At least for the past quarter century, many psychologists have seriously considered the possibility that untutored humans as well as other animals are capable of acquiring and using statistical knowledge about the structure of the environment. Peterson and Beach (1967) called people "intuitive statisticians", and Kelley (1967) proposed that people are "intuitive scientists". In the context of experimental paradigms investigating classical conditioning, other theorists have suggested that lower animals operate as intuitive statisticians (e.g., Gallistel, 1990; Miller & Schactman, 1985). Although there has in fact been broad agreement that various forms of causal induction depend on the implicit computation of statistical information, the question of precisely what is computed has yet to be resolved. In the field of animal conditioning, as well as in human categorization and causal induction, various theorists have proposed that animals perform some implicit computation of statistical contingency: the difference between the proportion of events for which an effect occurs when a factor is present and that proportion when it is absent.

In all these fields, the contingency approach has been contrasted with the associationist approach exemplified by the connectionist learning rule incorporated in the Rescorla and Wagner (R-W) model of conditioning (Rescorla & Wagner, 1972). The R-W model is directly related to a number of issues that lie at the very core of cognitive science. The R-W model was originally proposed as a model of classical conditioning in animals; however, a number of researchers have extended the model to account for apparently higher-order learning in humans, such as categorization and causal induction. The R-W learning rule is equivalent to the least mean squares (LMS) learning rule commonly used to adjust the weights on links in connectionist networks (Widrow & Hoff, 1960; see Sutton & Barto, 1981). Gluck and Bower (1988), for example, have applied an adaptive connectionist network using the LMS rule to model data on human categorization (also Estes, Campbell, Hatsopoulos & Hurwitz, 1989). Similarly, Shanks (1991) applied a connectionist implementation of the R-W model to attempt to account for effects of cue competition in a task involving classification of diseases on the basis of symptoms (see also Chapman & Robbins, 1990; Wasserman, 1990). Because of its apparent simplicity and evident generality, the R-W model remains highly influential as an approach to inductive learning in adaptive systems.

Some theorists have argued that the R-W model can account for phenomena involving cue competition and other cue interactions that cannot be explained by contingency. We will argue, however, that these claimed advantages for the R-W model over contingency theory disappear when the concept of contingency is suitably generalized along lines suggested by a number of philosophers and psychologists. In fact, the R-W model can itself be analysed as a mechanism that computes contingency under a certain restricted condition that we will discuss later. For such cases, the R-W model is successful at predicting cue competition, although even here its successes are qualified for domains in which the adaptive system operates on representations coded in terms of the probabilities of events, for which the additivity assumption underlying the model is inappropriate. Outside of cases that satisfy the restricted condition, the R-W model does not compute contingency, and in such situations the model appears to be empirically inadequate. In contrast, a generalized contingency theory can explain a number of the phenomena that contradict predictions of the R-W model.

In this chapter we present a contingency analysis of the successes and failures of the R-W model. Our theoretical analyses may provide a framework for understanding the weakness of an important associationist model. It hopefully will guide future research concerning how statistical regularity is computed by adaptive systems to infer the causal structure of their environments in the course of learning, on the basis of which predictions are made.

It has long been argued that contingency is a component of the normative criterion for inferring a causal link between a factor and an effect (e.g., Kelley, 1967; Rescorla, 1968; Salmon, 1980). Cheng and Novick (1990) proposed an extended version of contingency theory, which they termed the probabilistic contrast model (PCM), as a descriptive account of the use of statistical regularity in human causal induction. The model, which applies to events describable by discrete variables, assumes that one of the initial criteria for identifying potential causes is perceived temporal priority (causes must be perceived to precede their effects). The model assumes that potential causes are then evaluated by contrasts computed over a focal set. (We will use the terms "contrast and "contingency" interchangably.) The focal set for a contrast is a contextually-determined set of events that the reasoner selects to use as input to the computation of that contrast. It is often not the universal set of events, contrary to what has been assumed by previous contingency theories in psychology. Consider the set of events for inferring what causes a forest fire. A reasoner will normally restrict their focal set to terrestrial events, in which oxygen is always present, and will not consider events that occur in oxygen-free outer space.

Using the events in the focal set, a main-effect contrast specifying a potential cause i is defined as

,(1)

where pi is the proportion of events for which the effect occurs when factor i is present, and p is the proportion of events for which the effect occurs when factor i is absent. (The proportions are estimates of the corresponding conditional probabilities.) If Dpi is noticeably different from 0, i is perceived as a cause. Note that if a factor i is constantly present within the focal set, the second term in the contrast, p, cannot be calculated. Thus in our forest-fire example, it will be impossible to compute a contrast for oxygen, since this factor is never absent within the focal set of terrestrial events; as a result, oxygen will not be considered a cause of the fire (even though people would agree, if probed, that the presence of oxygen was in fact necessary for the fire to have occurred).

Contrasts can be either positive, in which case the cause is excitatory; or they can be negative, in which case the cause is inhibitory. For example, smoking presumably has a positive contrast with respect to lung cancer, and hence be viewed an an excitatory cause of the disease; whereas exercise would have a negative contrast with respect to heart disease, and hence be viewed as an inhibitory cause for the disease. Confidence in the assessment of a contrast is presumed to increase monotonically with the number of cases observed.

Cheng (1994) shows that for situations in which alternative causes occur and act independently of i, a positive main-effect contrast for i gives an estimate of the causal power of i, as represented by the probability with which i produces the effect. This estimate is unbiased when alternative causes are absent within the focal set and/or the alternative factors present do not produce the effect. To the extent that these conditions are violated, the contrast for i tends to be an understimate of the power of i. In the extreme case in which some alternative cause is always present within the focal set and it always produces the effect, the contrast for i , which is zero, is uninterpretable.

Cheng's (1994) derivation also shows that for situations in which alternative causes do not occur independently of i, the main-effect contrast for i is confounded by the influence of these causes, and is not interpretable as an estimate of the power of i. To eliminate this confounding, it is therefore important to compute what we will call conditional contrast -- the contrast for the candidate factor conditional on holding constant the status of one (or more) other factors. A number of philosophers have proposed conditional contrasts as a criterion for inferring causality (cf. Cartwright, 1983, 1989; Reichenbach, 1956; Salmon, 1980; Suppes, 1970).

Main-effect contrasts assess the causal status of each factor considered individually. However, it is also possible for combinations of factors to influence the effect in ways that could not be predicted by the independent influences of the individual factors. Such situations involve interactions between factors, which can be assessed by a generalization of main-effect contrasts (Cheng, 1994). For example, a two-way interaction contrast specifying the conjunction of potential causal factors i and j is defined as

,(2)

where p, as before, denotes the observed proportion of cases in which the effect occurs when a potential contributing factor is either present or absent, as denoted by its subscripts. If Dpij is noticeably greater than zero, then i and j combine to produce the effect.[1] A two-way interaction contrast is thus based on a difference of differences -- here, the contrast for i when j is present minus the contrast for i when j is absent -- with the non-additivity of probabilities taken into account by the product term in Equation 2. Suppose, for example, that there are two drugs, A and B, which are safe when taken individually but usually fatal when taken together. The contrast for drug A with respect to death will therefore be high when B is also present, but zero when B is absent. The product of the two proportions given the sole presences of A and of B will also be zero. Accordingly, the interaction contrast (difference between the above two contrasts plus the product) will be high.

Notice that each of the two constituent contrasts in an interaction contrast is a conditional contrast. Also notice that conditional contrasts can be described in terms of variations of the focal set. We could say that in the focal set of events in which drug B is administered, the contrast for drug A is high; whereas in the focal set of events in which drug B is not administered, the contrast for drug A is zero. Furthermore, both of these conditional contrasts will differ from the unconditional contrast for A. This unconditional contrast is equivalent to the main-effect contrast for drug A over the focal set of all events involving the presence or absence of drug B.

Cheng and Novick (1990, 1991, 1992) have provided support for contrasts computed over an accurately identified focal set as a descriptive model of human causal inference. The model successfully predicts simple and conjunctive causal attributions and explains a number of empirical phenomena involving human causal attributions that had previously been considered biases. To illustrate the role of focal sets, consider the psychological distinction between causes and enabling conditions. In our example about what causes a forest fire, people might consider a lightning strike as the cause, but they will view the presence of oxygen as merely an enabling condition. Although oxygen is necessary for the fire, it is constant in the relevant focal set so that a contrast cannot be computed. Notice that in a different context, which evokes a focal set within which the presence of oxygen may vary (for example, a special laboratory intended to be oxygen-free), oxygen would be considered the cause of a fire that breaks out when oxygen leaks into that environment. The assessment of causation thus depends on pragmatic contextual influences. In terms of PCM, a potential causal factor that covaries with the effect (i.e., has a noticeable contrast with respect to the effect) within the contextually-determined focal set (e.g., lightning with respect to forest fires in the context of a forest) will be viewed as a cause; whereas a factor that is constant within that focal set (e.g., oxygen in a forest), but is known to covary with the effect in some other focal set (e.g., oxygen covaries with fire in special environments in which the occurrence of oxygen varies) is viewed as an enabling condition. As will be elaborated later, an enabling condition can be distinguished from an alternative cause that happens to be constantly present in the current focal set. We will return to the challenge that the distinction between causes and enabling conditions poses for R-W.

The most central claim of contingency theory, which is reflected in PCM, is that causal attributions depend not only on the probability of the effect given the presence of a cue, but also on the probability of the effect given the absence of the cue. In other words, a cue is viewed as causal only if its presence makes a difference to the probability of the effect. However, theorists have often resisted the notion that humans and other animals implicitly tally information about what happens in the absence of a potential cause. In particular, associationist models of animal conditioning have eschewed any direct representation of cause-absent information. One apparent reason for the reluctance to posit representations of cause-absent information is that any event could potentially be defined in terms of an indefinitely large number of absent factors. It would indeed be bizarre to suppose, for example, that your understanding of this passage might be caused by the (presumed) absence of ravens in the room in which you now sit. The generalized contingency model addresses this problem by restricting the initial tabulation of cause-absent information to those factors that are plausible causes according to prior knowledge or according to observed pairing with the effect. The challenge for associationist models has been to account for apparent influences of contingency on learning without introducing representations of the absence of potential causal factors. As we will see, the empirical successes and failures of the R-W model can be differentiated by when it succeeds or fails to implicitly tally cause-absent information.

Rescorla-Wagner Model

The most influential associationist theory of conditioning over the past two decades has been the R-W model. Interestingly, Rescorla (1968) was a harbinger of the importance of contingency in classical conditioning, which he demonstrated with elegant experiments showing that conditioning depends on events that occur in the absence as well as the presence of a cue. Nonetheless, he then went on to develop an associationist model that avoided postulation of representations of the absences of cues.The R-W model represents the learning of an association between cue i (e.g., a tone that is present in the current event) and outcome j (e.g., shock) by a change in the strength of a link between two elemental units, one representing cue i, and the other representing outcome j. (Cue i and outcome j are traditionally termed the "conditioned stimulus" and the "unconditioned stimulus", respectively.) For any cue i that is present during the event, strength is revised according to the rule

,(3)

where is the change in associative strength between cue unit i and outcome unit j as a result of the current event, and are rate parameters that respectively depend on the salience of i and j, and is the desired output corresponding to the actual outcome. Typically, if the outcome is present, is defined as 1; if the outcome is absent, this value is defined as 0. , defined as the sum of the current strengths of links to unit j from all units representing the n cues present in that event, is the actual output of the network predicting the outcome. If cue i is not present during the event, the associative strength of its cue unit remains unchanged. (The absence of a cue is not represented by any unit.) Learning continues until there is no discrepancy between the desired and actual outputs (averaged over a number of trials). In addition to the particular stimuli present (e.g., a tone), the cues are assumed to include one that represents a context present in every event (e.g., the conditioning cage). In causal terms, each cue i is a potential cause, and j is the effect. The strengths that are updated according to Equation (3) are equivalent to weights on the links in a two-layered connectionist network, with the predicting cues being represented on the input layer and the predicted outcome on the output layer.

A major attraction of R-W is its ability to explain the effects of interaction between cues. For example, it predicts the phenomenon of blocking (e.g., Kamin, 1969; Rescorla, 1981). Let P be a previously trained predictive cue (i.e., the presence of P has been paired with the outcome and the absence of P has been paired with the absence of the outcome). Consider the situation in which a novel cue, R, in combination with P, is paired with the outcome. It has been shown that, despite the positive unconditional contrast for R, the learning of this cue is blocked if it is presented only in combination with P. According to R-W, learning occurs only when there is some discrepancy between the predicted and actual outcomes. Because a predictive cue fully predicts the outcome as a consequence of prior pairings, no conditioning would accrue to R.

Rescorla (1968) demonstrated that no conditioning accrues at asymptote to a cue if the effect occurs equally often in its absence as in its presence. The R-W model explains this effect of contingency by the reduction of learning to the varying cue as a result of the strength that accrues to the constant context cue. More generally, the greater the strength of the context cue, the more it reduces the strength of the varying cue.

A second effect of cue interaction explained by R-W concerns the phenomenon of conditioned inhibition. It has been shown that a novel cue, I, acquires inhibitory associative strength when it -- in combination with a predictive cue P -- is paired with the absence of the outcome. In comparison, a novel cue that by itself is paired with the absence of the outcome acquires zero strength. According to R-W, the combination of P and I is initially expected to produce the outcome (due to the summing of the positive strength of P and the zero strength of I). A discrepancy between the predicted and actual outcomes therefore arises when the combination is paired with the absence of the outcome. This discrepancy leads to a reduction in the strength of I, which therefore becomes negative. (Although the strength of P will also be reduced on such trials, P will regain its strength on other trials in which the outcome continues to be predicted by the occurrence of P in the absence of I.) At asymptote the negative strength of I offsets the positive strength of P, leading to a net expectation of 0 on trials on which the combination of P and I is presented.

Limitations of the R-W Model

Despite its notable successes, the R-W model has several well-known limitations (see Gallistel, 1990; Holland, Holyoak, Nisbett & Thagard, 1986; Miller & Matzel, 1987). First, whereas the model predicts that the strength of a conditioned inhibitor should be revised upward toward 0 when it is presented alone without reinforcement, in fact such a procedure fails to extinguish the conditioned inhibitor (Zimmer-Hart & Rescorla, 1974). Second, the R-W model is unable to account for apparent changes in the associative strength of a cue that occur over a period in which that cue has not been presented (because the model only updates the strengths of cues that are present on a trial). For example, even though presentation of an inhibitor without reinforcement does not reduce its inhibitory power, extinction of the excitatory cue with which the inhibitor had been paired during acquisition can effectively weaken the inhibitor (Kaplan & Hearst, 1985; Kasprow, Schachtman & Miller, 1987; Miller & Schachtman, 1985). It is as if when the animal learns that the excitor no longer signals danger, it also loses confidence that the previously-paired inhibitor signals safety, even though the inhibitor has not been presented during the extinction phase. Similarly, a cue that was initially "overshadowed" by a more potent excitor will later gain excitatory potential during its absence if the overshadowing cue is extinguished (Kaufman & Bolles, 1981; Matzel, Schachtman & Miller, 1985; see Miller & Matzel, 1987). These "indirect" effects on conditioning cannot be explained by R-W, according to which the associative strength of a cue that is absent should not be updated (see Equation 3). Third, the R-W model does not explain the learned irrelevance of a cue that has been randomly paired with an unconditioned stimulus (i.e., the effect) before the effect is made contingent on the cue. Contradicting R-W, the conditioning of such a cue, relative to a novel cue, is severely retarded. These types of cues are predicted by the model to be equivalent, because they both should begin the conditioning phase with zero associative strength. Fourth, the R-W model predicts that the learning of the novel cue in the blocking paradigm described above will be completely blocked at asymptote. Available empirical results regarding causal induction by humans show, however, that blocking is only partial (e.g., Chapman & Robbins, 1990; Shanks, 1991; Shanks & Dickinson, 1987; Waldmann & Holyoak, 1992).

Learned Irrelevance

According to PCM, conditioning cue i requires creating a difference between pi and p. The rate at which this difference is created by any single event will be slower for a cue that has been randomly paired with an outcome than for a novel cue, because the cue that had been randomly-paired, unlike the novel cue, begins conditioning with large denominators in the two proportions. The impact of any single event in the conditioning phase, therefore, is smaller for the old than the novel cue. Suppose that a cue was present on 100 trials and absent on another 100 trials, and the outcome occurred on 40 trials of each type. The resulting contrast would be 0. Now suppose that in a subsequent conditioning phase, the outcome always occurs on 5 trials in which this pre-exposed cue is present and it never occurs on 5 trials on which this cue is absent. These 10 trials, together with the 200 initial trials, will lead to a contrast of 5/105 (i.e., 45/105 - 40/105). In comparison, for a novel cue, the same 10 trials in the conditioning phase will lead to a contrast of 1 (i.e., 5/5 - 0/5). The result, then, will be a marked attenuation of the rate of conditioning for the pre-exposed cue relative to a novel cue.

Conditional Contingencies and the Interpretation of Cue Interaction

When multiple potential causal factors are present, philosophers and computer scientists have proposed that assessment of causal relations should not be based on contingencies computed over the universal set of events (Cartwright, 1979, 1989; Pearl, 1988; Reichenbach, 1956; Salmon, 1980, 1984; Simpson, 1951; Suppes, 1970, 1984), because in these situations unconditional contingencies do not reflect what people intuitively judge to be normative causal inferences. In particular, people distinguish between a genuine cause and a spurious cause -- a factor that is contingently related to the effect but is not a cause of it. Unconditional contingencies do not reflect this difference.

These theorists have proposed that normatively, if a factor is known to be a cause of an effect, then determining the causal status of another factor requires that the contingency of the latter be calculated separately conditional on the presence and on the absence of that cause (a test of "conditional independence").[3] Testing for conditional independence is analogous to comparing experimental to control conditions in standard experimental design, where extraneous variables are kept constant across conditions. Although this criterion has not been uncontested among philosophers (e.g., Cartwright, 1989; Salmon, 1984), the prevalent adoption of the analogous principle of experimental design gives an indication of its normative appeal. One important difference between conditional contrasts and comparisons involving experimental design is that conditional contrasts includes observational situations, which generally provide less firm support for causal inferences. In terms of PCM, the adoption of the criterion of conditional contrasts involves computing contrasts for a potential causal factor separately for focal sets that are restricted to events in which the known cause is (a) present, and (b) absent, rather than computing them over the universal set of events.

We will consider the interpretation of tests of conditional independence, describe a process model for assessing conditional independence, and illustrate the explanation of cue interaction effects according to conditional contingencies in terms of this process model.

Let us first consider the interpretation of some possible outcomes of the test of conditional independence for a target factor that has a positive unconditional contingency with the effect (i.e., a possible excitatory cause). For example, suppose we are assessing possible causes of cancer, and that smoking cigarettes is an established cause. Now we observe that coffee drinking is also statistically relevant to cancer, in that the probability of cancer is higher for people who drink over five cups per day than for those who drink less coffee. However, let us further suppose that people who drink large quantities of coffee also tend to smoke. To tease apart the influence of coffee drinking from that of smoking, it is desirable to calculate the conditional contingency between coffee drinking and cancer separately for cases involving the presence versus the absence of smoking. The following are four possible outcomes that will be relevant in interpreting blocking and similar cue interaction effects:

Case 1: If both conditional contingencies for the target factor are positive, then the target factor will be interpreted as a genuine cause. For example, if coffee drinking increases the risk of cancer both for smokers and for non-smokers, then coffee drinking will be interpreted as a genuine cause (unless it turned out to be confounded with some other cause of cancer, such as eating fatty foods).

Case 2: If contingencies for the target factor conditional on both the presence and the absence of the established cause are zero, then that factor will be interpreted as a spurious cause. It is said to be "screened off" (i.e., normatively blocked) from the effect by the conditionalizing cause. For our example, the statistical link between coffee drinking and cancer would be attributed entirely to the confounding between coffee drinking and smoking.

Case 3: If the effect always occurs in the presence of the established cause, regardless of whether the target factor occurs (therefore, the contingency conditional on the presence of the established cause is zero), but the contingency conditional on the absence of the causal factor is positive, then the target factor will be interpreted as a genuine cause. This situation would arise if smoking always caused cancer, so that coffee drinking did not increase the risk of cancer for smokers, but did increase the risk for non-smokers. In this situation coffee drinking would be interpreted as a genuine cause of cancer. As noted earlier, the zero contingency for a candidate factor (coffee drinking) in the presence of an alternative factor that always produces the effect (smoking) does not give an interpretable estimate of the causal power of the candidate factor. In other words, it would likely be attributed to a ceiling effect (i.e., smoking by itself generates the maximal cancer risk, so that the detrimental impact of coffee drinking is masked for smokers).

Case 4: If the contingency of the target factor conditional on the presence of the established cause is positive, but the effect never occurs in the absence of the established cause (therefore, the contingency conditional on the absence of the known cause is zero), then the two factors will be interpreted as interacting to produce the effect (see Equation 2). Such an interaction would exist if coffee drinking in combination with smoking increases the risk of cancer for smokers, but had no effect on the probability of cancer for non-smokers.

One problem that complicates the test of conditional independence is that the information required for computing the two conditional contingencies is not always available. Recall that in the blocking paradigm, a novel cue R is paired with the outcome only when a predictive cue P is also present. Table 1 gives a schematic representation of the typical probability of the outcome for the two cues. The \x\to(P) and R cell receives no information, and the outcome always occurs when P is present. Because P is known to have a positive contingency with respect to the outcome, the status of R should be based on conditional contrasts. When the focal set is restricted to events in which P is present (the top row), R has a zero contrast. When the focal set is restricted to events in which P is absent (the bottom row), however, the contrast for R cannot be computed. Because this cue is never presented in the absence of P in this paradigm, pR \x\to(P) is undefined due to division by zero. As Waldmann and Holyoak (1992) noted, because the level of the effect produced by P is already at ceiling, it is impossible to determine whether the redundant cue R is a spurious cause (Case 2 above), or a genuine cause (Case 3). Given that relevant information is missing in the blocking design, subjects who adopt the criterion of conditional independence would be uncertain about the predictive status of the redundant cue, as opposed to being certain that this cue is not predictive, as implied by the R-W learning rule.[4]

The above analyses of the informativeness of conditional contingency tests apply in the case of possible excitatory causes, but not in that of possible inhibitory causes. A test of a target factor in the absence of all established causes cannot demonstrate that the factor is an inhibitor, because unless some excitatory cause is operating the impact of an inhibitor will be obscured by a cellar effect. That is, if the outcome is not being produced by some excitatory cause, an inhibitor cannot achieve a non-zero contingency. We assume that as a general principle based on a preference for cognitive simplicity, a factor will not be deemed causal unless positive evidence for a causal interpretation is obtained. Accordingly, the default interpretation of a zero contingency is that the factor is non-causal (rather than inhibitory). This assumption is supported by the fact that simply presenting a cue alone without reinforcement, while another cue presented alone is reinforced, generally does not yield strong conditioned inhibition (Baker, 1977). The former cue has a negative unconditional contingency, but its contingency conditional on the presence of the latter cue cannot be computed due to the lack of information on the frequency of the effect when both cues are present. Thus for a candidate inhibitory factor, the most informative tests will involve computation of its contingency conditional on the presence of a single excitor, coupled with the absence of all other known causal factors. If there is more than one known excitor, it will be desirable to perform separate tests for the candidate factor conditional on the presence of each excitor in turn. If the candidate yields a negative contingency conditional on the presence of an excitor, it will be interpreted either as a main-effect inhibitory cause or as a component of an inhibitory interaction.

Conditioned Inhibition and "Indirect" Extinction of Associative Strength

PCM can account both for the acquisition of conditioned inhibition as well as for the failure to extinguish a conditioned inhibitor by presenting it alone without the outcome. Table 2 schematically represents the typical probability of the outcome for the two cues in the learning phase of the conditioned-inhibition paradigm. When P is presented alone, the outcome occurs, but when P and I are presented in combination, the outcome does not occur. No information is received about the \x\to(P) and I cell (the empty one in the table) during the learning phases. Notice that the set of events so far shows a negative conditional contrast for I conditional on the presence of P (i.e., pI |P < p\x\to(I)|P). Therefore, PCM predicts that I will become inhibitory. Now consider an extinction phase in which I is presented alone without the outcome. The \x\to(P) and I cell would then be filled in with the information that the probability of the outcome in the presence of I alone is zero. This information will have no impact on the crucial conditional contingency -- that of I in the presence of P -- and hence will not yield extinction (Zimmer-Hart & Rescorla, 1972).

A Process Model for Assessing Conditional Dependence and Independence

We have so far been describing contingency theory at the computational level of analysis. Here we will describe an algorithmic instantiation of the theory. This process model is based on PCM, with extensions to specify which conditional contingencies are computed. Contingency analysis can of course be evaluated independently of this particular instantiation, but this model will serve to provide a detailed illustration of how contingency theory might account for cue interactions.

A plausible psychological model of causal inference based on contingency analysis must specify mechanisms that would allow people to decide (a) what cues should be used to conditionalize others, (b) what conditional tests to perform once a set of conditionalizing cues has been selected, and (c) how to integrate the resulting contingency information to make causal assessments of the cues. In situations where there is no guidance from prior knowledge, every cue is potentially causal. Given n binary cues, exhaustively conditionalizing the contingencies for each target cue on every combination of the presence and absence of the other cues requires computing 2n - 1.n contingencies. Given processing limitations, it is crucial to specify how people select which contingencies to compute. It is also likely that many of the cue combinations that would be relevant to a contingency analysis will never actually occur. Accordingly, it is necessary to specify which contingencies will be computed in the face of missing information.

Let us first consider the selection of conditionalizing cues. The ideal set of conditionalizing cues would include all and only those that are actually causal. Given the limitations of knowledge, the best people could do is to select as conditionalizing cues those that they currently believe to be plausible causes. In cases where prior knowledge is relevant, such knowledge would be used to establish certain cues as likely causes, and the contingencies for other cues would then be conditionalized on the (perhaps tentatively) established causes. If such prior knowledge is lacking, people may nonetheless use some heuristic criterion to select an initial set of conditionalizing cues. A simple heuristic that might be employed is to include any cue that is noticeably associated with the effect. That is, people may follow the tacit rule, "If the effect is likely to occur when the cue occurs, tentatively assume that the cue may be causal." Contingencies are not computed in this initial phase of selecting conditionalizing cues; rather, people simply identify a pool of cues that have been paired with the effect, which will be treated as an initial set of plausible causes. There is some evidence for such an initial phase of cue selection based on positive associations. For example, Rescorla (1972) found that a cue that was randomly paired with the outcome (i.e., one that was associated with the outcome but non-contingent with it) appeared to initially acquire associative strength, which eventually disappeared after several sessions of training. The association heuristic suggested here implies that this phase implicitly ignores the possibility of cues being interactive or inhibitory causes. The sole presence of an inhibitory cause, for example, would be perceived as a lack of association.

Contingency assessment will occur in the subsequent phase, in which people will compute the conditional contingencies of all cues based on the set of conditionalizing cues identified in the initial phase. In Cheng and Novick's (1990) terminology, the set of conditionalizing cues defines the focal sets for contingency computations. The initial set of conditionalizing cues can be dynamically updated if contingency assessments indicate that cues that at first appeared to be plausible causes are in fact spurious, or that cues initially viewed as causally irrelevant are in fact causal. That is, after conditional contrasts are calculated based on the initial set of conditionalizing cues, these contrasts will be used to update that set of cues. Cues in that set that have zero or low contrasts may be dropped and other cues outside that set that have noticeable positive or negative contrasts may be added. Changes in the set of conditionalizing cues will in turn change the relevant conditional contingencies for all cues, which may alter subsequent causal assessments. The entire assessment process will thus be iterative. If the values of the cues stabilize as the process iterates, the process will return these values and stop. Otherwise, the process will stop after an externally determined number of iterations.

In assessing conditional contingencies, heuristics will be required to determine which tests (of those possible given the cue combinations that are actually presented) should in fact be performed. We assume, based on the arguments presented earlier, that people will prefer to conditionalize the contingency for each target factor on the simultaneous absence of all conditionalizing cues. If this is not possible, then they will try to select a focal set in which as many conditionalizing cues are absent as possible, while the rest of the conditionalizing cues are constantly present. In general, application of the contingency analysis will be necessarily constrained by the information actually provided by observation.

In addition to specifying what cues are selected to form the conditionalizing set, and which conditional contingencies are computed, a process model must specify a response mechanism that translates the calculated contingencies into causal judgments. If all conditionalizing cues can be kept either absent or present, and there are no ceiling effects for excitatory cues, the confidence associated with the contrast values based on these focal sets will be relatively high. But if the experimental design omits cases that would be relevant in assessing the conditional dependence or independence for a target factor, such that there are ceiling effects or some of the conditionalizing cues cannot be kept constant, the confidence associated with the contrast values based on these focal sets will be relatively low. In such experiments, if subjects are not given the choice of withholding judgment, they may base their causal assessments on a mixture of the best available focal sets, for example, the unconditional as well as the conditional contingencies for cues. Mean ratings over subjects may therefore reflect some mixture of the evidence provided by conditional and unconditional contingencies.

When subjects do not all use one and the same focal set to compute contingencies, the mean causal judgment about a cue (averaged across subjects in an experimental condition) should reflect some mixture of assessments based on the multiple focal sets used. These may include the universal focal set of all events in the experiment (i.e., unconditional contingencies) and various more restricted focal sets (i.e., conditional contingencies). The response mechanism must then account for how multiple contingencies are integrated. The clearest situation is that in which the relevant unconditional and conditional contingencies for a factor are all computable and equal to zero, in which case subjects should be certain that the factor is non-causal. Beyond this limiting case, we make no claim about the exact quantitative mapping between contingency values and subjects' responses. Our assumption is that subjects' causal estimate will increase monotonically with a non-negatively weighted function of the contingency values of their focal sets. Individual subjects may compute and integrate multiple contingencies for a cue (e.g., by simple averaging). Alternatively, each subject may use only one focal set but different subjects may use different focal sets, in which case the mean ratings may mask distributions that are in fact multimodal. We will refer to the assumption that causal ratings may be based on multiple contingencies (calculated either by individual subjects or by different subjects) as the "mixture-of-focal-sets" hypothesis. As we will see, this hypothesis helps to explain circumstances in which partial rather than complete blocking is observed.

Computing contingency conditional on the presence of an alternative cause raises the problem of how an alternative cause that happens to be constantly present in the current focal set can be distinguished from an enabling condition. To distinguish between them, Cheng and Novick (1992) refined their definition of an enabling condition as follows. Let i be a factor that is constantly present in the current focal set. Factor i is an enabling condition for a cause j in that focal set if i covaries with the effect in another focal set, and j no longer covaries with the effect in a focal set in which i is constantly absent. In contrast, i is an alternative to cause j if i covaries with the effect in another focal set, and there exists a focal set in which i is constantly absent, but j continues to covary with the effect in that set.

To summarize, our proposed process model assumes that subjects will (a) identify as initial conditionalizing cues those that are noticeably associated with the effect; (b) compute contingencies for each target factor conditional on the absence of as many conditionalizing cues as possible, dynamically revising the set of conditionalizing cues in the process; and then (c) use the computed conditional contingencies and/or unconditional contingencies to produce causal assessments for the cues.

Interpreting Blocking, Partial Blocking, and Other Cue Interaction Effects

We will now consider how generalized contingency theory, in particular as implemented in the process model we have described, can account for blocking, partial blocking, overshadowing, retroactive extinction of overshadowing, and other cue-interaction effects.

Blocking and partial blocking. In the standard blocking design illustrated in Table 1, the unconditional contingency is higher for the predictive cue P than for the redundant cue R (because the outcome sometimes occurs in the absence of R, but never in the absence of P), although the contingency is positive for both. Thus even subjects who compute contingency over the universal focal set would be expected to show at least partial blocking (i.e., the higher response strength for P than R, both strengths being positive). It is possible, however, to design an experiment in which unconditional contingency is held constant for two cues, and yet have their causal status differ. Such designs have been used in classical conditioning experiments, as well as in experiments on causality judgments by humans (Chapman & Robbins, 1990, Experiment 1; Shanks, 1991, Experiment 2). The design used by Shanks is schematized in Table 3. After being presented with a series of "case histories" (patterns of patients' symptoms associated with various fictitious diseases), subjects were asked to rate how strongly they associated each symptom with each disease, using a 0-100 rating scale. In what Shanks' termed the "contingent" set, the compound-cue AB signaled the presence of Disease 1 (15 trials) but symptom C by itself does so as well (15 trials). However, cue B by itself signaled the absence of the disease (15 trials), as did the absence of A, B, and C (45 trials). In the "non-contingent"[5] set, compound-cue DE signaled the presence of Disease 2 (15 trials), as did the presence of cue E alone (15 trials). In contrast, cue F alone signaled the absence of the disease (15 trials), as did the joint absence of D, E, and F (45 trials). The critical comparison is between the association rating given to symptom A for Disease 1 and the association rating given to symptom D for Disease 2. Although the contingency computed over the entire set of events presented for both relations is .8 (see Table 4), the R-W model predicts that because D is paired with a better predictor, E, subjects should rate D as less associated than the corresponding symptom A, which is only paired with a nonpredictor, B. This difference was observed. In other words, the rating given to a cue was reduced if a competing cue was a better predictor of the relevant disease, even though unconditional contingency was equated. But although subjects gave higher mean ratings to A than to D (59 versus 34, respectively), even cue D received modestly positive ratings; whereas the R-W model predicts that at asymptote the strength of the association between D and Disease 2 should be 0 (see Melz, Cheng, Holyoak & Waldmann, 1993). Shanks' experiment is representative of several other cases in which human subjects show only partial blocking, rather than complete blocking as the R-W model would predict (e.g., Chapman & Robbins, 1990; Shanks & Dickinson, 1987; Waldmann & Holyoak, 1992).

For Disease 2 (see the right half of Table 4), cues D and E will be selected as conditionalizing cues. Because D never occurs in the absence of E, its contingency can only be calculated conditional on the presence of E. For this focal set (enclosed by the dashed rectangle in the illustration), the conditional contingency for D with respect to Disease 2 is 0. The difference between the computed contingency for cue A with respect to Disease 1 (1.0) and that for cue D with respect to Disease 2 (0) provides an explanation for cue competition -- the lower ratings given to D than A.

In addition, Cue E has a contingency of 1.0 conditional on the absence of D (rows 1-3 and 5-6 on the right in Table 4). All other cues have a contingency of 0 with respect to Disease 2 in the absence of cues D and E.

Now consider how partial blocking might arise. As we mentioned, the R-W model predicts that associative learning of a novel cue in the blocking paradigm will be completely blocked at asymtote; yet all available empirical results regarding humans show that blocking is not complete. The above contingency of 0 for cue D was conditional on the presence of cue E. However, in the presence of E, the effect always occurs. Since it was not possible to conditionalize the contingency for D on the absence of E, subjects should be uncertain of the interpretation of the contingency value of 0. Accordingly, at least some subjects may assess the unconditional contingency for D (i.e., over the universal set of events), which is 0.8. Assuming subjects' causal ratings reflect a mixture (either within individual subjects or across subjects) of these two contingencies, D will receive a relatively low but positive mean rating. That is, it will be partially blocked. Moreover, the prediction of cue competition remains, since the contingency for A (1.0) is still higher than the mixture of the contingencies for D (0.8 and 0). In sum, for situations in which there is no focal set that allows an unambiguous interpretation, if there is a mixture of focal sets, either within subjects or across subjects, contingency theory predicts partial blocking in addition to cue competition.

Overshadowing and retroactive reduction of overshadowing. A similar contingency interpretation can be provided for experiments that have demonstrated that a salient predictive cue acquires greater strength than a less salient cue that is perfectly correlated with it (i.e., the salient cue overshadows the less salient cue); and that extinguishing the salient competing predictor can increase the excitatory power of the previously overshadowed cue (Kaufman & Bolles, 1981; Matzel, Schachtman & Miller, 1985).[6]

When two cues are perfectly correlated with each other, the association of the salient cue with the outcome is likely to be noticed earlier than the association of the less salient cue. Accordingly, the former cue will be selected earlier than the latter as a conditionalizing cue. It follows that the subject will initially attempt to conditionalize the contingency of the non-salient or "pallid" cue on the state of the salient cue, but not vice versa. But due to the absence of information in this design regarding the occurrence of the outcome in the presence of one cue and the absence of the other cue, neither of the relevant contingencies for the pallid cue (i.e., those conditional on the presence and on the absence of the salient cue) can be computed. Accordingly, the subject will be uncertain about the causal status of the pallid cue during this phase. Meanwhile, given the positive unconditional contingency of the salient cue with respect to the outcome, this cue will be judged causal. Hence, it will be confirmed as a conditionalizing cue for computing the contingency of the pallid cue, whereas the pallid cue may never acquire that status with respect to the salient cue. In the subsequent phase that ensues in a retroactive paradigm, however, the salient cue is presented alone (and it is not followed by the outcome). Information then becomes available for computing the contingency of the pallid cue conditional on the presence of the salient cue. The resulting positive value for this conditional contingency predicts the increased causal strength of the pallid cue.[7]

Direction of causality. The above analyses of Shanks' results assume that subjects based their inferences on calculations based on probabilities of diseases conditional on the various symptoms. This seems likely for at least some subjects in view of the instructions and the learning procedure. The instructions did not make it clear, for example, whether a disease name referred to the cause of the associated symptoms, or was simply a label for them. However, if the causal direction is made salient to subjects, then the predictions of the R-W versus contingency approaches are very different indeed. The R-W model, although often interpreted as an account of causal induction, does not in fact draw any distinction between a context in which cues are interpreted as possible causes of an effect (the typical situation involving predictive learning), and a context in which cues are interpreted as possible effects of a common cause (diagnostic learning). Diagnostic tasks require reasoning in a backward causal direction (e.g., from symptoms, which are effects, to underlying diseases, which are interpreted as causes of the symptoms).

Waldmann and Holyoak (1992) have shown that the degree of cue competition differs radically depending on whether people interpret the cues as the causes of an effect to be predicted, or as the effects of a cause to be diagnosed. In their Experiment 3, Waldmann and Holyoak exposed subjects to a series of trials in which states of previously unfamiliar cues (buttons connected to an alarm system) were paired with states of the alarm system. Each button had two settings, "on" and "off", as did the alarm system. Subjects in a predictive condition were told that pressing one or more buttons would cause the alarm to go on. In this condition the states of the buttons were thus characterized as possible causes, and the states of the alarm system were characterized as possible effects. In contrast, subjects in a diagnostic condition were told that one or more of the buttons signaled whether or not the alarm system was on. Notice that the direction of causality was reversed according to this cover story relative to that according to the cover story in the predictive condition. As in the predictive condition, however, subjects saw only the state of the buttons, had to respond by predicting the state of the alarm, and then received feedback as to the actual state of the alarm. The presented cues and the required responses were thus equated across the conditions.

The experimental design in both conditions included two phases, corresponding to a standard blocking paradigm. Phase 1 established a certain button (P) as a perfect predictor of the state of the alarm. A second button (C) was constantly set to the value off; and a third button (U) varied in a fashion that was uncorrelated with the state of the alarm. Phase 2 maintained these same contingencies, but also added a fourth button (R) that was always on when P was on and off when P was off. Thus if subjects learned to predict the state of the alarm from the states of the buttons according to the R-W rule, then in both conditions learning should have have been blocked in Phase 2 for button R by the associative strength that would already have accrued to button P in Phase 1.

The most important findings involve the predictiveness ratings obtained after Phase 2 of the experiment (panel B of Figure 1). According to the R-W model, the associative strength acquired for the redundant button R should have approached 0 in both the predictive and diagnostic conditions (as should also have happened for the non-contingent buttons C and U). That is, associative learning for cue R should have been entirely blocked by the prior strength of cue P. However, a very different prediction follows from causal contingencies. If people tend to compute contingencies from causes to effects, rather than from effects to causes -- even when the causal direction is opposite to the order of cue-outcome presentation -- then contingency theory predicts that no blocking will be observed in the diagnostic condition. For in the diagnostic context the redundant cue, button R, is not an alternative possible cause, the contingency of which should be conditionalized on the status of the established predictor, button P; rather, the state of R is simply a second possible effect of the same cause. If alternative effects, unlike alternative causes, are given separate contingency analyses, then no cue competition should be observed. And indeed, Waldmann and Holyoak found that while button P was rated higher than button R in the predictive condition (9.7 and 4.3, respectively), in the diagnostic condition buttons P and R were given high and statistically equal ratings (9.6 and 8.7, respectively). This interaction between causal direction and the difference in the mean ratings for buttons P and R was highly significant. In addition, the results indicated that even in the predictive condition blocking for button R was only partial: the rating for cue R was significantly higher than the ratings for the non-contingent cues C and U. The latter finding is consistent with other evidence that blocking is only partial in human causal induction, as we discussed earlier.

In sum, evidence from studies of human causal induction using paradigms formally similar to blocking studies in animal conditioning has revealed phenomena that are inconsistent with the predictions of the R-W model, but interpretable in terms of a contingency theory such as PCM.

One of the primary attractions of the R-W model is the apparent simplicity and generality of its learning algorithm. However, the simplicity of the model can be questioned (Gallistel, 1990); and whether or not it is simple, its wide range of empirical shortcomings indicates that it is simplistic. It may be instructive to consider when and why the R-W model fails to acount for phenomena concerning conditioning and causal induction.

First, the R-W model does not represent cause-absent information -- in particular, the proportion of trials on which the outcome occurs in the absence of a cue. To understand why the lack of representation of the cause-absent proportion is a weakness, let us first consider the reason for the model's successes. On the basis of interaction among cues that are defined solely in terms of their presence, the model is able to account for a number of apparent effects of cause-absent information: it accounts for the role of contingency (Rescorla, 1968), the acquisition of conditioned inhibition (e.g., Chapman & Robbins, 1990), blocking (e.g., Rescorla, 1981), and other cue interaction effects (e.g., Wagner, Logan, Haberlandt & Price, 1968). In each of these cases, two or more cues that had an identical cause-present proportion, but a different conditional or unconditional cause-absent proportion, have been observed to elicit different behavior, as predicted by the model.

Two properties of the model allow it to arrive at these predictions. First, the model indirectly tallies the cause-absent proportion with respect to a target cue in terms of the cause-present proportion of one or more other cues that are present when the target cue is absent; these surrogate cues therefore acquire weights that reflect the cause-absent proportion of the target cue. In some applications of the model, the surrogate cue is one that represents the context, which is constantly present (and hence is present on occasions when the target cue is absent). Second, on trials when the two cues are both present, the strength of the target cue is adjusted towards the difference between the cause-present proportion of the target cue and the cause-present proportion of the surrogate cues (i.e., the cause-absent proportion of the target cue), potentially yielding contingency as the asymptotic output. In sum, R-W relies on the pairing of cues to transmit the indirectly tallied cause-absent proportion.

Cheng (1994) presents a derivation of when the R-W model does and does not compute conditional contingencies at asymptote. Her analysis shows that it does so for a type of design with multiple cues in which every combination of cues except the one with a single cue can be characterized as a proper superset of all sets with fewer cues (i.e., the cue combinations are nested). In such designs, the strengths of the cues in each combination sum to the relative frequency of the outcome for that combination, implying that, for any combination with multiple cues, the strength of the cue in it that does not belong to the next smaller combination is equal to the contingency of that cue conditional on the presence of the cues in the smaller combination (i.e., the rest of the cues in the larger combination).

Cheng (1994) also presents a derivation of the conditions under which conditional contingencies estimate the causal power of a cue. Her analysis of the R-W model and of conditional contingencies shows that in some nested designs, the conditional contingencies computed by the R-W model give an estimate of causal power, whereas in others, the conditional contingencies computed by this model do not give such an estimate. Those situations in which the R-W model estimates causal power include Kamin's (1969) blocking design, unconditional contingency (Rescorla, 1968), and the acquisition of conditioned inhibition. In these situtations, the R-W model is successful in predicting the observed results (see Cheng, 1994, however, for an explanation of the partial success of the R-W model in predicting the amount of blocking). Those designs in which the R-W model does not estimate causal power include the extinction of conditioned inhibition, retroactive unblocking, and the retroactive reduction of overshadowing (see Miller & Matzel, 1987). In these situations, the R-W model fails to predict the observed results.

A second problem with the R-W model is that the causal or conditioning strength of a cue with respect to an effect is represented by a single parameter -- the associative strength of the link between the cue and the outcome. The model therefore loses information about sample size, leading to its failure to account for learned irrelevance, and more generally, people's sensitivity to reliability as a function of sample size (Koslowski, 1989; Nisbett, Krantz, Jepson, & Kunda, 1983). Moreover, the R-W model does not offer any way to represent the difference between lack of certainty about a causal association and high certainty that such an association has some medium strength. In contrast, the outcome of a contingency analysis can include not only a definite evaluation of the causal status of a cue, but also uncertainty about its status. Uncertainty naturally falls out of PCM when a relevant contingency is not computable, as in the case of the redundant cue in the blocking paradigm.

For the same reason, the R-W model cannot account for causal assessments that result from comparing the distinct causal status that a cue has in different focal sets. In particular, the R-W model cannot represent the distinction between a cause and an enabling condition, nor that between an enabling condition and a causally irrelevant cue. This deficit arises because the status of an enabling condition results from the cue being causal in one focal set and having a non-computable contingency in another focal set.

This last point brings up the related problem of the need to specify (potentially multiple) focal sets. Our explanations of enabling conditions and of partial blocking provide examples of the use of such an assumption. One might ask: would the R-W model be able to explain these phenomena if it is amended with the assumption of computation over multiple focal sets? With respect to blocking (see Table 1), R-W predicts that a redundant cue, R, should have zero associative strength, regardless of which focal set is adopted. For none of the focal sets that arise in a contingency analysis is there ever a discrepancy between the expected outcomes based on R-W and the target outcomes for any trial on which R is present. (See the Appendix for derivations of the asymptotic weights of the cues assuming various focal sets.) Considering either the focal set in which the predictive cue is always present (i.e., the top row in Table 1) or the universal focal set (i.e., the entire table), the outcome is completely predicted by P. Considering the focal set in which the predictive cue is always absent (i.e., the bottom row), R is never present. Therefore, the strength of R is never revised from zero. In sum, even when amended with the concept of a focal set, the R-W model, unlike our process model, cannot predict the partial blocking of R. Nor can it predict a possible multimodal distribution of judgments regarding R. With respect to the status of an enabling condition, because the R-W model does yield a definite value of strength for a constant cue, it cannot yield the uncertainty that leads to the reliance on the status of the cue in another focal set. Even if the model is applied to a focal set in which the cue is constant and another in which it varies, the result will be two strengths for that cue. It is not clear how this result (i.e., an enabling condition represented by two strengths) can be distinguished from that involving a cause that has different strengths in different focal sets.

Finally, the R-W model does not provide any way to distinguish the case in which cues are possible causes (predictive learning) from that in which cues are possible effects (diagnostic learning). That is, cues and outcomes are defined with respect to their roles as stimuli presented versus responses made, rather than with respect to their conceptual roles as causes or effects. However, a cause can be either a stimulus or a response (as can an effect). As a result, the R-W model is unable to explain interactions between perceived causal direction and cue competition (Waldmann & Holyoak, 1992).

It is not obvious to us that any of the above shortcomings of the R-W model can be readily amended. Contingency theory provides a basis for formulating alternative models of how natural adaptive systems operate as intuitive statisticians.

2. What is referred to here as the "universal set" is actually the pragmatically-restricted set of events that occur in the conditioning experiment (i.e., a small subset of the "truly" universal set of all events known to the subject). This contextual delimitation of the largest relevant focal set implies that even the cases in the "cause and effect both absent" cell are restricted to a small finite number.

3. When there are multiple known causes, assessing the status of a potential causal factor normatively requires computing its contingencies exhaustively conditionalizing on every combination of the presence and absence of the other cues. We do not mean to imply that a test of conditional independence is the only process for differentiating between genuine and spurious causes (Lien & Cheng, 1992).

4. The prediction of uncertainty does not generalize to situations in which the representation of the target phenomenon does not have a maximum value, as does the probability of a phenomenon.

5. Because the critical cues in Shanks "noncontingent conditions" were contingently related to the respective diseases by the conventional definition, the labels for his stimulus sets in Experiments 1 and 2 -- "contingent condition" and "noncontingent condition" -- do not conform to conventional usage.

6. It should be noted, however, that analogous conditioning experiments with animals that attempted to find indirect effects of increasing (rather than decreasing) the excitation of a previously-paired cue have failed to obtain such effects (see Miller & Matzel, 1987). However, "retroactive blocking" -- reduction in the causal value of one cue as a result of increasing the apparent predictiveness of another cue with which it had been previously paired -- has been observed in studies of causal induction by humans (Chapman, 1991; Shanks, 1985). These effects, however, have been relatively small in magnitude.

7. Consideration of the unconditional contingency for the pallid cue yields the same prediction. As the salient cue becomes extinguished, it no longer maintains its conditionalizing status. The positive unconditional contingency of the pallid cue then becomes interpretable as evidence that the latter is in fact causal.

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Table 1: Probability of the outcome for cues P and R in the blocking paradigm.

Table 2: Probability of the outcome in the learning phase of the conditioned inhibition paradigm.

Table 3: Conditions, Trial Types, Number of Trials, and Percentage of Correct Diagnoses for Experiments 2 of Shanks (1991) (Adapted from Shanks,1991).

Condition         Trial     TypeTrials    % Correct

"contingent"      C -> D1   15            100

                  AB -> D1  15            100

                  B -> 0    15            94

"non-contingent"  DE -> D2  15            100

                  E -> D2   15            100

                  F -> 0    15            94

Focal Sets for Cue A with     Focal Sets for Cue D with
  Respect to Disease 1          Respect to Disease 2

     Proportion of                Proportion of
      Cases with                   Cases with
     CueDisease 1                 Cue Disease 2

       ABCDEF  (out of 15)    ABCDEF  (out of 15)
      
       oo+ooo    15/15        oo+ooo     0/15
       ++oooo    15/15        ++oooo     0/15
Trial  o+oooo     0/15        o+oooo     0/15
type   ooo++o     0/15        ooo++o    15/15
       oooo+o     0/15        oooo+o    15/15
       ooooo+     0/15        ooooo+     0/15 

 Contingency for Cue A:        Contingency for Cue D:

    Universal set                 Universal set
  15/15 - 15/75 = .8            15/15 - 15/75 = .8

 Focal set in which              Focal set in which
  cue C is absent                 cue E is present
  15/15 - 0/60 = 1.0            15/15 - 15/15 = 0

Deriving Asymptotic Weights

Because our derivation is based on asymptotic weights, we first describe a method for deriving such weights. To obtain the asymptotic weights of a network according to the R-W model, we note the equivalence between the R-W learning rule and the least-mean-squares (LMS) rule of Widrow and Hoff (1960) (cf. Sutton & Barto, 1981). The Rescorla-Wagner/Widrow-Hoff rule implements an iterative algorithm for computing the solution to a set of linear equations, defined by the set of stimulus-response patterns presented to the network. A pattern is a configuration of stimuli and response deterministically describing a set of trials. If the input stimulus patterns are linearly independent, then the updating rule will discover a unique solution. Even if the stimulus patterns are not linearly independent, the network will still converge provided that the learning rate is sufficiently small, and that the various stimulus patterns occur with sufficient frequency in the input sequence. The network will converge so as to minimize the sum of the squared errors over the stimulus patterns. That is, the equation

(4)

will be minimized, where p is the index for a particular stimulus-response pattern, pp is the frequency of pattern p, ip is the learning rate associated with pattern p (bj and gj , respectively, for the presence and the absence of outcome j), lp is the desired output for the outcome of the pattern (usually either 0 or 1), and is the actual output for the pattern, which is equal to the sum of the weights Vi associated with every present cue i for the pattern. If the reinforcement learning rate bj is equal to the nonreinforcement learning rate gj, the ip term may be omitted from the equation. We assume that the learning rates bi and gi are equal in the rest of this paper.

Thus, the asymptotic weights of a network according to the R-W model can be calculated analytically by minimizing the sum of the squared errors given by Equation (4). This minimum value may be obtained by setting the partial derivatives with respect to each weight to 0, and solving the resulting set of equations.

A Predictive Cue and a Redundant Cue

First, consider the network corresponding to the design summarized by Table 1. There are three patterns of trials: When P and R are both present, the outcome always occurs; when P is present and R is absent, the outcome also always occurs; but when P and R are both absent, the outcome never occurs. Because the pp and ip terms do not affect our result, for simplicity of exposition we assume that the three patterns occur with equal frequency, and that the learning rates for the various patterns are equal, so that we omit pp and ip from Eq. (4). Applying the equation to all events in the table (the universal focal set),

We see that E will have its lowest value when VP + VR = 1 and VP = 1. Therefore, the aymptotic solution for this network is VP = 1 and VR = 0. That is, the redundant cue R is completely blocked. Note that the pattern involving the joint absence of P and R does not lead to any error terms, because no cue is present. Therefore, applying the R-W model to the focal set consisting of trials in which P is always present yields the same asymptotic solution for VP and VR.

Adding a Constant Context Cue

Applications of the R-W model often assume that there is a constantly present context cue, K. The error for the universal set is then

We see that E will have its lowest value when VK = 0, VP = 1, and VR = 0. That is, R is completely blocked.

If we adopt the focal set consisting of trials on which P is constantly present, we drop the third error term above, obtaining

By inspection, E will be at a minimum when

(5)

(6)

There is no unique solution for VP and VK in this case. But, subtracting Eq. (6) from Eq. (5) yields the solution VR = 0. Thus, R is completely blocked in this as well as all of the above networks.