Toward Boundedly Rational Analysis

Toward Boundedly Rational Analysis Thomas Icard (icard@cmu.edu) Department of Philosophy, Baker Hall 135 Carnegie Mellon, Pittsburgh, PA 15213-3890 USA Abstract The Bayesian program in cognitive science has been subject to criticism, due in part to puzzles about the role of rationality and approximation. While somewhat sympathetic with these con- cerns, I propose that a thoroughgoing boundedly rational anal- ysis strategy can answer to some of them. Through simulation results I illustrate the method by showing how one can retrod- ict recently reported results about particle filter models of cat- egorization (Sanborn et al., 2010). I also introduce new obsta- cles that surface once we take bounded rationality seriously. Specifically, again through simulation, I show that the analy- sis of optimal sampling from Vul et al. (2014) is interestingly complicated by the introduction of agents capable of metarea- soning. Under broad conditions, such agents outperform all uniform k-sampling agents. This motivates the computational study of boundedly rational metareasoning in its own right. Keywords: rational analysis, bounded rationality, algorithmic level, sampling, categorization, metareasoning. Anderson (1990) proposed his often-rehearsed six steps comprising the rational analysis strategy: 1. Precisely specify the goals of the cognitive system. 2. Develop a formal model of the environment to which the system is adapted. 3. Make minimal assumptions on computational limitations. 4. Derive the optimal behavior given items 1 through 3. 5. Examine the empirical literature to see if the predictions of the behavioral function are confirmed. 6. If the predictions are off, iterate. He then illustrated the methodology with four example do- mains: memory retrieval, category learning, judging causal strength, and problem solving (i.e., decision making). In each case, he was able to show a good fit to a wide range of data, demonstrating that the strategy can be successful. The Rational Analysis Strategy The program of rational analysis, pioneered by Marr and Pog- gio (1976), and greatly extended by Anderson (1990), seeks to understand cognition in terms of rational solutions to un- derlying problems the mind is assumed to be solving. Many cognitive phenomena can be characterized as inference prob- lems under uncertainty, where some latent state must be in- ferred on the basis of observed information. For such prob- lems Bayesian methods provide a robust and well understood notion of optimality or rationality (DeGroot, 2004). Bayesian models of cognition have become increasingly popular in re- cent years, and have been applied to phenomena as diverse as vision, causal learning, language understanding, and in- tuitive physics (see Griffiths et al. 2008; Tenenbaum et al. 2011). This work typically understands inference as condi- tionalization on a probability distribution assumed to capture the subject’s ‘intuitive model’ of the situation or domain. Rational analysis, and the Bayesian instantiation thereof, is sometimes used to show in what sense a given behavior can be understood as rational. Even when we have a mechanistic understanding of how some cognitive function works, a ratio- nal analysis can shed light on why it works the way it does, often generating new testable predictions (e.g., Movellan and McClelland 2001). A more ambitious use of the method is in guiding our search for mechanisms in the first place. Indeed, one of the motivations behind Marr and Poggio’s and An- derson’s proposals was to narrow down the search space of cognitive models, by assuming that the right model must be one that at least approximately solves the underlying (e.g., in- ference) problem. In that way, progress on the computational level problem (Marr, 1982)—in addition to being worthwhile in its own right—may also promise progress in the search for more mechanistic, biologically detailed, models of cognition. Anderson’s Rational Model of Categorization As a running example, consider Anderson’s (1990; 1991) analysis of categorization. One may object to some of the assumptions behind the model, but categorization per se is not our focus—it is merely intended as a useful illustration. A subject is assumed to observe a sequence of objects with various combinations of features; upon observing a new ob- ject, the subject needs to make an inference about an unob- served feature. Let Y i be the value of the feature of interest for the ith observed object, and let X i be the ith vector of values for the remaining features. Let us furthermore abbreviate the conjunction of the first n values, Y 1 , . ,Y n and X 1 , . , X n , as Y N and X N , respectively. The problem facing the subject is to infer the value of Y n , given observations of X N and Y N−1 , i.e., to find the value of Y n that maximizes the posterior probability P(Y n | X N , Y N−1 ) of Y n given X N and Y N−1 . Anderson argued that the ideal way to determine such probabilities would be to consider all possible clusterings of the first n objects, figuring out the probability of each, and using the clusterings to determine the probability of Y n given X n . In other words, one must determine the value of a latent variable Z n , which corresponds to a clustering of the n ob- jects observed so far. Then we can determine the posterior probabilities by summing over the possible clusterings: P(Y n | X N , Y N−1 ) = ∑ P(Y n | Z n ) P(Z n | X N , Y N−1 ) . Z n P(Z n | X N , Y N−1 ) is given in terms of Bayes Rule: P(Z n | X N , Y N−1 ) ∝ P(X N , Y N−1 | Z n ) P(Z n ) .