Causal Learning from Biased Sequences
Causal Learning from Biased Sequences David Danks (ddanks@cmu.edu) Department of Philosophy, Carnegie Mellon University, 135 Baker Hall Pittsburgh, PA 15213 USA; and Institute for Human & Machine Cognition, 40 S. Alcaniz St. Pensacola, FL 32502 USA Samantha Schwartz (sschwartz@andrew.cmu.edu) Carnegie Mellon University, 135 Baker Hall Pittsburgh, PA 15213 USA both algorithmic and computational—correspond to maximum likelihood estimates of parameters in specific Bayesian network structures. An overview of these connections can be found in Danks (in press). In this paper, we focus on a salient feature of essentially all of the algorithmic theories (that are currently considered viable). They all predict some sensitivity to the order of case presentation: different orderings of the same set of cases will (sometimes) lead to different responses at the end of the sequence. Various theories predict different order effects, and so biased sequences have been used to test the various algorithmic-level theories. In contrast, essentially all of the standard computational- level theories, including power PC, conditional ∆P, and standard Bayesian network learning (whether Bayesian updating or constraint-based) assume that the observed cases are independently distributed. That is, they assume that the probability of observing a case does not depend on the previous trial(s). As a result, they make no prediction about order effects. This does not mean that they explicitly predict the absence of order effects. Rather, since a basic assumption is violated (e.g., if a sequence is biased in certain ways), these theories do not make any clear, determinate predictions. Abstract Multiple psychological theories of causal learning provide case-by-case updating rules: given my current causal beliefs about the world and a novel case, how should I change those beliefs? Most of these theories predict some type of order effect: biased and unbiased sequences of cases will lead to different final causal beliefs, even if the overall statistics are identical. This paper describes an experiment that (i) finds only small order effects that (ii) are not dependent on the number of observed cases, and in which (iii) observed patterns of belief change during the sequences are not explained by various proposed algorithmic theories. Keywords: Causal models; biased observations; learning. Introduction Causal knowledge and beliefs play a significant role in much of our everyday cognition. A range of theories have been proposed over the past fifteen years to explain human causal learning, and in particular, learning from sequences of observations or manipulations of the world. That is, they predict the inference of causal relationships from a sequence of individual cases, each of which is either observed or produced by the learner. Algorithmic theories offer explicit, case-by-case updating rules. Most notably, this group of theories includes standard associationist models (e.g., Pearce, 1994; Rescorla & Wagner, 1972). In contrast, computational theories aim to predict our stable, long-run causal beliefs. These theories range from the purely probabilistic (e.g., conditional ∆P model of Cheng & Novick, 1992; Spellman, 1996), to theories with more robust metaphysics based on observed probabilities. This last group includes both Cheng’s (1997) power PC theory, and various theories based on causal Bayesian network structure inference (a partial list includes Danks, Griffiths, & Tenenbaum, 2003; Gopnik, Glymour, Sobel, Schulz, Kushnir, & Danks, 2004; Griffiths & Tenenbaum, in press; Lagnado & Sloman, 2004; Steyvers, Tenenbaum, Wagenmakers, & Blum, 2003; Tenenbaum & Griffiths, 2001; Waldmann & Martignon, 1998). These types of theories are connected in at least two ways. First, the long-run behaviors of many algorithmic theories are characterized by (independently proposed) computational theories. Second, many of these theories— Primacy vs. Recency Consider the following three distinct types of correlations within some sequence of observed cases: • Pos/Neg: The first half of the sequence has a positive correlation between variables C and E, and the second half has a negative correlation; • Neg/Pos: The first half has a negative correlation, and the second half has a positive correlation; and • Even: C and E are uncorrelated during the sequence. There are two natural types of order effects. A primacy effect occurs if the initial cases have a greater weight in the inference process than later ones. In that case, the first sequence should result in a positive perceived causal strength, the second in a negative causal strength, and the third with a zero causal strength. In contrast, a recency effect occurs when the later cases have a greater impact on perceived causal strength than earlier ones. The response profile of a recency effect in