Learning from Games: Inductive Bias and Bayesian Inference

Learning from Games: Inductive Bias and Bayesian Inference Michael H. Coen 1,2 and Yue Gao 2 Department of Biostatistics and Medical Informatics 1 Department of Computer Sciences 2 University of Wisconsin, Madison, WI 53706 {mhcoen, gao} @ cs.wisc. edu Abstract A classic problem in understanding human intelligence is determining how people make inductive inferences when presented with small amounts of data. We examine this question in the context of the guess-the-next-number game, where players are presented with short series of numbers and asked to guess the next one in the sequence. Our approach is unique in that we use a stochastic context free grammar to model the mathematical operations that generate a given sequence. The individual probabilities in this grammar are learned by observing people play this game, and thereby, they capture some of the mathematical inductive bias of our sample population. We then use this framework to solve novel sequence guessing problems computationally, mirroring human performance. Our goal is to better understand how people approach math problems by examining the space of mathematical functions they find easiest to both generate and recognize. We are also interested in tracking how this changes over time as functions of education and age. Finally, we examine how our results confirm a large body of psychological observations about how people approach mathematics problems. Keywords: Inductive Bias; Mathematical Modeling; Stochastic Context Free Grammars; Bayesian inference. Introduction People regularly make inferences by induction, even when presented with very small amounts of information. The purpose of our work is to understand more about how this occurs in mathematical reasoning, especially when these inferences are remarkably consistent and have little formal justification. We explore this problem in the context of the guess-the-next-number game. For example, suppose we present someone with the sequence [1, 2, 4] and ask him to guess the next number in the series. If he were to suggest 8, stating this set corresponded to “powers of 2,” we would presumably find this a plausible explanation. On the other hand, were he to suggest the number is 6, explaining the sequence seems to be “1 followed by the even numbers,” we might find that answer less satisfactory. 1 Of course, this determination is highly arbitrary. Because inductive inferences are not logically entailed, they must rest upon some set of assumptions, known as an inductive bias (Mitchell 1980). In the absence of such a bias, all inferences consistent with a finite dataset are equally valid, We will see, however, that appeals to minimum description length (e.g., Rissanen 1978) do not capture human preferences in making these judgments. as observed by Hume (1739) and formalized by Wolpert (1996). While this is true in a theoretical sense, people surely do have preferences; in other words, it is quite reasonable to suppose that we find some answers more plausible than we do others, even in the absence of objective justifications. Although there is no doubt that humans generalize using an inductive bias, formally characterizing it can be challenging. For example, even in domains where one believes that Occam’s Razor is the principle guiding human induction (Myung and Pitt 1997), the numerous formulations of this classic notion of parsimony may lead to diametrically opposed conclusions. (For an interesting discussion of this issue in elucidating the innate mathematical abilities of infants, see Carey 2002). In this paper, we explore a domain where we can formally model inductive bias using probabilities in a stochastic context free grammar, which appear to capture how people do math “in their heads. ” Because our notion of inductive bias is rigorously defined, we can also trace how it changes over time. Thus, we expect grammar school, high school, and college students to have very different inductive biases, which can of course be further characterized by their fields of study. We are also interested in tracking how these mathematical biases change as people age. More generally, this work provides a window in the types of mathematical operations with which people are most comfortable as well as the types of operations that perhaps require more careful instruction. Figure 1. Examining the guess-the-next-number game. While the examples in (a) and (b) are fairly straightforward, the sequence in (c) is somewhat more ambiguous.