume 10, issue 7).
  • homework solutions

    • 9/20: Bayesian Models

  • outline and exercises
  • code
  • homework solutions and code

    • 9/27: Bayes nets and MCMC

  • outline
  • exercise

    • 10/4: Decision-making

  • outline
  • code
  • exercise
  • Reading: Jones, M. (in press). The diffusion model of speeded choice, from a rational perspective. To appear in W. Batchelder, H. Colonius, E. Dzhafarov, & J. Myung (Eds.), New Handbook of Mathematical Psychology, Vol. 2. Cambridge University Press.

    10/11: Fitting diffusion models

  • See outline and code from previous week
  • exercise

    • 10/18: Back-propagation

  • outline
  • code
  • exercise

    • 10/25: Hopfield networks

  • outline
  • code
  • exercises and solutions

    • 11/1: Boltzmann machines

  • outline
  • code
  • exercises

    • 11/8: More Boltzmann machines

    11/15: Categorization

  • Reading - Sections 1 and 2 (later sections optional)
  • outline
  • code
  • exercises

    • 11/29: Categorization and model translation

  • exercises

    • 12/6: Reproducing-kernel Hilbert space

  • outline

    • 12/13: Function learning

  • Reading: Griffiths, T. L., Lucas, C. G., Williams, J. J., & Kalish, M. L. (2009). Modeling human function learning with Gaussian processes. Advances in Neural Information Processing Systems, 21, 553-560.
  • code

    University Policies (standard on all course syllabi)

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