Generalization,similarity, and Bayesian inference

Shepard has argued that a universal law should govern generalization across different domains of perception and cognition, as well as across organisms from different species or even different planets. Starting with some basic assumptions about natural kinds, he derived an exponential decay function as the form of the universal generalization gradient, which accords strikingly well with a wide range of empirical data. However, his original formulation applied only to the ideal case of generalization from a single encountered stimulus to a single novel stimulus, and for stimuli that can be represented as points in a continuous metric psychological space. Here we recast Shepard's theory in a more general Bayesian framework and show how this naturally extends his approach to the more realistic situation of generalizing from multiple consequential stimuli with arbitrary representational structure. Our framework also subsumes a version of Tversky's set-theoretic model of similarity, which is conventionally thought of as the primary alternative to Shepard's continuous metric space model of similarity and generalization. This unification allows us not only to draw deep parallels between the set-theoretic and spatial approaches, but also to significantly advance the explanatory power of set-theoretic models.     

Bayesian generalization with circular consequential regions

Generalization–deciding whether to extend a property from one stimulus to another stimulus–is a fundamental problem faced by cognitive agents in many different settings. Shepard (1987) provided a mathematical analysis of generalization in terms of Bayesian inference over the regions of psychological space that might correspond to a given property. He proved that in the unidimensional case, where regions are intervals of the real line, generalization will be a negatively accelerated function of the distance between stimuli, such as an exponential function. These results have been extended to rectangular consequential regions in multiple dimensions, but not for circular consequential regions, which play an important role in explaining generalization for stimuli that are not represented in terms of separable dimensions. We analyze Bayesian generalization with circular consequential regions, providing bounds on the generalization function and proving that this function is negatively accelerated.     

Shepard's universal law supported by honeybees in spatial generalization

An animal that is rewarded for a response in one situation (the S+) is likely to respond to similar but recognizably different stimuli, the ubiquitous phenomenon of stimulus generalization. On the basis of functional analyses of the probabilistic structure of the world, Shepard formulated a universal law of generalization, claiming that generalization gradients, as a function of the appropriately scaled distance of a stimulus from S+, should be exponential in shape. This law was tested in spatial generalization in honeybees. Based on theoretically derived scales, generalization along both the dimensions of the distance from a landmark and the direction to a landmark followed Shepard's law. Support in an invertebrate animal increases the scope of the law, and suggests that the ecological structure of the world may have driven the evolution of cognitive structures in diverse animals.     

Mental rotation and perceptual uprightness

Performance in Cooper and Shepard’s (1973) mental rotation task was examined in the context of a model that defined the extent to which alphabet letters could be tilted from their normal orientation and still be perceptually upright. For letters with a broad range of orientations for which they remain perceptually upright, a nonlinear effect of orientation on reaction time was obtained (as in Cooper and Shepard). However, for letters with a narrow range of orientations for which they remain perceptually upright, reaction time was linearly related to orientation. The results supported the hypothesis that subjects in the Cooper and Shepard task would mentally rotate alphabet letters only when they were presented in orientations for which they were not perceptually upright.     

Mapclus: A mathematical programming approach to fitting the adclus model

We present a new algorithm, MAPCLUS (MAthematicalProgrammingCLUStering), for fitting the Shepard-Arabie ADCLUS (forADditiveCLUStering) model. MAPCLUS utilizes an alternating least squares method combined with a mathematical programming optimization procedure based on a penalty function approach, to impose discrete (0,1) constraints on parameters defining cluster membership. This procedure is supplemented by several other numerical techniques (notably a heuristically based combinatorial optimization procedure) to provide an efficient general-purpose computer implemented algorithm for obtaining ADCLUS representations. MAPCLUS is illustrated with an application to one of the examples given by Shepard and Arabie using the older ADCLUS procedure. The MAPCLUS solution uses half as many clusters to achieve nearly the same level of goodness-of-fit. Finally, we consider an extension of the present approach to fitting a three-way generalization of the ADCLUS model, called INDCLUS (INdividualDifferencesCLUStering).We are indebted to Scott A. Boorman, W. K. Estes, J. A. Hartigan, Lawrence J. Hubert, Carol L. Krumhansl, Joseph B. Kruskal, Sandra Pruzansky, Roger N. Shepard, Edward J. Shoben, Sigfrid D. Soli, and Amos Tversky for helpful discussions of this work, as well as the anonymous referees for their suggestions and corrections on an earlier version of this paper. We are also grateful to Pamela Baker and Dan C. Knutson for technical assistance. The research reported here was supported in part by LEAA Grant 78-NI-AX-0142 and NSF Grants SOC76-24512 and SOC76-24394.     

Spatial factors in visual attention: a reply to Crassini

Distances between stimuli and, derivatively, compactness of stimulus subsets are pervasive determiners of discrimination and classification performance. Contrary to Crassini (1986), these factors are sufficient to account for the major patterns in the chronometric data of Podgorny and Shepard (1978, 1983). Other factors, such as the dichotomous one distinguishing what Crassini terms unitary and nonunitary subsets, may exert some additional influence. But a convincing demonstration would require formulation of a quantitative model capable of being pitted against the distance-based model of Podgorny and Shepard within the context of their entire body of data.     

A generalization of the Yule-Simon model,with special reference to word association tests and neural cell assembly formation

A generalization of the Yule-Simon model is suggested and such related questions as inversion problems and nonequilibrium behavior are solved. The generalization follows the work of Haight and Jones (Journal of Mathematical Psychology, 1974, 11, 237–244) and thus special reference is made to word association tests. Some new possible applications of the presented model are offered, namely, in the field of neural cell assemblies.     

Similarity, kernels, and the triangle inequality

Similarity is used as an explanatory construct throughout psychology and multidimensional scaling (MDS) is the most popular way to assess similarity. In MDS, similarity is intimately connected to the idea of a geometric representation of stimuli in a perceptual space. Whilst connecting similarity and closeness of stimuli in a geometric representation may be intuitively plausible, Tversky and Gati [Tversky, A., & Gati, I. (1982). Similarity, separability, and the triangle inequality. Psychological Review, 89(2), 123-154] have reported data which are inconsistent with the usual geometric representations that are based on segmental additivity. We show that similarity measures based on Shepard’s universal law of generalization [Shepard, R. N. (1987). Toward a universal law of generalization for psychologica science. Science, 237(4820), 1317-1323] lead to an inner product representation in a reproducing kernel Hilbert space. In such a space stimuli are represented by their similarity to all other stimuli. This representation, based on Shepard’s law, has a natural metric that does not have additive segments whilst still retaining the intuitive notion of connecting similarity and distance between stimuli. Furthermore, this representation has the psychologically appealing property that the distance between stimuli is bounded.     

Generalization of conditioned fear along a dimension of increasing fear intensity

The present study investigated the extent to which fear generalization in humans is determined by the amount of fear intensity in nonconditioned stimuli relative to a perceptually similar conditioned stimulus. Stimuli consisted of graded emotionally expressive faces of the same identity morphed between neutral and fearful endpoints. Two experimental groups underwent discriminative fear conditioning between a face stimulus of 55% fear intensity (conditioned stimulus, CS+), reinforced with an electric shock, and a second stimulus that was unreinforced (CS−). In Experiment 1 the CS− was a relatively neutral face stimulus, while in Experiment 2 the CS− was the most fear-intense stimulus. Before and following fear conditioning, skin conductance responses (SCR) were recorded to different morph values along the neutral-to-fear dimension. Both experimental groups showed gradients of generalization following fear conditioning that increased with the fear intensity of the stimulus. In Experiment 1 a peak shift in SCRs extended to the most fear-intense stimulus. In contrast, generalization to the most fear-intense stimulus was reduced in Experiment 2, suggesting that discriminative fear learning procedures can attenuate fear generalization. Together, the findings indicate that fear generalization is broadly tuned and sensitive to the amount of fear intensity in nonconditioned stimuli, but that fear generalization can come under stimulus control. These results reveal a novel form of fear generalization in humans that is not merely based on physical similarity to a conditioned exemplar, and may have implications for understanding generalization processes in anxiety disorders characterized by heightened sensitivity to nonthreatening stimuli.Fear generalization occurs when a fear response acquired to a particular stimulus transfers to another stimulus. Generalization is often an adaptive function that allows an organism to rapidly respond to novel stimuli that are related in some way to a previously learned stimulus. Fear generalization, however, can be maladaptive when nonthreatening stimuli are inappropriately treated as harmful, based on similarity to a known threat. For example, an individual may acquire fear of all dogs after an aversive experience with a single vicious dog. In this case, recognizing that a novel animal is related to a feared (or fear-conditioned) animal is made possible in part by shared physical features to the fear exemplar, such as four legs and a tail. On the other hand, fear generalization may be selective for those features that are associated with natural categories of threat; a harmless dog may not pose a threat, but possesses naturally threatening features common to other threatening animals, such as sharp teeth and claws. Moreover, the degree to which an individual fearful of dogs responds with fear may be related to either the physical similarity to the originally feared animal (e.g., from a threatening black dog to another black dog), or the intensity of those threatening features relative to the originally feared animal (e.g., sharp teeth from one animal to sharp teeth of another animal). Therefore, fear generalization based on perceptual information may occur via two routes—similarity to a learned fear exemplar along nonthreatening physical dimensions or along dimensions of fear relevance. Given that fear generalization often emerges as a consequence of conditioning or observational learning, it is important to determine which characteristics of novel stimuli facilitate fear generalization and the extent to which generalization processes can be controlled.Early explanations of stimulus generalization emphasized that an organism''s ability to generalize to nonconditioned stimuli is related to both the similarity and discriminability to a previously conditioned stimulus (CS) (Hull 1943; Lashley and Wade 1946). While Lashley and Wade (1946) argued that generalization was simply a failure of discriminating between a nonconditioned stimulus (CS−) and the reinforced CS (CS+), contemporary views contend that generalization enables learning to extend to stimuli that are readily perceptually distinguished from the CS (Pearce 1987; Shepard 1987; McLaren and Mackintosh 2002). This latter view has been supported by empirical studies of stimulus generalization in laboratory animals (Guttman and Kalish 1956; Honig and Urcuioli 1981). In these studies, animals were reinforced for responding to a CS of a specific physical quality such as color, and then tested with several different values along the same stimulus dimension as the CS (e.g., at various wavelengths along the color spectrum). Orderly gradients of responses are often reported that peak at or near the reinforced value and decrease as a function of physical similarity to the CS along the stimulus dimension (Honig and Urcuioli 1981). Further generalization was shown to extend from the CS+ to discriminable nonconditioned stimuli, suggesting that generalization is not bound to the organism''s ability to discriminate stimuli (Guttman and Kalish 1956, 1958; Shepard 1987).Interestingly, when animals learn to distinguish between a CS+ and a CS−, the peak of behavioral responses often shift to a new value along the dimension that is further away from the CS− (Hanson 1959). For instance, when being trained to discriminate a green CS+ and an orange CS−, pigeons will key peck more to a greenish-blue color than the actual CS+ hue. Intradimensional generalization of this sort is reduced when animals are trained to discriminate between two or more stimulus values that are relatively close during conditioning (e.g., discriminating a green-yellow CS+ from a green-blue CS−), suggesting that the extent of generalization can come under stimulus control through reinforcement learning (Jenkins and Harrison 1962). Spence (1937) described the transposition of response magnitude as an effect of interacting gradients of excitation and inhibition formed around the CS+ and CS−, respectively, which summate to shift responses to values further from the inhibitory CS− gradient. In all, early theoretical and empirical treatments of stimulus generalization in nonhuman animals revealed that behavior transfers to stimuli that are physically similar, but can be discriminated from a CS, and that differential reinforcement training can both sharpen the stimulus gradient and shift the peak of responses to a nonreinforced value.Although this rich literature has revealed principles of generalization in nonhuman animals, few studies of fear generalization have been conducted in humans (for review, see Honig and Urcuioli 1981; Ghirlanda and Enquist 2003). Moreover, the existing human studies have yet to consider the second route through which fear responses may generalize—via gradients of fear relevance. While a wide range of neutral stimuli, such as tones or geometric figures, can acquire fear relevance through conditioning processes, other stimuli, such as threatening faces or spiders, are biologically prepared to be fear relevant (Lanzetta and Orr 1980; Dimberg and Öhman 1996; Whalen et al. 1998; Öhman and Mineka 2001). Compared with fear-irrelevant CSs, biologically prepared stimuli capture attention (Öhman et al. 2001), are conditioned without awareness (Öhman et al. 1995; Öhman and Soares 1998), increase brain activity in visual and emotional processing regions (Sabatinelli et al. 2005), and become more resistant to extinction when paired with an aversive unconditioned stimulus (US) (Öhman et al. 1975). Although the qualitative nature of the CS influences acquisition and expression of conditioned fear, it is unknown how generalization proceeds along a gradient of natural threat. For instance, human studies to date have all tested variations of a CS along physically neutral stimulus dimensions, such as tone frequency (Hovland 1937), geometric shape (Vervliet et al. 2006), and physical size (Lissek et al. 2008). These investigations implicitly assume that the generalization gradient is independent of the conditioned value (equipotentiality principle). In other words, since the stimuli are all equally neutral prior to fear learning, fear generalization operates solely as a function of similarity along the reinforced physical dimension. However, since fear learning is predisposed toward fear-relevant stimuli, generalization may be selective to those shared features between a CS+ and CS− that are associated with natural categories of threat. Examining generalization using fear-relevant stimuli is thus important to gain better ecological validity and to develop a model system for studying maladaptive fear generalization in individuals who may express exaggerated fear responses to nonthreatening stimuli following a highly charged aversive experience (i.e., post-traumatic stress disorder or specific phobias).To address this issue, the present study examined generalization to fearful faces along an intradimensional gradient of fear intensity. A fearful face is considered a biologically prepared stimulus that recruits sensory systems automatically for rapid motor responses (Öhman and Mineka 2001), and detecting fearful faces may be evolutionarily selected as an adaptive response to social signals of impending danger (Lanzetta and Orr 1980; Dimberg and Öhman 1996). During conditioning, an ambiguous face containing 55% fear intensity (CS+) was paired with an electric shock US, while a relatively neutral face (11% fear intensity) was explicitly unreinforced (CS−) (Experiment 1). Skin conductance responses (SCR) were recorded as a dependent measure of fear conditioning. Before and following fear conditioning, SCRs were recorded in response to face morphs of the same actor expressing several values of increasing fear intensity (from 11% to 100%; see Fig. 1). A total of five values along the continuum were used: 11% fear/88% neutral, 33% fear/66% neutral, 55% fear/44% neutral, 77% fear/22% neutral, and 100% fear. For clarity, these stimuli are herein after labeled as S1, S2, S3, S4, and S5, respectively.Open in a separate windowFigure 1.Experimental design. (A) Pre-conditioning included six presentations of all five stimulus values without the US. (B) Fear conditioning involved discriminative fear learning between the S3, paired with the US (CS+), and either the unreinforced S1 (Experiment 1) or the unreinforced S5 (Experiment 2) (CS−). (C) The generalization test included nine presentations of all five stimuli (45 total), with three out of nine S3 trials reinforced with the US. Stimuli are not drawn to scale.Testing generalization along an intradimensional gradient of emotional expression intensity allows for an examination of the relative contributions of fear intensity and physical similarity on the magnitude of generalized fear responses. If fear generalization is determined purely by the perceptual overlap between the CS+ and other morph values, without regard to fear intensity, then we would expect a bell-shaped generalization function with the maximum SCR centered on the reinforced (intermediate) CS+ value (S3), less responding to the directly adjacent, but most perceptually similar values (S2 and S4), and the least amount of responding to the most distal and least perceptually similar morph values (S1 and S5). This finding would be in line with stimulus generalization reported along fear-irrelevant dimensions (Lissek et al. 2008) and in stimulus generalization studies using appetitive instrumental learning procedures (Guttman and Kalish 1956). If, however, fear generalization is biased toward nonconditioned stimuli of high fear intensity, then an asymmetric generalization function should result with maximal responding to the most fear-intense nonconditioned stimuli. This finding would suggest that fear generalization is selective to the degree of fear intensity in stimuli, similar to studies of physical intensity generalization gradients in nonhuman animals (Ghirlanda and Enquist 2003). We predicted that the latter effect would be observed, such that the magnitude of SCRs will disproportionately generalize to stimuli possessing a greater degree of fear intensity than the CS+ (Experiment 1). A secondary goal was to determine whether fear generalization to nonconditioned stimuli can be reduced through discriminative fear learning processes. Therefore, a second group of participants was run for whom the CS− was the 100% fearful face (Experiment 2). In this case, we predicted that discriminative fear conditioning between the CS+ (55% intensity) and the most fear-intense nonconditioned stimulus would sharpen the generalization gradient around the reinforced CS+ value, and that responses to the most fear-intense stimulus would decrease relative to Experiment 1. Moreover, this discriminative fear-learning process may provide evidence that fear generalization is influenced by associative learning processes and is not exclusively driven by selective sensitization to stimuli of high fear relevance (Lovibond et al. 1993). Finally, we were interested to discover whether generalization processes would yield subsequent false memory for the intensity of the CS+ in a post-experimental retrospective report. In sum, the present study has implications for understanding how fear generalization is related to the degree of fear intensity of a nonconditioned stimulus, the extent to which discrimination training efforts can thwart the generalization process, and how fear generalization affects stimulus recognition.     

Measurement foundations for multiattribute psychophysical theories based on first order polynomials

The class of first order polynomial measurement representations is defined, and a method for proving the existence of such representations is described. The method is used to prove the existence of first order polynomial generalizations of expected utility theory, difference measurement, and additive conjoint measurement. It is then argued that first order polynomial representations provide a deep and far reaching characterization of psychological invariance for subjective magnitudes of multiattributed stimuli. To substantiate this point, two applications of first order polynomial representation theory to the foundations of psychophysics are described. First, Relation theory, a theory of subjective magnitude proposed by Shepard (Journal of Mathematical Psychology, 1981, 24, 21–57) and Krantz (Journal of Mathematical Psychology, 1972, 9, 168–199), is generalized to a theory of magnitude for multiattributed stimuli. The generalization is based on a postulate of context invariance for the constituent uniattribute magnitudes of a multiattribute magnitude. Second, the power law for subjective magnitude is generalized to a multiattribute version of the power law. Finally, it is argued that a common logical pattern underlies multiattribute generalizations of psychological theories to first order polynomial representations. This abstract pattern suggests a strategy for theory construction in multiattribute psychophysics.     

Relative attention in judgments of heterogeneous similarity

Shepard’s (1964) study on similarity of stimuli with clearly discernible dimensions was repeated with some modifications, the most important being that the Os had to make numerical similarity estimates of the stimulus pairs. The overall outcome did not deviate much from Shepard’s findings. By using quantitative estimates and choosing stimulus series so that the two dimensions were negatively correlated, data for each 0 could be analyzed separately with a partial correlation technique. It was found that the more an 0 attended to one dimension the less he attended to the other. The shifts in attention seemed to be random rather than regular. Consequently, the meaningfulness of a contention like Shepard’s as to the nonexistence of a metric is questioned.     

Circularity in relative pitch judgments for inharmonic complex tones: The Shepard demonstration revisited,again

The Shepard illusion, in which the presentation of a cyclically repetitive sequence of complex tones composed of partials separated by octave intervals (Shepard, 1964) gives the illusion of an endlessly increasing sequence of pitch steps, is often cited as evidence for octave equivalence. In this paper, evidence is presented which demonstrates that this illusion can be produced using (inharmonic) complex tones whose partials are separated by equal ratios other than octaves. Therefore, the illusion is not evidence for octave equivalence.     

Similarity, feature discovery, and the size principle

In this paper we consider the “size principle” for featural similarity, which states that rare features should be weighted more heavily than common features in people’s evaluations of the similarity between two entities. Specifically, it predicts that if a feature is possessed by n objects, the expected weight scales according to a 1/n law. One justification of the size principle emerges from a Bayesian analysis of simple induction problems ( [Tenenbaum and Griffiths, 2001a] and [Tenenbaum and Griffiths, 2001b]), and is closely related to work by Shepard (1987) proposing universal laws for inductive generalization. In this article, we (1) show that the size principle can be more generally derived as an expression of a form of representational optimality, and (2) present analyses suggesting that across 11 different data sets in the domains of animals and artifacts, human judgments are in agreement with this law. A number of implications are discussed.     

Internalization: a metaphor we can live without

Shepard has supposed that the mind is stocked with innate knowledge of the world and that this knowledge figures prominently in the way we see the world. According to him, this internal knowledge is the legacy of a process of internalization; a process of natural selection over the evolutionary history of the species. Shepard has developed his proposal most fully in his analysis of the relation between kinematic geometry and the shape of the motion path in apparent motion displays. We argue that Shepard has made a case for applying the principles of kinematic geometry to the perception of motion, but that he has not made the case for injecting these principles into the mind of the percipient. We offer a more modest interpretation of his important findings: that kinematic geometry may be a model of apparent motion. Inasmuch as our recommended interpretation does not lodge geometry in the mind of the percipient, the motivation of positing internalization, a process that moves kinematic geometry into the mind, is obviated. In our conclusion, we suggest that cognitive psychologists, in their embrace of internal mental universals and internalization may have been seduced by the siren call of metaphor.     

A parallel-process model of mental rotation

It is argued that some of the phenomena identified with analog processes by Shepard can be understood as resulting from a parallel-process algorithm running on a processor having many individual processing elements and a restricted communication structure. In particular, an algorithm has been developed and implemented which models human behavior on Shepard's object rotation and comparison task. The algorithm exhibits computation times which increase linearly with the angle of rotation. Shepard found a similar linear function in his experiments with human subjects. In addition, the intermediate states of the computation are such that if the rotation process were to be interrupted at any point, the object representation would correspond to that of the actual object at a position along the rotation trajectory. The computational model presented here is governed by three constraining assumptions: (a) that it be parallel: (b) that the communication between processors be restricted to immediate neighbors: (c) that the object representation be distributed across a large fraction of the available processors. A method of diagnosing the correct axis of rotation is also presented.     

Previous Conceptions of the Typical Group Member and the Contact Hypothesis

Allport's (1954) contact hypothesis predicted that pleasant contact with a member of a negatively stigmatized group would change attitudes both toward the specific person interacted with (specific attitude change) and also toward the group as a whole (generalization). Many previous studies of the contact hypothesis have demonstrated specific attitude change. In previous studies that demonstrated generalization, attitude change toward the group as a whole might have occurred because participants changed their opinions about what constituted a "typical" group member. Although this postulated mechanism of attitude change is difficult to test directly, our study sought indirect support by showing that preexisting conceptions of the typical group member differ in a way that affects the extent of generalization. Students who initially conceived of the typical person with AIDS (PWA) as an abstraction displayed greater generalization following pleasant contact with a PWA than did students who initially conceived of the typical PWA as a specific person. The generalization part of Allport's contact hypothesis may thus be related to recent research on social categorization.     

Stimulus generalization and the peak shift with movement stimuli

Stimulus generalization is suggested as an alternative method for examination of the "novelty" problem in motor learning. These experiments demonstrated that stimulus generalization occurs using simple movements as stimuli. The phenomenon of the "peak shift" in post-discrimination generalization gradients was also examined. The first experiment demonstrated that a peak shift occurred using linear movements as stimuli and that the magnitude of the peak shift increased as the difference between the training stimuli decreased. The second experiment showed similar results when the stimuli consisted of a range of movements rather than single movement length. The final experiment provided evidence that perception of movement length is influenced by the magnitude of an immediately preceding movement. The relevance of these studies to current motor-learning theory is discussed.     

A comparison of generalization functions and frame of reference effects in different training paradigms.

Six experiments were carried out to compare go/no-go and choice paradigms for studying the effects of intradimensional discrimination training on subsequent measures of stimulus generalization in human subjects. Specifically, the purpose was to compare the two paradigms as means of investigating generalization gradient forms and frame of reference effects. In Experiment 1, the stimulus dimension was visual intensity (brightness); in Experiment 2, it was line orientation (line-angle stimuli). After learning to respond (or to respond "right") to stimulus value (SV) 4 and not to respond (or to respond "left") to SV2 (in Experiment 1) or SV1 (in Experiment 2), the subjects were tested for generalization (recognition) with an asymmetrical set of values ranging from SV1 to SV11. Go/no-go training produced peaked gradients, whereas choice training produced sigmoid gradients. The asymmetrical testing resulted in a gradual shift of the peak of responding (go/no-go group) or in the point of subjective indifference (PSI; choice group) toward the central value of the test series; thus, both paradigms revealed a frame of reference effect. The results were comparable for the quantitative (intensity) and the qualitative (line-angle) stimulus dimensions. Experiment 3 compared the go/no-go procedure with a yes/no procedure in which subjects responded "right" to SV4 and "left" to all other intensities and found no differences between these procedures. Thus the difference in gradient forms in go/no as opposed to (traditional) choice paradigms depends on whether one or two target stimuli are used in training. In Experiment 4, in which visual intensity was used, the shift in the PSI following choice training varied positively with the range of asymmetrical test stimuli employed. In Experiment 5, also with visual intensity, the magnitude of the peak shift following go/no-go training varied as a function of overrepresenting a high or a low stimulus value during generalization testing. Experiment 6, with line angles, showed that the PSI following choice training varies in a similar way. The frame of reference effects obtained in these experiments are consistent with an adaptation-level model.     

What is learned in sequential learning? An associative model of reward magnitude serial-pattern learning

A computational model of sequence learning is described that is based on pairwise associations and generalization. Simulations by the model predicted that rats should learn a long monotonic pattern of food quantities better than a nonmonotonic pattern, as predicted by rule-learning theory, and that they should learn a short nonmonotonic pattern with highly discriminable elements better than 1 with less discriminable elements, as predicted by interitem association theory. In 2 other studies, the model also simulated behavioral "rule generalization," "extrapolation," and associative transfer data motivated by both rule-learning and associative perspectives. Although these simulations do not rule out the possibility that rats can use rule induction to learn serial patterns, they show that a simple associative model can account for the classical behavioral studies implicating rule learning in reward magnitude serial-pattern learning.     

Multivariate stochastic processes compatible with “aspect” models of similarity and choice

Various recent works have developed feature or aspect models of similarity and preference. These models are more concerned with the fine detail of the judgment process than were prior models, but nevertheless they have not in general developed an underlying stochastic process compatible with the assumed structure. In this paper, we show that a particular class of multivariate stochastic processes, namely those associated with the Marshall-Olkin multivariate exponential distribution, generates several of these models. In particular, such stochastic processes (appropriately interpreted) yield Tversky's elimination by aspects model, Edgell and Geisler's (normal) additive random aspects model, and Shepard and Arabie's additive cluster model.This work was supported by Natural Science and Engineering Research Council of Canada Grant A8124 to A.A.J. Marley.