Mathematical Modeling of Cognition

Wed. 10-12:30, Muen D430

Matt Jones – mcj@colorado.edu

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

Group discussion.  The most important contribution each person can make is to our discussions of the readings.  I believe a seminar course in which everyone actively participates can be the most productive and educational forum in grad school (often for the instructor as well).  Bringing together the various backgrounds and training of everyone in the room generally leads to a much richer perspective than would otherwise be possible.  This course will be slightly less discussion-oriented than other graduate seminars, but the philosophy still applies.  Asking frequent questions and offering insights during tutorials will also help a lot to keep the class together.

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 10.

Grading

Group discussion: 50%
Modeling project: 50%

Office hours

I won’t keep regular office hours this semester, but I’m available most of the time and always happy to talk.  Send me an email (mcj@colorado.edu) or stop by my office (Muen E229).  I’m around most of the time except Tuesday and Thursday afternoons.

Request for Articles

Please suggest articles in your area of study that present mathematical models you would like to understand better.  Alignment with a topic area below is ideal but not necessary.  Although I can easily compile readings for all of our topics, the course will be more engaging if the subject domains reflect student interests.  This request stands throughout the semester but I would most appreciate suggestions during the first week. 

Topics

The schedule of topics will be adjusted as we go, to accommodate variation in how much time we choose to spend on each topic and to allow for additions suggested by class members.  For some topics we may spend a whole meeting on tutorial and then read an application article for discussion the following week.  In other cases we will cover a topic in a single week.  So far, we have the following tentative ordering of topics.

  1. Game Theory
  2. Bayesian inference; Diffusion models
  3. Model fitting and evaluation
  4. Scaling, Thurstonian models, Signal Detection Theory
  5. Reinforcement Learning
  6. Idealized neural networks
  7. Dynamic Systems
  8. Analogy
  9. Mixture of Experts

Schedule

This will be updated as we go.  Readings listed with each date should be read in advance of that class meeting.

8/25 – Introduction

9/1 – Game theory
Colman, A. M. (2003). Cooperation, psychological game theory, and limitations of rationality in social interaction. Behavioral and Brain Sciences, 26, 139–198.
*Just read the target article (pp. 139-153) and any of the commentaries that interest you.

9/8 – Overview of Bayesian modeling (no reading)

9/15 – Discussion of Bayesian modeling
Chater, N., & Manning, C.D. (2006). Probabilistic models of language processing and acquisition. TRENDS in Cognitive Science, 7, 335-344.
Kemp, C., Tenenbaum, J. B., Niyogi, S. & Griffiths, T. L. (2010). A probabilistic model of theory formation. Cognition, 114(2), 165-196.

9/22 – Model evaluation (no reading)

9/29 – Model evaluation cont.

10/6 – Perceptual scaling
Nosofsky, R.M. (1992). Similarity scaling and cognitive process models. Annual Review of Psychology, 43, 25-53.

10/13 – Perceptual scaling cont.

10/20 – Reinforcement Learning

10/27 – Reinforcement Learning cont.
Read Chapter 3. Also go over Chapter 2 to the extent you need to review last week.
Sutton, R.S., & Barto, A.G. (1998). Reinforcement Learning: An Introduction. Cambridge: MIT Press.

11/3 – Did we really do three weeks of RL?

11/10 – Neural networks: Perceptrons and basic learning rules

11/17 – Neural networks: Backpropagation, attractor networks
Rumelhard, D.E., Hinton, G.E., & Williams, R.J. (1986). Learning representations by back-propagating errors. Nature, 323, 533-536.
Hopfield, J.J. (1982). Neural networks and physical systems with emergent collective computational abilities. PNAS, 79, 2554-2558.

12/1 – Analogy
Gentner, D. (1983). Structure mapping: A theoretical framework for analogy. Cognitive Science, 7, 155-170.
Larkey, L., & Love, B. C. (2003). CAB: Connectionist analogy builder. Cognitive Science, 27, 781-794.

12/8 – Cancelled for NIPS conference (“extra credit” for attending)

Supplementary Readings (from Ramesh)

    Utility theory
    Game theory
    Bayesian models
    Model evaluation
    Scaling
    Reinforcement learning
    Neural networks
    Analogy

University Policies (standard on all course syllabi)

CU Policy for Students with Disabilities 

If you qualify for accommodations because of a disability, please submit to me a letter from Disability Services in a timely manner so that your needs be addressed. Disability Services determines accommodations based on documented disabilities. Contact: 303-492-8671, Willard 322, and www.Colorado.EDU/disabilityservices

CU Sexual Harassment Policy 

The University of Colorado at Boulder policy on Discrimination and Harassment, the University of Colorado policy on Sexual Harassment and the University of Colorado policy on Amorous Relationships apply to all students, staff and faculty. Any student, staff or faculty member who believes (s)he has been the subject of discrimination or harassment based upon race, color, national origin, sex, age, disability, religion, sexual orientation, or veteran status should contact the Office of Discrimination and Harassment (ODH) at 303-492-2127 or the Office of Judicial Affairs at 303-492-5550. Information about the ODH, the above referenced policies and the campus resources available to assist individuals regarding discrimination or harassment can be obtained at http://www.colorado.edu/odh

CU Religious Observance Policy 

Campus policy regarding religious observances requires that faculty make every effort to deal reasonably and fairly with all students who, because of religious obligations, have conflicts with scheduled exams, assignments or required attendance. Please notify the instructor of anticipated conflicts as early in the semester as possible so that there is adequate time to make necessary arrangements. See full details at http://www.colorado.edu/policies/fac_relig.html

CU Classroom Behavior Policy 

Students and faculty each have responsibility for maintaining an appropriate learning environment. Those who fail to adhere to such behavioral standards may be subject to discipline. Professional courtesy and sensitivity are especially important with respect to individuals and topics dealing with differences of race, culture, religion, politics, sexual orientation, gender, gender variance, and nationalities. Class rosters are provided to the instructor with the student's legal name. I will gladly honor your request to address you by an alternate name or gender pronoun. Please advise me of this preference early in the semester so that I may make appropriate changes to my records. See policies at

http://www.colorado.edu/policies/classbehavior.html 

http://www.colorado.edu/studentaffairs/judicialaffairs/code.html#student_code

CU Honor Code

All students of the University of Colorado at Boulder are responsible for knowing and adhering to the academic integrity policy of this institution. Violations of this policy may include: cheating, plagiarism, aid of academic dishonesty, fabrication, lying, bribery, and threatening behavior. All incidents of academic misconduct shall be reported to the Honor Code Council (honor@colorado.edu; 303-725-2273). Students who are found to be in violation of the academic integrity policy will be subject to both academic sanctions from the faculty member and non-academic sanctions (including but not limited to university probation, suspension, or expulsion). Other information on the Honor Code can be found at

http://www.colorado.edu/policies/honor.html 

http://www.colorado.edu/academics/honorcode/