on 17-Apr-2017 (Mon)

3. Multiple Levels of Computational Cognitive Modeling

he goal of this chapter is to illustrate the kinds of computational models of cogni- tion that we can build if we assume that hu- man learning and inference approximately follow the principles of Bayesian probabilis- tic inference and to explain some of the mathematical ideas and techniques under- lying those models.

Bayesian models have ad- dressed animal learning (Courville, Daw, & Touretzky, 2006), human inductive learn- ing and generalization (Tenenbaum, Grif- fiths, & Kemp, 2006), visual scene percep- tion (Yuille & Kersten, 2006), motor con- trol (Kording & Wolpert, 2006), seman- tic memory (Steyvers, Griffiths, & Dennis, 2006), language processing and acquisition (Chater & Manning, 2006; Xu & Tenen- baum, 2007), symbolic reasoning (Oaksford &Chater, 2001), causal learning and in- ference (Steyvers et al., 2003; Griffiths & Tenenbaum, 2005, 2007a), and social cog- nition (Baker, Tenenbaum, & Saxe, 2007), among other topics.

To us, the big question is this: How does the human mind go beyond the data of experience? In other words, how does the mind build rich, abstract, veridical models of the world given only the sparse and noisy data that we observe through our senses?

he Bayesian framework for probabilis- tic inference provides a general approach to understanding how problems of induc- tion can be solved in principle and per- haps how they might be solved in the hu- man mind. Let us give a few examples. Vision researchers are interested in how the mind infers the intrinsic properties of a ob- ject (e.g., its color or shape) as well as its role in a visual scene (e.g., its spatial re- lation to other objects or its trajectory of motion). These features are severely under- determined by the available image data. For instance, the spectrum of light wavelengths reflected from an object’s surface into the observer’s eye is a product of two unknown spectra: the surface’s color spectrum and the spectrum of the light illuminating the scene. Solving the problem of “color constancy” – inferring the object’s color given only the light reflected from it, under any conditions of illumination – is akin to solving the equa- tion y = a × b for a given y,withoutknow- ing b. No deductive or certain inference is possible. At best, we can make a reasonable guess, based on some expectations about which values of a and b are more likely a priori. This inference can be formalized in a Bayesian framework (Brainard & Free- man, 1997), and it can be solved reasonably well given prior probability distributions for natural surface reflectances and illumination spectra.

2. The Basics of Bayesian Inference

2.1. Bayes’ Rule

2.2. Comparing Hypotheses

2.3. Parameter Estimation

2.4. Model Selection

2.5. Summary

3. Graphical Models

3.1. Bayesian Networks

3.2. Representing Probability Distributions over Propositions

3.3. Causal Graphical Models

3.4. Example: Causal Induction from Contingency Data

4. Hierarchical Bayesian Models

4.1. Example: Learning about Feature Variability

4.2. Example: Property Induction

5. Markov Chain Monte Carlo

5.1. Example: Inferring Topics from Text

6. Conclusion

Bayesian Models of Cognition

1. Introduction

he focus of this chapter will be on problems in higher-level cognition: inferring causal structure from patterns of statistical correlation, learning about categories and hidden properties of objects, and learning the meanings of words. This focus is partly a pragmatic choice, as these topics are the sub- ject of our own research and hence we know them best. But there are also deeper rea- sons for this choice. Learning about causal relations, category structures, or the proper- ties or names of objects are problems that are very close to the classic problems of induction that have been much discussed and puzzled over in the Western philo- sophical tradition. Showing how Bayesian methods can apply to these problems thus illustrates clearly their importance in un- derstanding phenomena of induction more generally. These are also cases where the im- portant mathematical principles and tech- niques of Bayesian statistics can be applied in a relatively straightforward way. They thus provide an ideal training ground for readers new to Bayesian modeling.

Bayesian principles dictate how rational agents should update their be- liefs in light of new data, based on a set of as- sumptions about the nature of the problem at hand and the prior knowledge possessed by the agents.

Bayesian models are typically formulated at Marr’s (1982)levelof“com- putational theory,” rather than the algorith- mic or process level that characterizes more traditional cognitive modeling paradigms

limits on the phenomena that are valuable to study within aBayesianparadigm.Somephenomenawill surely be more satisfying to address at an al- gorithmic or neurocomputational level. For example, that a certain behavior takes peo- ple an average of 450 milliseconds to pro- duce, measured from the onset of a visual stimulus, or that this reaction time increases when the stimulus is moved to a different part of the visual field or decreases when the same information content is presented audi- torily, are not facts that a rational computa- tional theory is likely to predict.

not all computational-level models of cog- nition may have a place for Bayesian anal- ysis. Only problems of inductive inference, or problems that contain an inductive com- ponent, are naturally expressed in Bayesian terms. Deductive reasoning, planning, or problem solving, for instance, are not tra- ditionally thought of in this way. However, Bayesian principles are increasingly coming to be seen as relevant to many cognitive ca- pacities, even those not traditionally seen in statistical terms (Anderson, 1990; Oaksford &Chater, 2001), because of the need for people to make inherently underconstrained inferences from impoverished data in an un- certain world

The more complex methods can support multiple hierarchi- cally organized layers of inference, struc- tured representations of abstract knowledge, and approximate methods of evaluation that can be applied efficiently to data sets with many thousands of entities. For the first time, we now have practical methods for developing computational models of human cognition that are based on sound proba- bilistic principles and that can also capture something of the richness and complexity of everyday thinking, reasoning, and learning.

Finally, probabilistic models can be used to advance and perhaps resolve some of the great theoretical debates that divide tradi- tional approaches to cognitive science. The history of computational models of cogni- tion exhibits an enduring tension between models that emphasize symbolic represen- tations and deductive inference, such as first-order logic or phrase structure gram- mars, and models that emphasize continu- ous representations and statistical learning, such as connectionist networks or other as- sociative systems. Probabilistic models can be defined with either symbolic or continu- ous representations, or hybrids of both, and help to illustrate how statistical learning can be combined with symbolic structure. More generally, we think that the most promis- ing routes to understanding human intelli- gence in computational terms will involve deep interactions between these two tradi- tionally opposing approaches, with sophis- ticated statistical inference machinery oper- ating over structured symbolic knowledge representations.

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