# Probabilistic Models of Cognition

By Noah D. Goodman, Joshua B. Tenenbaum, Timothy J. O'Donnell, and the Church Working Group^{[1]}.^{[2]}

(This tutorial is based on the ESSLLI Tutorial, created by Goodman, Tenenbaum, and O'Donnell.)

What is thought? How can we describe the intelligent inferences made in everyday human reasoning and learning? How can we engineer intelligent machines? The *computational theory of mind* aims to answer these questions starting from the hypothesis that the mind is a computer, mental representations are computer programs, and thinking is a computational process—running a computer program.

But what *kind* of program? A natural assumption is that this program take the inputs—percepts from the senses, facts from memory, etc—and compute the outputs—the intelligent behaviors. Thus the mental representations that lead to thinking are functions from inputs to outputs.
However, this input-output view suffers from a combinatorial explosion: we must posit an input-output program for each task in which humans draw intelligent inferences. A different approach is to assume that mental representations are more like *theories*: pieces of knowledge that can support many inferences in many different situations. For instance, Newton's theory of motion makes predictions about infinitely many different configurations of objects and can be used to reason both forward in time and from the consequences of an interaction to the initial state. The generative approach posits that mental representations are more like theories in this way: they capture more general descriptions of how the world works—hence, the programs of the mind are models of the world that can be used to make many inferences.
^{[3]}

A *generative model* describes a process, usually one by which observable data is generated. Generative models represent knowledge about the causal structure of the world—simplified, "working models" of a domain. These models may then be used to answer many different questions, by conditional inference. This contrasts to a more procedural or mechanistic approach in which knowledge represents the input-output mapping for a particular question directly. While such generative models often describe how we think the "actual world" works, there are many cases where it is useful to have a generative model even if there is no "fact of the matter". A prime example of the latter is in linguistics, where generative models of grammar can usefully describe the possible sentences in a language by describing a process for constructing sentences.

It is possible to use deterministic generative models to describe *possible* ways a process could unfold, but due to sparsity of observations or actual randomness there will often be many ways that our observations could have been generated. How can we choose amongst them? Probability theory provides a system for reasoning under exactly this kind of uncertainty. *Probabilistic* generative models describe processes which unfold with some amount of randomness, and probabilistic inference describes ways to ask questions of such processes. This tutorial is concerned with the knowledge that can be represented by probabilistic generative models and the inferences that can be drawn from them.

In order to make the idea of generative models precise we want a formal language that is designed to express the kinds of knowledge individuals have about the world. This language should be *universal* in the sense that it should be able to express any (computable) process. We build on the <math>\lambda</math>-calculus (as realized in functional programming languages) because the <math>\lambda</math>-calculus describes computational processes and captures the idea that what is important is causal dependence—in particular the <math>\lambda</math>-calculus does not focus on the sequence of **time**, but rather on which events influence which other events. We introduce randomness into this language to construct a *stochastic* <math>\lambda</math>-calculus, and describe conditional inferences in this language.

# Generative Models

## Generative Models: Exercises

## Church Basics: Exercises

# Conditioning

## Conditioning: Exercises

# Patterns of Inference

## Patterns of Inference: Exercises

# Learning as Conditional Inference

## Learning as Conditional Inference: Exercises

# Occam's Razor

## Occam's Razor: Exercises

# Hierarchical Models

# Mixture Models

# Non-Parametric Models

# Recursive Models

# Inference about inference: Nested query

- ↑ Contributors include: Andreas Stuhlmueller, John McCoy, Tomer Ullman, Long Ouyang.
- ↑ The construction and ongoing support of this tutorial are made possible by grants from the Office of Naval Research and the James S. McDonnell Foundation.
- ↑ Of course, the process by which inferences are drawn from a "model" or "theory" can, and should, also be described as a computational process. It is, however, useful to separate computational descriptions of knowledge and the inferences that can be drawn from knowledge, from computational descriptions of the process of inference. This is similar to Marr and Poggio's notion of "levels of analysis" (see Marr, 1982).