In unidimensional absolute identification tasks, participants identify stimuli that vary along a single dimension. Performance is surprisingly poor compared with discrimination of the same stimuli. Existing models assume that identification is achieved using long-term representations of absolute magnitudes. The authors propose an alternative relative judgment model (RJM) in which the elemental perceptual units are representations of the differences between current and previous stimuli. These differences are used, together with the previous feedback, to respond. Without using long-term representations of absolute magnitudes, the RJM accounts for (a) information transmission limits, (b) bowed serial position effects, and (c) sequential effects, where responses are biased toward immediately preceding stimuli but away from more distant stimuli (assimilation and contrast). Keywords: absolute identification, relative judgment Miller (1956) drew attention to a curious phenomenon. People have great difficulty identifying stimuli from a set that varies along a single psychological continuum, even though their ability to discriminate pairs of stimuli from the set suggests that they should be very good at the identification task. This phenomenon can be seen across a wide range of stimulus attributes-the frequency and loudness of tones, the strength of tastes and smells, the magnitude of lengths and areas, the hue and brightness of colors, and the intensity and numerousness of cutaneous stimulation-which suggests some common and fundamental source of the limitation. In an absolute identification task, participants are required to identify, with a unique label, stimuli drawn from a set of items that vary along only a single continuum. Typically, stimuli are evenly psychologically spaced. A stimulus's label is normally its ordinal position within the set. Three key phenomena, which we review in more detail below, are observed. First, there is a severe limit in the information transmitted from stimulus to response (i.e., the size of the set for which members can be identified perfectly) even when adjacent stimuli are perfectly discriminable. Second, a bow effect is observed when identification accuracy is plotted against stimulus, with an advantage for the smallest and largest stimuli. Third, there are strong sequential effects, whereby the stimuli on previous trials exert a strong bias on the response to the current stimulus. Many theoretical accounts have been offered for one or more of these phenomena. Nearly all of these models have in common the assumption that in an absolute identification task a representation of the absolute magnitude of the current stimulus is compared with some long-term representations of the absolute magnitudes of either other stimuli from the set, particular anchor values, or particular criterial values. However, in a review for the centenary issue of Psychological Review, Shiffrin and Nosofsky (1994) concluded that, since Miller's (1956) classic article, "a fully unified account of the numerous range, edge, and sequential effects has not been achieved" (p. 359). Here, in contrast to existing models (excepting Laming, 1984), we offer a relative judgment model (RJM) of absolute identification. The RJM does not use long-term representations of absolute magnitudes. Instead, it uses the difference between the current stimulus and the previous stimulus, in conjunction with the feedback from the previous trial, to generate a response. Thus, the magnitude of the current stimulus is judged relative to the magnitude of only the immediately preceding stimulus (hence the name RJM). In this article, we review existing models of absolute identification and show that none offers a complete account of the phenomena described above. We then show that the RJM offers a unified account of these phenomena and present new experimental evidence that supports the model. We begin with a review of the key empirical results. Empirical Results in Absolute Identification If one uses multivariate information transmission as a dependent variable (McGill, 1954) , it is possible to measure the information transmitted 1 in an absolute identification task. If performance in an absolute identification task were perfect, the information transmitted 1 Information transmitted is a measure of the amount of association between the input (stimulus) and the output (response) of a channel (the participant). The amount of information transmitted does not describe the nature of the association, and the analysis makes no assumptions about the form of the association. Miller (1956) began with a motivation for using this measure. would grow as the number of stimuli was increased. For example, perfect identification of two equally probable stimuli carries one bit of information, identification of four stimuli carries two bits, identification of eight stimuli carries three bits, and so on. However, the information transmitted from stimulus to response (sometimes, channel capacity) in an absolute identification task seems to be limited to very few bits (see Table 1 ), corresponding to perfect identification of very few stimuli across a very wide range of stimulus attributes (see Garner, 1962; Laming, 1984; Miller, 1956 , for reviews). Figure 1 shows information transmitted as a function of the number of stimuli in the set (with range of the stimuli held constant) for data from Garner (1953) and Pollack (1952) . (In all figures that present data, data are collapsed across participants.) With a small number of stimuli, obviously less information must be transmitted, but as the number of stimuli increases, the information transmitted from stimulus to response does not continue to increase. Although an increase in the range of stimuli (number held constant), and hence the separation of the stimuli, produces an initial increase in information transmitted, the increase is a negatively accelerated function of range and quickly reaches an asymptote once adjacent stimuli are discriminable (Alluisi & Sidorsky, 1958; Braida & Durlach, 1972; Eriksen & Hake, 1955a; Pollack, 1952) . When accuracy is plotted as a function of the rank of the stimulus within the stimulus set, a characteristic bow is observed in the resulting serial position curve (e.g., Kent & Lamberts, 2005; Lacouture & Marley, 2004; Murdock, 1960; W. Siegel, 1972) . Performance on stimuli at the ends of the range is better than performance on midrange stimuli even though, when presented in isolation, any two adjacent stimuli may be perfectly discriminable. As for information transmission, once stimuli are pairwise perfectly discriminable, increased spacing of items leads, at best, to only slight improvements in accuracy (Braida & Durlach, 1972; Brown, Neath, & Chater, 2002; Gravetter & Lockhead, 1973; Hartman, 1954; Lacouture, 1997; Luce, Green, & Weber, 1976; Pollack, 1952) . Figure 2 shows the very similar stimulus-response Figure 2 shows, increasing the stimulus spacing had almost no effect on performance. Increasing the number of stimuli in an absolute identification task increases the size of the bow effect (Alluisi & Sidorsky, 1958; Durlach & Braida, 1969; Lacouture & Marley, 1995; Pollack, 1953; W. Siegel, 1972; Weber, Green, & Luce, 1977) . Figure 3 shows the serial position curves obtained by Lacouture and Marley (1995; see also Kent & Lamberts, 2005; Lacouture, Li, & Marley, 1998) for different stimulus set sizes, with a larger bow effect for larger set sizes. Note that although Stimuli 5 and 6 can be nearly perfectly discriminated when they constitute the entire stimulus set, performance on these same stimuli drops considerably when they are identified within a larger stimulus set. Simply shifting all the stimuli along the dimension, so that each stimulus increases in value by a constant multiplicative factor, has no effect on the accuracy against stimulus magnitude curve (Lacouture, 1997). The bow effect remains even after extensive practice, although small improvements in accuracy are observed (Alluisi & Sidorsky, 1958; Hartman, 1954; Weber et al., 1977; but see Rouder, Morey, Cowan, & Pfaltz, 2004, for a larger practice effect). The bow effect is greatly reduced by correction for the asymmetry of errors on extreme versus interior stimuli (Weber et al., 1977) , which suggests that the restricted opportunity to make errors at the ends of the range is a major factor underlying the bow effect (see also Eriksen & Hake, 1957) . The bow effect is not due to response bias (at least, not response bias alone). In data in which end responses are not used more frequently than central responses, the effect is still observed (W. Siegel, 1972) . In our data from Experiment 1, the bow is observed although there is a bias against responding with extreme categories. We know of no absolute identification experiment in which strong sequence effects (where the response to the current stimulus was shown to depend on previous stimuli and responses) were not found. Of course, when performance in an absolute identification task is perfect, there are no sequential dependencies. Thus, the existence of sequential dependencies is likely to provide a useful insight into processing in an absolute identification task. The most salient sequential effect is that the response given to the current stimulus is shown to be assimilated to the immediately preceding stimulus (Garner, 1953; Holland & Lockhead, 1968; Hu, 1997; Lacouture, 1997; Lockhead, 1984; Long, 1937; Luce, Nosofsky, Green, & Smith, 1982; Petrov & Anderson, 2005; Purks, Callahan, Braida, & Durlach, 1980; Rouder et al., 2004; Staddon, King, & Lockhead, 1980; Stewart, 2001; Ward & Lockhead, 1970 , 1971 . In other words, participants are systematically  RELATIVE JUDGMENT MODEL biased to respond as if the current stimulus is nearer to the previous stimulus than it actually is. Figure 4 shows data from the feedback condition of Ward and Lockhead's (1970) absolute identification experiment. Stimuli were tones varying in loudness. The average error in responding on the current trial is plotted for each stimulus as a function of the stimulus on the previous trial. When the current stimulus was greater than the previous stimulus, the error was negative (i.e., the stimulus is underestimated); when the current stimulus was less than the previous stimulus, the error was positive. The five lines are approximately parallel, with positive slopes, demonstrating that assimilation took place for all combinations of current and previous stimuli (Lockhead, 1984) . Assimilation to preceding items is also observed in magnitude estimation tasks (e.g., Jesteadt, Luce, & Green, 1977) , in matching tasks (Stevens, 1975, p. 275), and in relative intensity judgment tasks (Lockhead & King, 1983) . The effect of stimuli further back in the sequence on the current response is the opposite-that is, there is a contrast effect (Holland & Lockhead, 1968; Lacouture, 1997; Ward & Lockhead, 1970 , 1971 . Assimilation to the previous trial and contrast to trials further back have been demonstrated within the same experiments, for the same participants. Figure 5 shows the average error on the current trial (averaged across all possible stimuli on the current trial) as a function of the stimulus k trials ago for data from Holland and Lockhead (1968) , Lacouture (1997) , and Ward and Lockhead (1970). As described above, assimilation was shown to the stimulus on the immediately preceding trial (k ϭ 1). Stimuli on less recent trials (k Ͼ 1) exhibited contrast, as shown by the reversal in the sign of the error. The contrast effect was smaller than the assimilation, and the error dependency reduced to zero with increased numbers of intervening trials. In the experiments on sequence effects discussed so far, stimulus, response, and feedback were all highly correlated. Which of these is the basis for assimilation (and contrast)? We focus on this question for the remainder of this section. The sequence effects observed are dependent on the quality of stimulus presentation. Ward and Lockhead (1971) examined performance in a standard absolute identification experiment using line length. When they increased task difficulty by reducing the luminance and duration of line length presentations, they observed more assimilation. In the difficult condition, accuracy was low, and therefore the correlation between stimuli and responses was reduced. Assimilation was demonstrated only to the previous stimulus and not the previous response. This suggests that assimilation to the previous response is only normally observed because the response is correlated with the previous stimulus (see also Garner, 1953; McGill, 1957; Mori, 1998) . Ward and Lockhead (1971) also observed assimilation to the previous trial's feedback but not to the previous response in a guessing task (although there was slight evidence of a small contrast effect to responses further back in the sequence). The guessing task was identical to an absolute identification experiment, except that the stimuli were omitted and therefore could not have been the cause of the assimilation observed. As the task was guessing, there was no correlation between the feedback and the responses. Thus, the observation of sequential effects only for the previous feedback but not the previous response in a task in which the two were not correlated also suggests that previous responses are not the locus of sequential effects. Manipulating the relative frequencies of the size of the differences between consecutive trials affects identification accuracy. In Lockhead (1970) . E n ϭ error in responding on trial n; S n ϭ rank of the stimulus presented on trial n. STEWART, BROWN, AND CHATER an absolute identification task, Luce et al. (1982) used four differently constrained sequences. In one condition, the sequence of trials was constrained so that the current stimulus was either identical to, one step softer than, or one step louder than the previous stimulus. This condition was called the small step (3) condition, because the current stimulus was chosen from one of three stimuli centered on the previous stimulus. In the small step (5) condition, the current stimulus was selected from five adjacent intensities centered on the previous stimulus. In the random condition, the sequence was random. In the large step condition, the current stimulus was at least four stimuli different from the previous stimulus. For all four sequence types, each intensity was equally frequent over the course of the whole experiment. From the identification confusion matrix, a measure of the confusibility, dЈ i,iϩ1 , of each loudness i with the adjacent loudness, i ϩ 1, was obtained. This method of analysis allows comparison of identification performance free from contamination by constraints imposed by the control of the sequences in each condition. 2 (The procedure for calculating dЈ i,iϩ1 is given in the Appendix.) When dЈ i,iϩ1 is plotted against stimulus magnitude, each condition shows a characteristic bow, with poorer performance for the middle of the range of signals (see the bottom panel in Figure 6 ). The key result is that the curves lie one above the other, such that tones are more confusable in the conditions in which the step size is larger: In order of decreasing identification performance, the curves are small step (3), small step (5), random step, and large step. Smaller transitions seem to lead to higher accuracy (see also Hu, 1997, and Petzold & Haubensak, 2001 , for similar findings). (The top panel of Figure 6 shows the corresponding bows in the accuracy serial position curves. Here, the ordering of the large step and random conditions is reversed, with better performance in the large step condition because of the restricted possibility of making mistakes imposed by the restricted set of possible responses on each trial.) Further work (Nosofsky, 1983b) , testing alternative hypotheses, is consistent with the idea that smaller transitions lead to greater accuracy. There are many existing accounts of some of the phenomena seen in absolute identification data. The extant models can be divided into four main classes: (a) models in which memories of recent stimuli are assimilated (Holland & Lockhead, 1968; Lockhead & King, 1983) , (b) modified Thurstonian models (Braida et al., 1984; Durlach & Braida, 1969; Luce et al., 1976; Purks et al., 1980; Treisman, 1985) , (c) limited response or processing capacity models (Lacouture & Marley, 1991 , 2004 Laming, 1984 Laming, , 1997 Marley & Cook, 1984 , 1986 , and (d) exemplar models Kent & Lamberts, 2005; Nosofsky, 1997; Petrov & Anderson, 2005) . Below, we briefly review each of these models and consider which of the phenomena outlined (limit in information transmitted, bow effects, and assimilation and contrast) are and are not accounted for by each model. Table 2 gives an overview of the scope of these models. Two themes emerge from this review. First, there are two different types of explanation as to why increasing the range of stimuli does not increase information transmitted. Some models assume a perceptual locus, and others assume the limit lies in the response process. The second theme is that current models that assume that long-term representations of absolute magnitudes are the basis for absolute identification do not provide a full account of sequential effects. Holland and Lockhead (1968) . In Holland and Lockhead's (1968) model, participants are assumed to generate a response by adding the judged distance between the current stimulus and the previous stimulus to the feedback from the previous trial. Assimilation and contrast are accounted for in terms of the contamination of the representations of the absolute magnitudes of stimuli. Specifically, the memory of the previous stimulus is assumed to be contaminated by the memories of earlier stimuli. Of the phenomena outlined above, Holland and Lockhead's (1968) model accounts for assimilation and contrast, but only on average. For example, consider a low-magnitude stimulus on the previous trial. The stimuli on preceding trials are likely to have been larger in magnitude, and, thus, when the previous stimulus is confused with them, the representation of its magnitude is an overestimate. This causes the difference between the current and previous stimuli to be underestimated on average (as the current stimulus is also likely to be larger than the previous stimulus) and leads to the current response being biased toward the previous  RELATIVE JUDGMENT MODEL stimulus (i.e., assimilation). Contrast also follows: On average, the current response is biased away from stimuli two or more trials ago because these stimuli are, on average, greater in magnitude than the (low-magnitude) stimulus on the immediately preceding trial. However, a detailed examination reveals this account to be unsatisfactory. Typically, assimilation is observed for all combinations of current and previous stimuli (e.g., Ward & Lockhead, 1970; our Experiment 1). Holland and Lockhead's model predicts contrast in some cases. For example, consider the case of absolute identification of 10 stimuli, with Stimulus 3 on the preceding trial followed by Stimulus 2 on the current trial. The confused representation of Stimulus 3 will be an overestimate (as earlier stimuli are likely to have been larger), and, thus, the difference between the current stimulus and the previous stimulus will be overestimated. This produces a contrast effect in which assimilation is observed. In addition to this difficulty for the model, Holland and Lockhead gave no account of the other phenomena listed in Table 2 . Lockhead and King (1983) . In Lockhead and King's (1983; see also Lockhead, 1984) model, two assumptions are made: (a) that successive stimuli assimilate in memory, and (b) that people compare each new stimulus with a collection of stimulus memories to determine a response. No psychological mechanism is chosen to motivate these assumptions; "the focus here is on a simple equation to fit the data" (Lockhead, 1984, p. 44). The equation is R n ϭ S n ϩ a 1 ͑S n Ϫ S nϪ1 ͒ ϩ a 2 ͑S n Ϫ S nϪ2 ͒ ϩ a 3 ͑S n Ϫ S nϪ3 ͒ ϩ . . . ϩ , where R n is the response on trial n, S n is the stimulus on trial n, a 1 Ͻ 0 for assimilation to S nϪ1 , a 2 Ͼ a 3 Ͼ . . . Ͼ 0 for decreasing contrast to less recent stimuli, and is a noise term. Although such a model can inevitably describe assimilation and contrast, no consideration is given to other absolute identification phenomena. In our application of Lockhead and King's model, we have found that such a model does not offer an account of the limit in information transmitted or the bow effect. In a Thurstonian account, presentation of a stimulus results in perception of an absolute magnitude, represented as a noisy value on an internal sensory scale. Criteria, or bounds, divide this scale into response categories. The criteria provide a long-term frame of reference for absolute magnitudes. There are important multidimensional extensions of this idea (e.g., Ashby & Townsend, 1986) . The source of variability in responding in the standard Thurstonian model is the noise in the representation of the stimulus on the internal sensory scale. A simple Thurstonian model can offer some account of the limit in information transmitted as the number of stimuli is increased with the range held constant (the "Range constant" columns in Table 2 ): As the number of stimuli is increased, the bounds become closer together, and the fixed magnitude noise on the sensory scale means that a stimulus is more likely to be classified incorrectly. The bow effect can also be explained because there is a limited ability to make mistakes for stimuli at the edges of the range. For example, if the smallest magnitude stimulus is greatly underestimated, then it will still be correctly classified into the first category. The invariance of these phenomena as range is increased (the "N constant" columns in Table 2 ) and an account of sequential effects require further modification of the model. We evaluate three modifications below. Durlach and Braida (1969) . Durlach and Braida (1969) modified the simple Thurstonian decision model outlined above to include an internal noise model. Durlach and Braida proposed that  STEWART, BROWN, AND CHATER memory can operate in one of two modes. Here we discuss only the context-coding mode, which applies to absolute identification. In the context-coding mode, the presented stimulus is compared with the general context of recent stimulus presentations. The context-coding mode adds an additional source of variability in responding (beyond the perceptual noise in the simple Thurstonian model) that results from "the inability of the subject to determine the context precisely and his inability to determine or represent the relation of the sensation to this context precisely" (p. 374). The standard deviation of the context-coding noise is assumed to be proportional to the range of the stimuli, though no psychological motivation is given for this assumption. The inclusion of a source of variability that grows with the stimulus range allows Durlach and Braida's (1969) preliminary theory of intensity resolution to account for the invariance of the absolute identification phenomena as range is increased (either through an increase in the number of stimuli with the spacing held constant or through an increase in the spacing). To account for bow effects, Braida et al. (1984) suggested that the general context is set by two anchors at either end of the range. The participant compares a stimulus with the general context by counting steps (which are some proportion of the distance between the anchors) using a noisy measurement unit. Thus, there is less variability for stimuli near one of the anchors, as the number of steps is small, and thus the cumulative error is small. No mechanism is offered to account for sequential effects. However, Purks et al. (1980) suggested that the distributions that represent signals are unaffected by the location of the previous signal but that the category boundaries are. By partitioning their data by the previous signal and fitting a Thurstonian decisionbound model to each partition, Purks et al. demonstrated that the separation between signal distributions was unaffected but that the locations of decision boundaries were, being shifted away from the previous signal. The next modification of the simple Thurstonian model extends this idea. Treisman (1985). Treisman (1985) used criterion-setting theory (Treisman & Williams, 1984) to maintain response criteria in a simple Thurstonian model. Two opposing short-term mechanisms act on the criteria on a trial-by-trial basis. A tracking mechanism, motivated by the assumption that objects in the real world tend to persist, moves criteria away from the currently perceived sensory effect, increasing the probability of a repetition of the previous response. A stabilizing mechanism acts to locate criteria nearer to the prevailing flux of sensory inputs, motivated by the assumption that criteria will be adjusted to maximize information transmitted. Tracking shifts are larger in magnitude than stabilizing shifts but decay more quickly. Thus, Treisman's model predicts assimilation to the immediately preceding stimulus when tracking shifts dominate but predicts contrast to less recent stimuli when stabilizing shifts dominate. However, this account does not fully explain the sequential biases. Treisman (1985, p. 192) stated that the magnitude of criteria movement decreases with the distance of the criteria from the stimulus. Therefore, the model would be expected to predict greater assimilation where previous and current stimuli are similar. However, assimilation is greater where stimuli differ more rather than less (see, e.g., Figure 4 ). For the same reason, Treisman's model predicts that the error in responding should be greater if the previous and current stimuli are similar: Luce et al. (1982) , Nosfosky (1983b ), Hu (1997 ), and Rouder et al. (2004 found the opposite result. The magnitude of the stabilizing shift is, unlike tracking shifts, not fixed but instead proportional to the distance between the sensory input and the nearest criterion. The magnitude of stabilization is thus proportional to the intercriterion distance and therefore also proportional to the range. This allows Treisman's (1985) model to explain why increasing the range of stimuli does not increase information transmitted. However, it also means that as the range is increased, stabilization should come to dominate, and it predicts a change in the pattern of assimilation and contrast that is not observed (e.g., Experiment 1 below). Treisman's (1985) model does not predict the gradual and smooth U-shaped pattern of the bow effect (instead, accuracy is approximately equal for all but the two most extreme stimuli). However, if criteria in the central region are more closely spaced, bow effects will be more U shaped. Such a spacing would also lead to a response bias for extreme stimuli, but the opposite pattern-a central tendency in responding-is typically observed (see Experiment 1). Luce et al. (1976). Luce et al. (1976) proposed a modified Thurstonian model in which an attention band roves over the stimulus range. Items falling in the band result in a less variable Thurstonian representation than those that do not. As the stimulus range is increased, the probability of stimuli falling inside the fixed-width band is reduced, causing a reduction in identification performance. This allows the model to predict a limit in performance as stimulus range is increased. With the additional assumption that the attention band dwells at the edges of the range of stimuli, bow or edge effects are accounted for, although no motivation for this assumption is given. The further assumption that attention tends to dwell on the location of the last stimulus explains the finding that there is typically reduced variation in responding when the previous stimulus is similar to the current stimulus (e.g., Luce et al., 1982) . However, the attention band model does not offer an account of the systematic bias in responding to the current stimulus by preceding stimuli (i.e., assimilation and contrast). Cook, Lacouture, and Marley (Lacouture & Marley, 1991 , 2004 Marley & Cook, 1984 , 1986 have presented three models of absolute identification that account for limits on information transmitted and bow effects by assuming a limited capacity process in either memory or response processes (not perceptual processes). These models can account for limits in performance as stimulus range is increased because they do not assume that the limit in information transmitted is perceptual. Marley and Cook (1984 Cook ( , 1986 and Karpiuk, Lacouture, and Marley (1997) . In Marley and Cook's (1984, 1986) models, perception is assumed to be absolute, with the location of the 3 Gravetter and Lockhead (1973) proposed a similar model, except that in their model noise is assumed to be proportional to the criterial range (i.e., the distance between the two most extreme category boundaries) rather than the distance between the two most extreme stimuli. Though these two distances are highly correlated in most situations, if the lowest and highest stimuli are placed more extremely, criterial range is found to be more appropriate. RELATIVE JUDGMENT MODEL stimulus represented accurately on a Thurstonian continuum. The exact location on the continuum is unavailable to the response process and must be deduced through comparison of the stimulus with the context in which it is presented. Marley and Cook assumed that the context, which comprises a set of elements, must be rehearsed. Each element's activation is incremented each time a pulse arrives from a Poisson pulse process, before its activation continues to decay exponentially. There is a fixed total rehearsal capacity, which they modeled by limiting the pulse rate across all elements. The location of the stimulus within the continuum of elements is the sum of the activation of the elements between the stimulus and known anchors (cf. Braida et al., 1984) . The anchors are assumed to be at or outside the location of the extreme stimuli. Marley and Cook (1984) showed that, under these assumptions, the variability of the total activity of elements to one side of the stimulus increases with the number of elements. Marley and Cook (1984) demonstrated that the model can account for (a) the asymptote in information transmission as stimulus range increases, with the number of stimuli held constant, or as the number of stimuli increases, with the spacing held constant, and (b) the bow effect. Karpiuk et al. (1997) extended the model to predict reaction time distributions. Marley and Cook provided no account of the sequential effects observed. Marley and Cook also pointed out that, if one were to extend their model to provide the necessary account by assuming that the range of the rehearsal is determined by the immediately preceding context, it is not clear how the model could explain assimilation and contrast without a further addition to the model. Lacouture and Marley (1991) . Lacouture and Marley (1991) demonstrated that a simple network model could provide a reasonable account of the limit in information transmitted. The model is a three-layer feed-forward network that learns by mean-variance back-propagation of error. Input vectors of adjacent stimuli overlap. For example, if Stimulus 5 is presented, Input Unit 5 will be activated and neighboring Input Units 4 and 6 will also be activated to a lesser extent. The model predicts the limit in information transmitted when the number of hidden units is one, although the observed characteristic shape of the information transmitted against set size (see our Figure 1 ) is not well reproduced. The model does, however, produce a good fit to Braida and Durlach's (1972) data, in which information transmitted was measured as stimulus range was varied (with set size held constant). Modeling of bow and sequence effects is not described. The model cannot provide an account of sequence effects without substantial modification because, once learning has reached asymptote, the representation and processing of a stimulus are independent of immediately preceding stimuli. Lacouture and Marley (1995 Marley ( , 2004 . Lacouture and Marley's (1995, 2004) mapping model is a feed-forward network with one single input unit, one single hidden unit, and an output unit for each response. The activation of the input unit represents the magnitude of the stimulus. Perception is assumed to be noisy, and repeated presentation of the same stimulus does not always lead to the same activation. The hidden unit normalizes this activation via a lower and an upper anchor value, so that the resulting activation falls within the range 0 to 1. Fixed magnitude noise that represents a noisy mapping process is then added, resulting in a limited channel capacity. With a large number of stimuli, the resulting set of possible mean hidden unit activations is closer together than for a smaller set, and, thus, the fixed magnitude noise has a greater effect on performance for larger sets. The mapping of the hidden unit activation onto output units acts to partition the unit interval into response categories. For each output unit, activation is accumulated over the course of the trial, with the corresponding response being emitted once the accumulator reaches a given threshold (Lacouture & Marley, 1995) . The assumption of repeated intratrial sampling of the output units allows the model to predict response times as well as accuracy, providing an extension over previous models. Lacouture and Marley (2004) replaced the accumulator and threshold with a leaky, competing accumulator (Usher & McClelland, 2001 ) to capture full correct response time distributions. The mapping model provides an account of the limit on information transmitted and of the bow effect for different set sizes. By incorporating the (unmotivated) assumption that, after a response is made, there will be less variation in the output of that unit and those immediately adjacent to it on the next trial, the model also accounts for the data from the sequence manipulation experiments (Luce et al., 1982) . However, this model suffers the same difficulty as the attention band model described above in accounting for sequence effects. Lacouture and Marley (2004) Laming 's (1984, 1997) RJM Laming (1984) described a model that accounts for the limit in information transmission and the effects of constraining possible jump sizes between successive stimuli (i.e., Luce et al., 1982) . The crucial assumption in Laming's model is that all judgments are relative to the immediately preceding context (i.e., that the differences between successive stimuli are used, not the absolute magnitudes of the stimuli). Further, Laming proposed that such relative judgments are limited. In particular, Laming suggested that the current stimulus can be judged as "much less than," "less than," "equal to," "more than," or "much more than" the previous stimulus. This judgment limit provides a limit in the information transmitted. Numerical estimates of the stimulus magnitudes are assigned such that they follow the same pattern. If the difference between stimuli is judged as "less than," for example, then the number assigned to the estimate of the magnitude of the current stimulus is less than the estimate of the magnitude of the previous stimulus. The ordering of Luce et al.'s (1982) conditions is also explained by Laming's model. Laming showed that the variability in responding depends mainly on the mean squared jump sizes in the sequence, and, as jump size predicts perfectly the ordering of performance in Luce et al.'s conditions, so does Laming's model. Laming (1984) did not offer an account of the bow in the serial position curve or of assimilation and contrast. Indeed, Laming stated that an additional principle is required to provide an account of the sequential effects observed in magnitude estimation and absolute identification. He suggested that researchers take into 888 STEWART, BROWN, AND CHATER account the prior expectations of the distribution of stimulus magnitudes as a candidate principle. Exemplar models (Medin & Schaffer, 1978; Nosofsky, 1986) assume a long-term memory for each stimulus's magnitude, together with the label associated with that stimulus. On presentation of a stimulus, the probability of a given response is given by the similarity of the presented stimulus to the memory of the stimulus associated with that response divided by the summed similarity of the presented stimulus to each stimulus memory. Brown et al. (2002) . Brown et al.'s (2002) model of scaleinvariant memory, perception, and learning has been applied to absolute identification data (as well as free, serial, and probed recall memory tasks). The model is an exemplar model of absolute identification and is equivalent to the generalized context model (GCM; Nosofsky, 1986) in its application to absolute identification. Exemplar models of absolute identification provide a reasonable account of bow effects. Bow effects are accounted for because items at the end of the range have fewer similar neighbors to be confused with. However, exemplar models do not predict the gradual bowing that is typically observed: Instead, all items tend to show almost identical levels of performance, except for superior performance on the very edge items. It is possible to provide a better fit by biasing the responses associated with more extreme stimuli. However, this bias for extreme responses is at odds with the central tendency in responding that is typically observed. Further, as described above, the bow effect is still observed in data in which each response is used equally often (W. Siegel, 1972) or in which middle responses are used more often (see Experiment 1). Exemplar models face a further problem. Recall that increasing the spacing of stimuli does not remove the bow effect and leads only to a slight improvement in accuracy. Exemplar models, however, predict a large improvement in accuracy, as items become more discriminable. Brown et al. (2002) introduced the assumption that discriminability is inversely related to stimulus range and showed that, with this additional assumption, the bow effects are invariant under stimulus range. Exemplar models do not predict any curves in dЈ without further assumptions. In their simplest form, exemplar models offer no account of sequence effects. When adapted to predict sequence effects, typically by the assumption that more recent exemplars are more available in memory and/or weighted more heavily in the subsequent decision process (e.g., Nosofsky & Palmeri, 1997; see also Elliott & Anderson, 1995) , the models do not correctly predict sequence effects observed in classification (Stewart & Brown, 2004; Stewart, Brown, & Chater, 2002) . Increased weighting of more recent items makes a prediction similar to assimilation, as repetition of the previous response is more likely. However, the criticism applied to the Thurstonian models applies: An increased probability of repetition is not equivalent to assimilation. Further, this modification provides no account of contrast. Nosofsky (1997). Nosofsky (1997) applied Nosofsky and Palmeri's (1997) exemplar-based random walk model, which is an extension of the GCM (Nosofsky, 1986) , to predict responses and reaction times in absolute identification. Stimuli are represented by normal distributions on a psychological continuum, with stimuli at the edges of the range assumed to be less variable. In this way, an account of bow or edge effects is built into the model. On presentation of a test item, the model assumes memories race to be retrieved. The probability of a memory being retrieved at a given time is a function of the exemplar's similarity to the test item and the exemplar's strength in memory. Once an item is retrieved, a counter for the associated category label is incremented, and all others are decremented. The remaining items then race again. When any counter falls too low, the response associated with the counter leaves the race. When a counter reaches a given threshold, the response associated with that counter is emitted, with the reaction time being a function of the sum of the times for each retrieval. No mechanism is outlined for prediction of sequence effects, and no explanation is offered on the invariance of the bow in serial position when stimulus range is altered. Petrov and Anderson (2005) . Petrov and Anderson (2005) presented a scaling model (ANCHOR) based on the ACT-R architecture (Anderson, 1990; Anderson & Lebière, 1998 ) that they applied to absolute identification and category rating. The perception of the absolute magnitude of a stimulus is compared with anchors or exemplars stored in memory. Perception is assumed to be stochastic. The selection of exemplars is also stochastic and depends on the similarity between the exemplars and the target stimulus and also on the frequency and recency with which each exemplar was previously used. Exemplars compete for selection. One exemplar is selected, and the associated response is retrieved. If there is a discrepancy between the exemplar magnitude and the percept magnitude, then an adjustment is applied to the response to correct it either up or down. The system is adaptive, and, after feedback, the location of the associated exemplar is assimilated toward the percept. Petrov and Anderson (2005) fitted the model to their own data from an absolute identification of nine stimuli. The model was able to fit simultaneously the information transmitted, central tendency in responding, assimilation (on average), and a small practice effect. The model did not predict bows in dЈ but was able to predict an accuracy advantage for end stimuli because of the limited opportunity for errors at the ends of the range. Petrov and Anderson (2005) did not model the effect of increasing the number of stimuli (with the range held constant) or the range of the stimuli (with the number held constant), and, thus, it is uncertain whether the model can account for the effects of these variables. However, the model does include noisy components that are independent of the spacing of the stimuli, so it may well be able to account for the effects without modification. Petrov and Anderson (2005) did not examine assimilation in detail. In Figure 7 , we show the predictions of ANCHOR (a) for the effect on the current response of the previous stimulus and current stimulus (top panel) and (b) for the effect of the stimulus at different lags (bottom panel). We used the parameters that Petrov and Anderson (2005, pp. 406, 416) reported as best fitting their data and ran 200 simulations of 450 trials. Although AN-CHOR does predict assimilation on average, it does not predict the detailed pattern that is normally observed (e.g., Figure 4 and Experiment 1), in which assimilation increases as the difference between the previous and the current stimuli increases. In AN-CHOR, sequential effects are caused by exemplars being weighted by their recency of use. Thus, ANCHOR fails to predict the more detailed pattern of assimilation for the same reason as Treisman's 889 RELATIVE JUDGMENT MODEL (1985) and Luce et al.'s (1976) models fail: Predicting that the response associated with the previous stimulus is more likely to be repeated is not the same as predicting that the current response will be biased toward the previous stimulus. Further, the model does not predict contrast to stimuli at lags of two or greater and instead predicts assimilation to stimuli at these lags. Kent and Lamberts (2005) . Kent and Lamberts (2005) presented an application of Lamberts's (2000) extended GCM (EGCM) to absolute identification. The EGCM differs from the GCM in assuming that the amount of information about a stimulus magnitude increases over the time course of the stimulus presentation as stimulus elements are sampled. The probability that sampling will be halted and a response given increases as more elements are sampled. The EGCM was able to predict bow in accuracy and mean correct reaction times as well as the complete reaction time distributions for individual stimuli. By allowing more generalization for larger set sizes and less sampling for larger set sizes, the EGCM could also predict changes in accuracy and reaction time as set size varied. An alternative, more parsimonious modification, in which the information contributed by each additional sample was a decreasing function of the number of samples, also allowed the model to account for set-size effects. Kent and Lamberts did not model the effect of changing the stimulus spacing. However, without altering the discriminability (cf. , the EGCM predicts that performance will increase greatly with increased spacing (limits in information transmitted also have not been modeled). The model does not predict any sequential effects.
In contrast, Adaptive Crowd Kernel Learning (A-CKL) REF ) selects queries by how likely they are to reduce uncertainty in the positions of each object in the learned embedding.
Absolute identification by relative judgment
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