## Graduate Summer School: Probabilistic Models of Cognition: The Mathematics of Mind

July 9 - 26, 2007

## Overview

“Probabilistic Models of Cognition: The Mathematics of Mind” will involve leaders from Cognitive Science and experts from Computer Science, Mathematics and Statistics, who are interested in making bridges to Cognitive Science. The goal is to develop a common mathematical framework for all aspects of cognition, and review how it explains empirical phenomena in the major areas of cognitive science – including vision, memory, reasoning, learning, planning, and language. The summer school is motivated by recent advances which offer the promise of modeling human cognition mathematically. These advances have occurred largely because the mathematical and computational tools developed for designing artificial systems are beginning to make an impact on theoretical and empirical work in Cognitive Science. In turn, Cognitive Science offers an enormous range of complex problems which challenge and test these theories.

The main theoretical theme of the summer school is to model cognitive abilities as sophisticated forms of probabilistic inference. The approach is “sophisticated” in at least three respects. First, the knowledge and beliefs of cognitive agents are modeled using sophisticated probability distributions defined over structured relational systems, such as graphs and generative grammars. Second, the learning and reasoning processes of cognitive agents are modeled using advanced mathematical techniques from statistical estimation, statistical physics, and stochastic differential equations. Third, the decision making processes of agents are modeled using techniques from decision theory and game theory.

The summer school is intended for graduate students and postdocs, as well as more senior researchers interested in focusing their efforts on these mathematical challenges and crucial applications. The program is organized as follows:

Week 1: Tutorials. Introduction to the conceptual foundations and basic mathematical and computational techniques. Topics include Bayesian probability theory, parameter estimation, graphical models (directed and undirected), inference, learning (parameters & structure), dynamical models, basic Bayesian decision theory, MCMC other unsupervised learning topics (e.g. EM, PCA/FA), model selection, and information maximization. These methods will be illustrated on simple cognitive examples. Computer software packages will be available so that students can implement these theories and apply them to model simple cognitive tasks.

Week 2: Core applications to cognitive science. This includes advanced methods such as probabilistic grammars and relational models, which have recently been successfully applied to language and vision and hierarchical reinforcement learning (which relates to how cognitive agents make decisions over time). Core applications will include how these mathematical techniques can be used to predict and explain cognitive phenomena, modeling reasoning over time, which relates to decision making experiments, and modeling information based exploration which accounts for cognitive reasoning experiments and aspects of visual search. All these core applications will emphasize themes and tools that are common to all aspects of cognitive science.

Week 3: Advanced topics. There has recently been considerable success in developing unsupervised methods for learning probabilistic models for language and vision which has major implications for cognitive development. Talks will take place on unsupervised learning of grammars for language and vision in tandem with research on modeling learnability and cognitive development. Advanced topics will also include modeling mutilmodal sensory interactions (e.g. between vision and audition) and sensorimotor integration, neuroeconomics which studies how decisions are made in brain and how this relates to decision theory and game theory. This will be supplemented with studies of advanced decision making.

### Organizing Committee

Josh Tenenbaum (Massachusetts Institute of Technology)
Alan Yuille (University of California, Los Angeles (UCLA))