Modeling the Role of Unobserved Causes in Causal Learning

Modeling the Role of Unobserved Causes in Causal Learning Christian C. Luhmann (christian.luhmann@vanderbilt.edu) Department of Psychology, 2 Hillhouse Ave New Haven, CT 06511 USA Woo-koung Ahn (woo-kyoung.ahn@yale.edu) Department of Psychology, 2 Hillhouse Ave New Haven, CT 06511 USA To compute the probability of the unobserved cause (u) in a situation with one observed cause (o) and one effect (e), BUCKLE applies Bayes theorem to the current beliefs about the strength of o and u (q o and q u respectively) and the prior belief about the probability of u being present (i.e., P(u) with we will always assume to be . 5). The following equations are for cases when q o and q u are believed to be generative in a current trial (see Luhmann & Ahn, 2006, for equations for other cases): P(u) • (1- q u ) P(u | o = 0,e = 0) = [ 1- P(u) ] + [ P(u) • (1- q u ) ] Abstract Current theories suggest that causal learning is based on covariation information. However, information about the presence/absence of events (particularly causes) is frequently unavailable, rendering them unobserved. The current paper presents a new model of causal learning, BUCKLE (Bidirectional Unobserved Cause LEarning), which extends existing models of causal learning by dynamically inferring information about unobserved causes. During the course of causal learning, BUCKLE continually computes the probability that an unobserved cause is present on each occasion and uses the results of these inferences to adjust the strengths of the unobserved, as well as observed, causes. P(u | o = 0,e = 1) = Keywords: Causal learning, inference, induction Introduction Current models of causal induction assume that the input available to reasoners comes in the form of covariation; how the causes vary with their effects. Thus, a learner observes whether the presence or absence of a causal candidate is followed by the presence or absence of an effect, and translates these observations into beliefs about causal relations. Yet, in the real world, covariation is often not available. For example, acquiring information about the presence/absence of causes sometimes requires special methods (e.g., genetic influences on cancer). Perhaps more commonly, causes are unobserved simply because learners cannot possibly consider all alternative causes of a particular event. For instance, we do not know all possible causes for gender discrepancy in science. Thus, lacking information about the presence/absence of causes seems to be the rule rather than the exception. This paper presents a new model of causal learning, BUCKLE, which attempts to capture how people learn causal relations when information about causes is missing. BUCKLE BUCKLE (Bidirectional Unobserved Cause LEarning) assumes that the learning environment always includes an unobserved cause and learns by performing two steps during each trial. The first step is to compute the probability that the unobserved cause is present. The second step is to adjust the strengths of each cause-effect relationship using an error-correction algorithm. P(u | o = 1,e = 0) = P(u | o =1,e =1) = P(u) • q u P(u) • q u (1− q o ) • P(u) • (1- q u ) q { o [ P(u) ] } + [ (1− q o ) • P(u) • (1- q u ) ] P(u) • [ q o + q u − ( q o • q u ) ] { q o • [ 1− P(u) ] } + { P(u) • [ q o + q u − ( q o • q u ) ] } In words, Equation 1, for instance, shows the probability that u is present when o and e are absent. The denominator is the probability of e being present given that o is absent, which occurs when either u is absent, or u is present but fails to cause e. The numerator of Equation 1 is the probability of the latter occurring (i.e., u being present). Once the probability of u is computed, the unobserved cause is treated just like an observed cause except that it is present with some probability. These equations allow BUCKLE to make several predictions. For example, the probability of u should vary as a function of trial type (i.e., whether o and e are present or absent). Also, note that Equation 2 is special. This equation suggests that people should believe that u present with a certainty (i.e., P(u) ≈ 1), when o is absent but e is present (i.e., what Luhmann & Ahn, 2003 call unexplained effects), because o and u are the only possible causes. BUCKLE’s second step is to use the observed and inferred information to adjust the strength of each causal relationship. BUCKLE learns via an error-correction algorithm. Information about the state of the causes (i.e., o and u) is first used to predict how likely the effect is given BUCKLE’s current causal beliefs (i.e., q o and q o ). This