The Conceptual Basis of Function Learning and Extrapolation: Comparison of Rule-Based and Associative-Based Models

Article excerpt

The purpose of this article is to provide a foundation for a more formal, systematic, and integrative approach to function learning that parallels the existing progress in category learning. First, we note limitations of existing formal theories. Next, we develop several potential formal models of function learning, which include expansion of classic rule-based approaches and associative-based models. We specify for the first time psychologically based learning mechanisms for the rule models. We then present new, rigorous tests of these competing models that take into account order of difficulty for learning different function forms and extrapolation performance. Critically, detailed learning performance was also used to conduct the model evaluations. The results favor a hybrid model that combines associative learning of trained input-prediction pairs with a rule-based output response for extrapolation (EXAM).

Human concepts are complex and varied and serve a myriad of purposes. One way in which concepts are used is in learning how to categorize people or things and infer properties from category membership. Historically, this view of concepts has dominated the theoretical and empirical literature in cognitive psychology. But this view of concepts is too restrictive; another important way in which concepts are used is in learning functional relationships between continuous variables and in making predictions about one variable on the basis of another (Bourne, Ekstrand, & Dominowski, 1971; UhI, 1963). There are many examples of such relationships that we encounter every day, such as predicting job performance on the basis of intelligence, anticipating mood level on the basis of stress intensity, forecasting interest rates on the basis of inflation rates, predicting harvest yields on the basis of amount of rainfall, and so on (Hammond, 1955; Huffman, 1960). Learning functional relations between causes and effects is fundamental to the formation of intuitive theories about how the world works, and these predictions guide subsequent decisions about how to control the world (Hammond & Stewart, 2001; also see Murphy & Medin, 1985). For example, in order to control the economy we need to know how increases in interest rates affect consumer spending, which in turn affects manufacturing and employment rates.

The literature reflects an imbalance in the amount of attention devoted to categorization relative to function learning, with extensive progress in both empirical and theoretical understanding of categorization (Estes, 1994; Lamberts & Shanks, 1997), and much less empirical interest and theoretical progress in understanding function learning. Given the importance of function learning for human conceptual activity, the dearth of theoretical development in this area is a serious omission.

The purpose of this article is to provide the foundations for a more formal, systematic, and integrative approach to function learning that parallels the existing progress in category learning. This is accomplished by developing mature and complete models on the basis of preliminary ideas from the function-learning literature. We first provide an overview of the initial theoretical approaches and highlight their limitations. Next, we develop a number of formal models that provide a more comprehensive specification of function learning than the initial models have. Finally, we evaluate and contrast how well the models account for a range of basic learning and transfer findings.

Before presenting the theoretical approaches, we need to describe briefly the function-learning paradigm. The present article focuses on single input-output functionlearning experiments in which a single cue χ is mapped by a continuous function F into a single criterion z. In a typical experiment, participants are initially provided a neutral cover story that verbally describes the experimental task but provides little or no direct information about the cue-criterion relation. …