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193 nips-2004-Theories of Access Consciousness

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Author: Michael D. Colagrosso, Michael C. Mozer

Abstract: Theories of access consciousness address how it is that some mental states but not others are available for evaluation, choice behavior, and verbal report. Farah, O’Reilly, and Vecera (1994) argue that quality of representation is critical; Dehaene, Sergent, and Changeux (2003) argue that the ability to communicate representations is critical. We present a probabilistic information transmission or PIT model that suggests both of these conditions are essential for access consciousness. Having successfully modeled data from the repetition priming literature in the past, we use the PIT model to account for data from two experiments on subliminal priming, showing that the model produces priming even in the absence of accessibility and reportability of internal states. The model provides a mechanistic basis for understanding the dissociation of priming and awareness. Philosophy has made many attempts to identify distinct aspects of consciousness. Perhaps the most famous effort is Block’s (1995) delineation of phenomenal and access consciousness. Phenomenal consciousness has to do with “what it is like” to experience chocolate or a pin prick. Access consciousness refers to internal states whose content is “(1) inferentially promiscuous, i.e., poised to be used as a premise in reasoning, (2) poised for control of action, and (3) poised for rational control of speech.” (p. 230) The scientiﬁc study of consciousness has exploded in the past six years, and an important catalyst for this explosion has been the decision to focus on the problem of access consciousness: how is it that some mental states but not others become available for evaluation, choice behavior, verbal report, and storage in working memory. Another reason for the recent explosion of consciousness research is the availability of functional imaging techniques to explore differences in brain activation between conscious and unconscious states, as well as the development of clever psychological experiments that show that a stimulus that is not consciously perceived can nonetheless inﬂuence cognition, which we describe shortly. 1 Subliminal Priming The phenomena we address utilize an experimental paradigm known as repetition priming. Priming refers to an improvement in efﬁciency in processing a stimulus item as a result of previous exposure to the item. Efﬁciency is deﬁned in terms of shorter response times, lower error rates, or both. A typical long-term perceptual priming experiment consists of a study phase during which participants are asked to read aloud a list of words, and a test phase during which participants must name or categorize a series of words, presented one at a time. Reaction time is lower and/or accuracy is higher for test words that were also on the study list. Repetition priming occurs without strategic effort on the part of participants, and therefore appears to be a low level mechanism of learning, which likely serves as the mechanism underlying the reﬁnement of cognitive skills with practice. In traditional studies, priming is supraliminal—the prime is consciously perceived. In the studies we model here, primes are subliminal. Subliminal priming addresses fundamental issues concerning conscious access: How is it that a word or image that cannot be identiﬁed, detected, or even discriminated in forced choice can nonetheless inﬂuence the processing of a subsequent stimulus word? Answering this question in a computational framework would be a signiﬁcant advance toward understanding the nature of access consciousness. 2 Models of Conscious and Unconscious Processing In contrast to the wealth of experimental data, and the large number of speculative and philosophical papers on consciousness, concrete computational models are rare. The domain of consciousness is particularly ripe for theoretical perspectives, because it is a signiﬁcant contribution to simply provide an existence proof of a mechanism that can explain speciﬁc experimental data. Ordinarily, a theorist faces skepticism when presenting a model; it often seems that hundreds of alternative, equally plausible accounts must exist. However, when addressing data deemed central to issues of consciousness, simply providing a concrete handle on the phenomena serves to demystify consciousness and bring it into the realm of scientiﬁc understanding. We are familiar with only three computational models that address speciﬁc experimental data in the domain of consciousness. We summarize these models, and then present a novel model and describe its relationship to the previous efforts. Farah, O’Reilly, and Vecera (1994) were the ﬁrst to model speciﬁc phenomena pertaining to consciousness in a computational framework. The phenomena involve prosopagnosia, a deﬁcit of overt face recognition following brain damage. Nonetheless, prosopagnosia patients exhibit residual covert recognition by a variety of tests. For example, when patients are asked to categorize names as famous or nonfamous, their response times are faster to a famous name when the name is primed by a picture of a semantically related face (e.g., the name “Bill Clinton” when preceded by a photograph of Hillary), despite the fact that they could not identify the related face. Farah et al. model face recognition in a neural network, and show that when the network is damaged, it loses the ability to perform tasks requiring high ﬁdelity representations (e.g., identiﬁcation) but not tasks requiring only coarse information (e.g., semantic priming). They argue that conscious perception is associated with a certain minimal quality of representation. Dehaene and Naccache (2001) outline a framework based on Baars’ (1989) notion of conscious states as residing in a global workspace. They describe the workspace as a “distributed neural system...with long-distance connectivity that can potentially interconnect multiple specialized brain areas in a coordinated, though variable manner.” (p. 13) Dehaene, Sergent, and Changeaux (2003) implement this framework in a complicated architecture of integrate-and-ﬁre neurons and show that the model can qualitatively account for the attentional blink phenomenon. The attentional blink is observed in experiments where participants are shown a rapid series of stimuli, which includes two targets (T1 and T2). If T2 appears shortly after T1, the ability to report T2 drops, as if attention is distracted. Dehane et al. explain this phenomenon as follows. When T1 is presented, its activation propagates to frontal cortical areas (the global workspace). Feedback connections lead to a resonance between frontal and posterior areas, which strengthen T1 but block T2 from entering the workspace. If the T1-T2 lag is sufﬁciently great, habituation of T1 sufﬁciently weakens the representation such that T2 can enter the workspace and suppress T1. In this account, conscious access is achieved via resonance between posterior and frontal areas. Although the Farah et al. and Dehaene et al. models might not seem to have much in common, they both make claims concerning what is required to achieve functional connectivity between perceptual and response systems. Farah et al. focus on aspects of the representation; Dehaene et al. focus on a pathway through which representations can be communicated. These two aspects are not incompatible, and in fact, a third model incorporates both. Mathis and Mozer (1996) describe an architecture with processing modules for perceptual and response processes, implemented as attractor neural nets. They argue that in order for a representation in some perceptual module to be assured of inﬂuencing a response module, (a) it must have certain characteristics–temporal persistence and well-formedness– which is quite similar to Farah et al.’s notion of quality, and (b) the two modules must be interconnected—which is the purpose of Dehaene et al.’s global workspace. The model has two limitations that restrict its value as a contemporary account of conscious access. First, it addressed classical subliminal priming data, but more reliable data has recently been reported. Second, like the other two models, Mathis and Mozer used a complex neural network architecture with arbitrary assumptions built in, and the sensitivity of the model’s behavior to these assumptions is far from clear. In this paper, we present a model that embodies the same assumptions as Mathis and Mozer, but overcomes its two limitations, and explains subliminal-priming data that has yet to be interpreted via a computational model. 3 The Probabilistic Information Transmission (PIT) Framework Our model is based on the probabilistic information transmission or PIT framework of Mozer, Colagrosso, and Huber (2002, 2003). The framework characterizes the transmission of information from perceptual to response systems, and how the time course of information transmission changes with experience (i.e., priming). Mozer et al. used this framework to account for a variety of facilitation effects from supraliminal repetition priming. The framework describes cognition in terms of a collection of information-processing pathways, and supposes that any act of cognition involves coordination among multiple pathways. For example, to model a letter-naming task where a letter printed in upper or lower case is presented visually and the letter must be named, the framework would assume a perceptual pathway that maps the visual input to an identity representation, and a response pathway that maps a identity representation to a naming response. The framework is formalized as a probabilistic model: the pathway input and output are random variables and microinference in a pathway is carried out by Bayesian belief revision. The framework captures the time course of information processing for a single experimental trial. To elaborate, consider a pathway whose input at time t is a discrete random variable, denoted X(t), which can assume values x1 , x2 , x3 , . . . , xnx corresponding to alternative input states. Similarly, the output of the pathway at time t is a discrete random variable, denoted Y (t), which can assume values y1 , y2 , y3 , . . . , yny . For example, in the letter-naming task, the input to the perceptual pathway would be one of nx = 52 visual patterns corresponding to the upper- and lower-case letters of the alphabet, and the output is one of ny = 26 letter identities. To present a particular input alternative, say xi , to the model for T time steps, we specify X(t) = xi for t = 1 . . . T , and allow the model to compute P(Y (t) | X(1) . . . X(t)). A pathway is modeled as a dynamic Bayes network; the minimal version of the model used in the present simulations is simply a hidden Markov model, where the X(t) are observations and the Y (t) are inferred state (see Figure 1a). In typical usage, an HMM is presented with a sequence of distinct inputs, whereas we maintain the same input for many successive time steps; and an HMM transitions through a sequence of distinct hidden states, whereas we attempt to converge with increasing conﬁdence on a single state. Figure 1b illustrates the time course of inference in a single pathway with 52 input and 26 output alternatives and two-to-one associations. The solid line in the Figure shows, as a function of time t, P(Y (t) = yi | X(1) = x2i . . . X(t) = x2i ), i.e., the probability that input i (say, the visual pattern of an upper case O) will produce its target output (the letter identity). Evidence for the target output accumulates gradually over time, yielding a speed-accuracy curve that relates the number of iterations to the accuracy of identiﬁcation. Y0 Y1 X1 Y2 X2 P(Output) 1 Y3 X3 (a) (b) 0.8 0.6 O 0.4 0.2 0 Q Time Figure 1: (a) basic pathway architecture—a hidden Markov model; (b) time course of inference in a pathway when the letter O is presented, causing activation of both O and the visually similar Q. The exact shape of the speed-accuracy curve—the pathway dynamics—are determined by three probability distributions, which embody the knowledge and past experience of the model. First, P(Y (0)) is the prior distribution over outputs in the absence of any information about the input. Second, P(Y (t) | Y (t − 1)) characterizes how the pathway output evolves over time. We assume the transition probability matrix serves as a memory with diffusion, i.e., P(Y (t) = yi |Y (t − 1) = yj ) = (1 − β)δij + βP(Y (0) = yi ), where β is the diffusion constant and δij is the Kronecker delta. Third, P(X(t) | Y (t)) characterizes the strength of association between inputs and outputs. The greater the association strength, the more rapidly that information about X will be communicated to Y . We parameterize this distribution as P(X(t) = xi |Y (t) = yj ) ∼ 1 + k γik αkj , where αij indicates the frequency of experience with the association between states xi and yj , and γik speciﬁes the similarity between states xi and xk . (Although the representation of states is localist, the γ terms allow us to design in the similarity structure inherent in a distributed representation.) These association strengths are highly constrained by the task structure and the similarity structure and familiarity of the inputs. Fundamental to the framework is the assumption that with each experience, a pathway becomes more efﬁcient at processing an input. Efﬁciency is reﬂected by a shift in the speedaccuracy curve to the left. In Mozer, Colagrosso, and Huber (2002, 2003), we propose two distinct mechanisms to model phenomena of supraliminal priming. First, the association frequencies, αij , are increased following a trial in which xi leads to activation of yj , resulting in more efﬁcient transmission of information, corresponding to an increased slope of the solid line in Figure 1b. The increase is Hebbian, based on the maximum activation achieved by xi and yj : ∆αij = η maxt P(X(t) = xi )P(Y (t) = yj ), where η is a step size. Second, the priors, which serve as a model of the environment, are increased to indicate a greater likelihood of the same output occurring again in the future. In modeling data from supraliminal priming, we found that the increases to association frequencies are long lasting, but the increases to the priors decay over the course of a few minutes or a few trials. As a result, the prior updating does not play into the simulation we report here; we refer the reader to Mozer, Colagrosso, and Huber (2003) for details. 4 Access Consciousness and PIT We have described the operation of a single pathway, but to model any cognitive task, we require a series of pathways in cascade. For a simple choice task, we use a percpetual pathway cascaded to a response pathway. The interconnection between the pathways is achieved by copying the output of the perceptual pathway, Y p (t), to the input of the response pathway, X r (t), at each time t. This multiple-pathway architecture allows us to characterize the notion of access consciousness. Considering the output of the perceptual pathway, access is achieved when: (1) the output representation is sufﬁcient to trigger the correct behavior in the response pathway, and (2) the perceptual and response pathways are functionally interconnected. In more general terms, access for a perceptual pathway output requires that these two condi- tions be met not just for a speciﬁc response pathway, but for arbitrary response pathways (e.g., pathways for naming, choice, evaluation, working memory, etc.). In Mozer and Colagrosso (in preparation) we characterize the sufﬁciency requirements of condition 1; they involve a representation of low entropy that stays active for long enough that the representation can propagate to the next pathway. As we will show, a brieﬂy presented stimulus fails to achieve a representation that supports choice and naming responses. Nonetheless, the stimulus evokes activity in the perceptual pathway. Because perceptual priming depends on the magnitude of the activation in the perceptual pathway, not on the activation being communicated to response pathways, the framework is consistent with the notion of priming occurring in the absence of awareness. 4.1 Simulation of Bar and Biederman (1998) Bar and Biederman (1998) presented a sequence of masked line drawings of objects and asked participants to name the objects, even if they had to guess. If the guess was incorrect, participants were required to choose the object name from a set of four alternatives. Unbeknownst to the participant, some of the drawings in the series were repeated, and Bar and Biederman were interested in whether participants would beneﬁt from the ﬁrst presentation even if it could not be identiﬁed. The repeated objects could be the same or a different exemplar of the object, and it could appear in either the same or a different display position. Participants were able to name 13.5% of drawings on presentation 1, but accuracy jumped to 34.5% on presentation 2. Accuracy did improve, though not as much, if the same shape was presented in a different position, but not if a different drawing of the same object was presented, suggesting a locus of priming early in the visual stream. The improvement in accuracy is not due to practice in general, because accuracy rose only 4.0% for novel control objects over the course of the experiment. The priming is ﬁrmly subliminal, because participants were not only unable to name objects on the ﬁrst presentation, but their fouralternative forced choice (4AFC) performance was not much above chance (28.5%). To model these phenomena, we created a response pathway with ﬁfty states representing names of objects that are used in the experiment, e.g., chair and lamp. We also created a perceptual pathway with states representing visual patterns that correspond to the names in the response pathway. Following the experimental design, every object identity was instantiated in two distinct shapes, and every shape could be in one of nine different visualﬁeld positions, leading to 900 distinct states in the perceptual pathway to model the possible visual stimuli. The following parameters were ﬁt to the data. If two perceptual states, xi and xk are the same shape in different positions, they are assigned a similarity coefﬁcient γik = 0.95; all other similarity coefﬁcients are zero. The association frequency, α, for valid associations in the perceptual pathway was 22, and the response pathway 18. Other parameters were β p = .05, β r = .01, and η = 1.0. The PIT model achieves a good ﬁt to the human experimental data (Figure 2). Speciﬁcally, priming is greatest for the same shape in the same position, some priming occurs for the same shape in a different position, and no substantial priming occurs for the different shape. Figure 3a shows the time course of activation of a stimulus representation in the perceptual pathway when the stimulus is presented for 50 iterations, on both the ﬁrst and third presentations. The third presentation was chosen instead of the second to make the effect of priming clearer. Even though a shape cannot be named on the ﬁrst presentation, partial information about the shape may nonetheless be available for report. The 4AFC test of Bar and Biederman provides a more sensitive measure of residual stimulus information. In past work, we modeled forced-choice tasks using a response pathway with only the alternatives under consideration. However, in this experiment, forced-choice performance must be estimated conditional on incorrect naming. In PIT framework, we achieve this using naming and 40 40 First Block Second Block 30 25 20 15 10 5 First Block 35 Percent Correct Naming Percent Correct Naming 35 Second Block 30 25 20 15 10 5 0 0 Control Objects Prime SHAPE: Same Objects POSITION: Same Same Different Different Different Second Same Different Control Control Objects Prime SHAPE: Same Objects POSITION: Same Same Different Different Different Second Same Different Control Figure 2: (left panel) Data from Bar and Biederman (1998) (right panel) Simulation of PIT. White bar: accuracy on ﬁrst presentation of a prime object. Black bars: the accuracy when the object is repeated, either with the same or different shape, and in the same or different position. Grey bars: accuracy for control objects at the beginning and the end of the experiment. forced-choice output pathways having output distributions N (t) and F (t), which are linked via the perceptual state, Y p (t). F (t) must be reestimated with the evidence that N (t) is not the target state. This inference problem is intractable. We therefore used a shortcut in which a single response pathway is used, augmented with a simple three-node belief net (Figure 3b) to capture the dependence between naming and forced choice. The belief net has a response pathway node Y r (t) connected to F (t) and N (t), with conditional distribution P (N (t) = ni |Y r (t) = yj ) = θδij + (1 − θ)/|Y r |, and an analogous distribution for P (F (t) = fi |Y r (t) = yj ). The free parameter θ determines how veridically naming and forced-choice actions reﬂect response-pathway output. Over a range of θ, θ < 1, the model obtains forced-choice performance near chance on the ﬁrst presentation when the naming response is incorrect. For example, with θ = 0.72, the model produces a forced-choice accuracy on presentation 1 of 26.1%. (Interestingly, the model also produces below chance performance on presentation 2 if the object is not named correctly—23.5%—which is also found in the human data—20.0%.) Thus, by the stringent criterion of 4AFC, the model shows no access consciousness, and therefore illustrates a dissociation between priming and access consciousness. In our simulation, we followed the procedure of Bar and Biederman by including distractor alternatives with visual and semantic similarity to the target. These distractors are critical: with unrelated distractors, the model’s 4AFC performance is signiﬁcantly above chance, illustrating that a perceptual representation can be adequate to support some responses but not others, as Farah et al. (1994) also argued. 4.2 Simulation of Abrams and Greenwald (2000) During an initial phase of the experiment, participants categorized 24 clearly visible target words as pleasant (e.g., HUMOR) or unpleasant (e.g., SMUT). They became quite familiar with the task by categorizing each word a total of eight times. In a second phase, participants were asked to classify the same targets and were given a response deadline to induce errors. The targets were preceded by masked primes that could not be identiﬁed. Of interest is the effective valence (or EV) of the target for different prime types, deﬁned as the error rate difference between unpleasant and pleasant targets. A positive (negative) EV indicates that responses are biased toward a pleasant (unpleasant) interpretation by the prime. As one would expect, pleasant primes resulted in a positive EV, unpleasant primes in a negative EV. Of critical interest is the ﬁnding that a nonword prime formed by recombining two pleasant targets (e.g., HULIP from HUMOR and TULIP) or unpleasant targets (e.g., BIUT from BILE and SMUT ) also served to bias the targets. More surprising, a positive EV resulted from unpleasant prime words formed by recombining two pleasant targets (TUMOR from TULIP and HUMOR ), indicating that subliminal priming arises from word fragments, not words as unitary entities, and providing further evidence for an early locus of subliminal priming. Note that the results depend critically on the ﬁrst phase of the experiment, which gave participants extensive practice on a relatively small set of words that were then used as and recombined to form primes. Words not studied in the ﬁrst phase (orphans) provided Probability 0.6 0.5 0.4 0.3 0.2 0.1 0 object, first presentation object, third presentation different object N(t) F(t) Yr(t) 1 50 1000 (a) (b) Time (msec) Figure 3: (a) Activation of the perceptual representation in PIT as a function of processing iterations Effective Valence on the ﬁrst (thin solid line) and third (thick solid line) presentations of target. (b) Bayes net for performing 4AFC conditional on incorrect naming response. 0.4 Experiment Model 0.3 0.2 0.1 0 targets hulip-type tumor-type orphans Figure 4: Effective valence of primes in the Abrams and Greenwald (2000) experiment for human subjects (black bars) and PIT model (grey bars). HULIP-type primes are almost as strong as target repetitions, and TUMOR-type primes have a positive valence, contrary to the meaning of the word. no signiﬁcant EV effect when used as primes. In this simulation, we used a three pathway model: a perceptual pathway that maps visual patterns to orthography with 200 input states corresponding both to words, nonwords, and nonword recombinations of words; a semantic pathway that maps to 100 distinct lexical/semantic states; and a judgement pathway that maps to two responses, pleasant and unpleasant. In the perceptual pathway, similarity structure was based on letter overlap, so that HULIP was similar to both TULIP and HUMOR, with γ = 0.837. No similarity was assumed in the semantic state representation; consistent with the previous simulation, β p = .05, β s = .01, β j = .01, and η = .01. At the outset of the simulation, α frequencies for correct associations were 15, 19, and 25 in the perceptual, semantic, and judgement pathways. The initial phase of the experiment was simulated by repeated supraliminal presentation of words, which increased the association frequencies in all three pathways through the ∆αij learning rule. Long-term supraliminal priming is essential in establishing the association strengths, as we’ll explain. Short-term subliminal priming also plays a key role in the experiment. During the second phase of the experiment, residual activity from the prime—primarily in the judgement pathway—biases the response to the target. Residual activation of the prime is present even if the representation of the prime does not reach sufﬁcient strength that it could be named or otherwise reported. The outcome of the simulation is consistent with the human data (Figure 4). When a HULIP -type prime is presented, HUMOR and TULIP become active in the semantic pathway because of their visual similarity to HULIP. Partial activation of these two practiced words pushes the judgement pathway toward a pleasant response, resulting in a positive EV. When a TUMOR-type prime is presented, three different words become active in the semantic pathway: HUMOR, TULIP, and TUMOR itself. Although TUMOR is more active, it was not one of the words studied during the initial phase of the experiment, and as a result, it has a relatively weak association to the unpleasant judgement, in contrast to the other two words which have strong associations to the pleasant judgement. Orphan primes have little effect because they were not studied during the initial phase of the experiment, and consequently their association to pleasant and unpleasant judgements is also weak. In summary, activation of the prime along a critical, well-practiced pathway may not be sufﬁcient to support an overt naming response, yet it may be sufﬁcient to bias the processing of the immediately following target. 5 Discussion An important contribution of this work has been to demonstrate that speciﬁc experimental results relating to access consciousness and subliminal priming can be interpreted in a concrete computational framework. By necessity, the PIT framework, which we previously used to model supraliminal priming data, predicts the existence of subliminal priming, because the mechanisms giving rise to priming depend on degree of activation of a representation, whereas the processes giving rise to access consciousness also depend on the temporal persistence of a representation. Another contribution of this work has been to argue that two previous computational models each tell only part of the story. Farah et al. argue that quality of representation is critical; Dehaene et al. argue that pathways to communicate representations is critical. The PIT framework argues that both of these features are necessary for access consciousness. Although the PIT framework is not completely developed, it nonetheless makes a clear prediction: that subliminal priming is can never be stronger than supraliminal priming, because the maximal activation of subliminal primes is never greater than that of supraliminal primes. One might argue that many theoretical frameworks might predict the same, but no other computational model is sufﬁciently well developed—in terms of addressing both priming and access consciousness—to make this prediction. In its current stage of development, a weakness of the PIT framework is that it is silent as to how perceptual and response pathways become ﬂexibly interconnected based on task demands. However, the PIT framework is not alone in failing to address this critical issue: The Dehaene et al. model suggests that once a representation enters the global workspace, all response modules can access it, but the model does not specify how the appropriate perceptual module wins the competition to enter the global workspace, or how the appropriate response module is activated. Clearly, ﬂexible cognitive control structures that perform these functions are intricately related to mechanisms of consciousness. Acknowledgments This research was supported by NIH/IFOPAL R01 MH61549–01A1. References Abrams, R. L., & Greenwald, A. G. (2000). Parts outweigh the whole (word) in unconscious analysis of meaning. Psychological Science, 11(2), 118–124. Baars, B. (1989). A cognitive theory of consciousness. Cambridge: Cambridge University Press. Bar, M., & Biederman, I. (1998). Subliminal visual priming. Psychological Science, 9(6), 464–468. Block, N. (1995). On a confusion about a function of consciousness. Brain and Behavioral Sciences, 18(2), 227–247. Dehaene, S., & Naccache, L. (2001). Towards a cognitive neuroscience of consciousness: basic evidence and a workspace framework. Cognition, 79, 1–37. Dehaene, S., Sergent, C., & Changeux, J.-P. (2003). A neuronal network model linking subjective reports and objective physiological data during conscious perception. Proceedings of the National Academy of Sciences, 100, 8520–8525. Farah, M. J., O’Reilly, R. C., & Vecera, S. P. (1994). Dissociated overt and covert recognition as an emergent property of a lesioned neural network. Psychological Review, 100, 571–588. Mathis, D. W., & Mozer, M. C. (1996). Conscious and unconscious perception: a computational theory. In G. Cottrell (Ed.), Proceedings of the Eighteenth Annual Conference of the Cognitive Science Society (pp. 324–328). Hillsdale, NJ: Erlbaum & Associates. Mozer, M. C., Colagrosso, M. D., & Huber, D. E. (2002). A rational analysis of cognitive control in a speeded discrimination task. In T. G. Dietterich, S. Becker, & Z. Ghahramani (Eds.), Advances in Neural Information Processing Systems 14. Cambridge, MA: MIT Press. Mozer, M. C., Colagrosso, M. D., & Huber, D. E. (2003). Mechanisms of long-term repetition priming and skill reﬁnement: A probabilistic pathway model. In Proceedings of the TwentyFifth Annual Conference of the Cognitive Science Society. Hillsdale, NJ: Erlbaum Associates.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract Theories of access consciousness address how it is that some mental states but not others are available for evaluation, choice behavior, and verbal report. [sent-5, score-0.411]

2 We present a probabilistic information transmission or PIT model that suggests both of these conditions are essential for access consciousness. [sent-7, score-0.162]

3 Having successfully modeled data from the repetition priming literature in the past, we use the PIT model to account for data from two experiments on subliminal priming, showing that the model produces priming even in the absence of accessibility and reportability of internal states. [sent-8, score-1.189]

4 The model provides a mechanistic basis for understanding the dissociation of priming and awareness. [sent-9, score-0.477]

5 Perhaps the most famous effort is Block’s (1995) delineation of phenomenal and access consciousness. [sent-11, score-0.168]

6 Phenomenal consciousness has to do with “what it is like” to experience chocolate or a pin prick. [sent-12, score-0.288]

7 Access consciousness refers to internal states whose content is “(1) inferentially promiscuous, i. [sent-13, score-0.3]

8 , poised to be used as a premise in reasoning, (2) poised for control of action, and (3) poised for rational control of speech. [sent-15, score-0.194]

9 A typical long-term perceptual priming experiment consists of a study phase during which participants are asked to read aloud a list of words, and a test phase during which participants must name or categorize a series of words, presented one at a time. [sent-22, score-1.092]

10 Repetition priming occurs without strategic effort on the part of participants, and therefore appears to be a low level mechanism of learning, which likely serves as the mechanism underlying the reﬁnement of cognitive skills with practice. [sent-24, score-0.532]

11 In traditional studies, priming is supraliminal—the prime is consciously perceived. [sent-25, score-0.664]

12 Subliminal priming addresses fundamental issues concerning conscious access: How is it that a word or image that cannot be identiﬁed, detected, or even discriminated in forced choice can nonetheless inﬂuence the processing of a subsequent stimulus word? [sent-27, score-0.712]

13 The domain of consciousness is particularly ripe for theoretical perspectives, because it is a signiﬁcant contribution to simply provide an existence proof of a mechanism that can explain speciﬁc experimental data. [sent-30, score-0.253]

14 However, when addressing data deemed central to issues of consciousness, simply providing a concrete handle on the phenomena serves to demystify consciousness and bring it into the realm of scientiﬁc understanding. [sent-32, score-0.301]

15 Farah, O’Reilly, and Vecera (1994) were the ﬁrst to model speciﬁc phenomena pertaining to consciousness in a computational framework. [sent-35, score-0.301]

16 For example, when patients are asked to categorize names as famous or nonfamous, their response times are faster to a famous name when the name is primed by a picture of a semantically related face (e. [sent-38, score-0.286]

17 They argue that conscious perception is associated with a certain minimal quality of representation. [sent-47, score-0.155]

18 Dehaene and Naccache (2001) outline a framework based on Baars’ (1989) notion of conscious states as residing in a global workspace. [sent-48, score-0.192]

19 The attentional blink is observed in experiments where participants are shown a rapid series of stimuli, which includes two targets (T1 and T2). [sent-55, score-0.175]

20 In this account, conscious access is achieved via resonance between posterior and frontal areas. [sent-62, score-0.221]

21 models might not seem to have much in common, they both make claims concerning what is required to achieve functional connectivity between perceptual and response systems. [sent-65, score-0.31]

22 focus on a pathway through which representations can be communicated. [sent-68, score-0.377]

23 Mathis and Mozer (1996) describe an architecture with processing modules for perceptual and response processes, implemented as attractor neural nets. [sent-70, score-0.368]

24 They argue that in order for a representation in some perceptual module to be assured of inﬂuencing a response module, (a) it must have certain characteristics–temporal persistence and well-formedness– which is quite similar to Farah et al. [sent-71, score-0.466]

25 First, it addressed classical subliminal priming data, but more reliable data has recently been reported. [sent-75, score-0.66]

26 The framework characterizes the transmission of information from perceptual to response systems, and how the time course of information transmission changes with experience (i. [sent-79, score-0.511]

27 used this framework to account for a variety of facilitation effects from supraliminal repetition priming. [sent-83, score-0.213]

28 The framework is formalized as a probabilistic model: the pathway input and output are random variables and microinference in a pathway is carried out by Bayesian belief revision. [sent-86, score-0.828]

29 To elaborate, consider a pathway whose input at time t is a discrete random variable, denoted X(t), which can assume values x1 , x2 , x3 , . [sent-88, score-0.377]

30 Similarly, the output of the pathway at time t is a discrete random variable, denoted Y (t), which can assume values y1 , y2 , y3 , . [sent-92, score-0.416]

31 For example, in the letter-naming task, the input to the perceptual pathway would be one of nx = 52 visual patterns corresponding to the upper- and lower-case letters of the alphabet, and the output is one of ny = 26 letter identities. [sent-96, score-0.695]

32 A pathway is modeled as a dynamic Bayes network; the minimal version of the model used in the present simulations is simply a hidden Markov model, where the X(t) are observations and the Y (t) are inferred state (see Figure 1a). [sent-104, score-0.377]

33 Figure 1b illustrates the time course of inference in a single pathway with 52 input and 26 output alternatives and two-to-one associations. [sent-106, score-0.445]

34 2 0 Q Time Figure 1: (a) basic pathway architecture—a hidden Markov model; (b) time course of inference in a pathway when the letter O is presented, causing activation of both O and the visually similar Q. [sent-118, score-0.923]

35 The exact shape of the speed-accuracy curve—the pathway dynamics—are determined by three probability distributions, which embody the knowledge and past experience of the model. [sent-119, score-0.455]

36 Second, P(Y (t) | Y (t − 1)) characterizes how the pathway output evolves over time. [sent-121, score-0.416]

37 We parameterize this distribution as P(X(t) = xi |Y (t) = yj ) ∼ 1 + k γik αkj , where αij indicates the frequency of experience with the association between states xi and yj , and γik speciﬁes the similarity between states xi and xk . [sent-127, score-0.322]

38 Fundamental to the framework is the assumption that with each experience, a pathway becomes more efﬁcient at processing an input. [sent-130, score-0.412]

39 In Mozer, Colagrosso, and Huber (2002, 2003), we propose two distinct mechanisms to model phenomena of supraliminal priming. [sent-132, score-0.207]

40 First, the association frequencies, αij , are increased following a trial in which xi leads to activation of yj , resulting in more efﬁcient transmission of information, corresponding to an increased slope of the solid line in Figure 1b. [sent-133, score-0.26]

41 The increase is Hebbian, based on the maximum activation achieved by xi and yj : ∆αij = η maxt P(X(t) = xi )P(Y (t) = yj ), where η is a step size. [sent-134, score-0.186]

42 In modeling data from supraliminal priming, we found that the increases to association frequencies are long lasting, but the increases to the priors decay over the course of a few minutes or a few trials. [sent-136, score-0.252]

43 4 Access Consciousness and PIT We have described the operation of a single pathway, but to model any cognitive task, we require a series of pathways in cascade. [sent-138, score-0.163]

44 For a simple choice task, we use a percpetual pathway cascaded to a response pathway. [sent-139, score-0.495]

45 The interconnection between the pathways is achieved by copying the output of the perceptual pathway, Y p (t), to the input of the response pathway, X r (t), at each time t. [sent-140, score-0.457]

46 Considering the output of the perceptual pathway, access is achieved when: (1) the output representation is sufﬁcient to trigger the correct behavior in the response pathway, and (2) the perceptual and response pathways are functionally interconnected. [sent-142, score-0.958]

47 In more general terms, access for a perceptual pathway output requires that these two condi- tions be met not just for a speciﬁc response pathway, but for arbitrary response pathways (e. [sent-143, score-1.063]

48 As we will show, a brieﬂy presented stimulus fails to achieve a representation that supports choice and naming responses. [sent-148, score-0.246]

49 Nonetheless, the stimulus evokes activity in the perceptual pathway. [sent-149, score-0.243]

50 Because perceptual priming depends on the magnitude of the activation in the perceptual pathway, not on the activation being communicated to response pathways, the framework is consistent with the notion of priming occurring in the absence of awareness. [sent-150, score-1.679]

51 1 Simulation of Bar and Biederman (1998) Bar and Biederman (1998) presented a sequence of masked line drawings of objects and asked participants to name the objects, even if they had to guess. [sent-152, score-0.263]

52 If the guess was incorrect, participants were required to choose the object name from a set of four alternatives. [sent-153, score-0.213]

53 Unbeknownst to the participant, some of the drawings in the series were repeated, and Bar and Biederman were interested in whether participants would beneﬁt from the ﬁrst presentation even if it could not be identiﬁed. [sent-154, score-0.249]

54 Accuracy did improve, though not as much, if the same shape was presented in a different position, but not if a different drawing of the same object was presented, suggesting a locus of priming early in the visual stream. [sent-159, score-0.594]

55 The priming is ﬁrmly subliminal, because participants were not only unable to name objects on the ﬁrst presentation, but their fouralternative forced choice (4AFC) performance was not much above chance (28. [sent-162, score-0.698]

56 To model these phenomena, we created a response pathway with ﬁfty states representing names of objects that are used in the experiment, e. [sent-164, score-0.583]

57 We also created a perceptual pathway with states representing visual patterns that correspond to the names in the response pathway. [sent-167, score-0.775]

58 Following the experimental design, every object identity was instantiated in two distinct shapes, and every shape could be in one of nine different visualﬁeld positions, leading to 900 distinct states in the perceptual pathway to model the possible visual stimuli. [sent-168, score-0.799]

59 If two perceptual states, xi and xk are the same shape in different positions, they are assigned a similarity coefﬁcient γik = 0. [sent-170, score-0.267]

60 The association frequency, α, for valid associations in the perceptual pathway was 22, and the response pathway 18. [sent-172, score-1.162]

61 Speciﬁcally, priming is greatest for the same shape in the same position, some priming occurs for the same shape in a different position, and no substantial priming occurs for the different shape. [sent-178, score-1.517]

62 Figure 3a shows the time course of activation of a stimulus representation in the perceptual pathway when the stimulus is presented for 50 iterations, on both the ﬁrst and third presentations. [sent-179, score-0.835]

63 The third presentation was chosen instead of the second to make the effect of priming clearer. [sent-180, score-0.559]

64 Even though a shape cannot be named on the ﬁrst presentation, partial information about the shape may nonetheless be available for report. [sent-181, score-0.16]

65 In past work, we modeled forced-choice tasks using a response pathway with only the alternatives under consideration. [sent-183, score-0.495]

66 White bar: accuracy on ﬁrst presentation of a prime object. [sent-186, score-0.241]

67 forced-choice output pathways having output distributions N (t) and F (t), which are linked via the perceptual state, Y p (t). [sent-189, score-0.378]

68 We therefore used a shortcut in which a single response pathway is used, augmented with a simple three-node belief net (Figure 3b) to capture the dependence between naming and forced choice. [sent-192, score-0.649]

69 The belief net has a response pathway node Y r (t) connected to F (t) and N (t), with conditional distribution P (N (t) = ni |Y r (t) = yj ) = θδij + (1 − θ)/|Y r |, and an analogous distribution for P (F (t) = fi |Y r (t) = yj ). [sent-193, score-0.587]

70 The free parameter θ determines how veridically naming and forced-choice actions reﬂect response-pathway output. [sent-194, score-0.154]

71 Over a range of θ, θ < 1, the model obtains forced-choice performance near chance on the ﬁrst presentation when the naming response is incorrect. [sent-195, score-0.354]

72 ) Thus, by the stringent criterion of 4AFC, the model shows no access consciousness, and therefore illustrates a dissociation between priming and access consciousness. [sent-202, score-0.699]

73 These distractors are critical: with unrelated distractors, the model’s 4AFC performance is signiﬁcantly above chance, illustrating that a perceptual representation can be adequate to support some responses but not others, as Farah et al. [sent-204, score-0.264]

74 2 Simulation of Abrams and Greenwald (2000) During an initial phase of the experiment, participants categorized 24 clearly visible target words as pleasant (e. [sent-207, score-0.33]

75 In a second phase, participants were asked to classify the same targets and were given a response deadline to induce errors. [sent-213, score-0.293]

76 The targets were preceded by masked primes that could not be identiﬁed. [sent-214, score-0.176]

77 Of interest is the effective valence (or EV) of the target for different prime types, deﬁned as the error rate difference between unpleasant and pleasant targets. [sent-215, score-0.435]

78 As one would expect, pleasant primes resulted in a positive EV, unpleasant primes in a negative EV. [sent-217, score-0.472]

79 Of critical interest is the ﬁnding that a nonword prime formed by recombining two pleasant targets (e. [sent-218, score-0.419]

80 Note that the results depend critically on the ﬁrst phase of the experiment, which gave participants extensive practice on a relatively small set of words that were then used as and recombined to form primes. [sent-224, score-0.208]

81 1 0 targets hulip-type tumor-type orphans Figure 4: Effective valence of primes in the Abrams and Greenwald (2000) experiment for human subjects (black bars) and PIT model (grey bars). [sent-237, score-0.294]

82 HULIP-type primes are almost as strong as target repetitions, and TUMOR-type primes have a positive valence, contrary to the meaning of the word. [sent-238, score-0.252]

83 In the perceptual pathway, similarity structure was based on letter overlap, so that HULIP was similar to both TULIP and HUMOR, with γ = 0. [sent-241, score-0.27]

84 The initial phase of the experiment was simulated by repeated supraliminal presentation of words, which increased the association frequencies in all three pathways through the ∆αij learning rule. [sent-249, score-0.489]

85 Long-term supraliminal priming is essential in establishing the association strengths, as we’ll explain. [sent-250, score-0.672]

86 Short-term subliminal priming also plays a key role in the experiment. [sent-251, score-0.66]

87 During the second phase of the experiment, residual activity from the prime—primarily in the judgement pathway—biases the response to the target. [sent-252, score-0.261]

88 Residual activation of the prime is present even if the representation of the prime does not reach sufﬁcient strength that it could be named or otherwise reported. [sent-253, score-0.481]

89 When a HULIP -type prime is presented, HUMOR and TULIP become active in the semantic pathway because of their visual similarity to HULIP. [sent-255, score-0.66]

90 Partial activation of these two practiced words pushes the judgement pathway toward a pleasant response, resulting in a positive EV. [sent-256, score-0.704]

91 When a TUMOR-type prime is presented, three different words become active in the semantic pathway: HUMOR, TULIP, and TUMOR itself. [sent-257, score-0.251]

92 Although TUMOR is more active, it was not one of the words studied during the initial phase of the experiment, and as a result, it has a relatively weak association to the unpleasant judgement, in contrast to the other two words which have strong associations to the pleasant judgement. [sent-258, score-0.442]

93 Orphan primes have little effect because they were not studied during the initial phase of the experiment, and consequently their association to pleasant and unpleasant judgements is also weak. [sent-259, score-0.457]

94 In summary, activation of the prime along a critical, well-practiced pathway may not be sufﬁcient to support an overt naming response, yet it may be sufﬁcient to bias the processing of the immediately following target. [sent-260, score-0.817]

95 5 Discussion An important contribution of this work has been to demonstrate that speciﬁc experimental results relating to access consciousness and subliminal priming can be interpreted in a concrete computational framework. [sent-261, score-1.024]

96 Although the PIT framework is not completely developed, it nonetheless makes a clear prediction: that subliminal priming is can never be stronger than supraliminal priming, because the maximal activation of subliminal primes is never greater than that of supraliminal primes. [sent-268, score-1.396]

97 One might argue that many theoretical frameworks might predict the same, but no other computational model is sufﬁciently well developed—in terms of addressing both priming and access consciousness—to make this prediction. [sent-269, score-0.633]

98 In its current stage of development, a weakness of the PIT framework is that it is silent as to how perceptual and response pathways become ﬂexibly interconnected based on task demands. [sent-270, score-0.453]

99 model suggests that once a representation enters the global workspace, all response modules can access it, but the model does not specify how the appropriate perceptual module wins the competition to enter the global workspace, or how the appropriate response module is activated. [sent-272, score-0.687]

100 Mechanisms of long-term repetition priming and skill reﬁnement: A probabilistic pathway model. [sent-349, score-0.906]

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When used with one-reason stopping and decision building blocks, the mean accuracy of the 362,880 (9!) cue orders is 70%, equivalent to the performance expected from Minimalist. The accuracy of the validity order, 74.2%, falls toward the upper end of the accuracy range (62-75.8%), but there are still 7421 cue orders that do better than the validity order. The frugality of the search orders ranges from 2.53 cues per decision to 4.67, with a mean of 3.34 corresponding to using Minimalist; TTB has a frugality of 4.23, implying that most orders are more frugal. Thus, there are many accurate and frugal cue orders that could be found—a satisficing decision maker not requiring optimal performance need only land on one. An ordering problem of this kind has been studied in computer science for nearly four decades, and can provide us with a set of potential heuristics to test. Consider the case of a set of data records arranged in a list, each of which will be required during a set of retrievals with a particular probability pi. On each retrieval, a key is given (e.g. a record’s title) and the list is searched from the front to the end until the desired record, matching that key, is found. The goal is to minimize the mean search time for accessing the records in this list, for which the optimal ordering is in decreasing order of pi. But if these retrieval probabilities are not known ahead of time, how can the list be ordered after each successive retrieval to achieve fast access? This is the problem of self-organizing sequential search [11], [12]. A variety of simple sequential search heuristics have been proposed for this problem, centering on three main approaches: (1) transpose, in which a retrieved record is moved one position closer to the front of the list (i.e., swapping with the record in front of it); (2) move-to-front (MTF), in which a retrieved record is put at the front of the list, and all other records remain in the same relative order; and (3) count, in which a tally is kept of the number of times each record is retrieved, and the list is reordered in decreasing order of this tally after each retrieval. Because count rules require storing additional information, more attention has focused on the memory-free transposition and MTF rules. Analytic and simulation results (reviewed in [12]) have shown that while transposition rules can come closer to the optimal order asymptotically, in the short run MTF rules converge more quickly (as can count rules). This may make MTF (and count) rules more appealing as models of cue order learning by humans facing small numbers of decision trials. Furthermore, MTF rules are more responsive to local structure in the environment (e.g., clumped retrievals over time of a few records), and transposition can result in very poor performance under some circumstances (e.g., when neighboring pairs of “popular” records get trapped at the end of the list by repeatedly swapping places). It is important to note that there are important differences between the selforganizing sequential search problem and the cue-ordering problem we address here. In particular, when a record is sought that matches a particular key, search proceeds until the correct record is found. In contrast, when a decision is made lexicographically and the list of cues is searched through, there is no one “correct” cue to find—each cue may or may not discriminate (allow a decision to be made). Furthermore, once a discriminating cue is found, it may not even make the right decision. Thus, given feedback about whether a decision was right or wrong, a discriminating cue could potentially be moved up or down in the ordered list. This dissociation between making a decision or not (based on the cue discrimination rates), and making a right or wrong decision (based on the cue validities), means that there are two ordering criteria in this problem—frugality and accuracy—as opposed to the single order—search time—for records based on their retrieval probability pi . Because record search time corresponds to cue frugality, the heuristics that work well for the self-organizing sequential search task are likely to produce orders that emphasize frugality (reflecting cue discrimination rates) over accuracy in the cue-ordering task. Nonetheless, these heuristics offer a useful starting point for exploring cue-ordering rules. 2.1 The cue-ordering rules We focus on search order construction processes that are psychologically plausible by being frugal both in terms of information storage and in terms of computation. The decision situation we explore is different from the one assumed by Juslin and Persson [10] who strongly differentiate learning about objects from later making decisions about them. Instead we assume a learning-while-doing situation, consisting of tasks that have to be done repeatedly with feedback after each trial about the adequacy of one’s decision. For instance, we can observe on multiple occasions which of two supermarket checkout lines, the one we have chosen or (more likely) another one, is faster, and associate this outcome with cues including the lines’ lengths and the ages of their respective cashiers. In such situations, decision makers can learn about the differential usefulness of cues for solving the task via the feedback received over time. We compare several explicitly defined ordering rules that construct cue orders for use by lexicographic decision mechanisms applied to a particular probabilistic inference task: forced choice paired comparison, in which a decision maker has to infer which of two objects, each described by a set of binary cues, is “bigger” on a criterion—just the task for which TTB was formulated. After an inference has been made, feedback is given about whether a decision was right or wrong. Therefore, the order-learning algorithm has information about which cues were looked up, whether a cue discriminated, and whether a discriminating cue led to the right or wrong decision. The rules we propose differ in which pieces of information they use and how they use them. We classify the learning rules based on their memory requirement—high versus low—and their computational requirements in terms of full or partial reordering (see Table 1). Table 1: Learning rules classified by memory and computational requirements High memory load, complete reordering High memory load, local reordering Low memory load, local reordering Validity: reorders cues based on their current validity Tally swap: moves cue up (down) one position if it has made a correct (incorrect) decision if its tally of correct minus incorrect decisions is ( ) than that of next higher (lower) cue Simple swap: moves cue up one position after correct decision, and down after an incorrect decision Tally: reorders cues by number of correct minus incorrect decisions made so far Associative/delta rule: reorders cues by learned association strength Move-to-front (2 forms): Take The Last (TTL): moves discriminating cue to front TTL-correct: moves cue to front only if it correctly discriminates The validity rule, a type of count rule, is the most demanding of the rules we consider in terms of both memory requirements and computational complexity. It keeps a count of all discriminations made by a cue so far (in all the times that the cue was looked up) and a separate count of all the correct discriminations. Therefore, memory load is comparatively high. The validity of each cue is determined by dividing its current correct discrimination count by its total discrimination count. Based on these values computed after each decision, the rule reorders the whole set of cues from highest to lowest validity. The tally rule only keeps one count per cue, storing the number of correct decisions made by that cue so far minus the number of incorrect decisions. If a cue discriminates correctly on a given trial, one point is added to its tally, if it leads to an incorrect decision, one point is subtracted. The tally rule is less demanding in terms of memory and computation: Only one count is kept, no division is required. The simple swap rule uses the transposition rather than count approach. This rule has no memory of cue performance other than an ordered list of all cues, and just moves a cue up one position in this list whenever it leads to a correct decision, and down if it leads to an incorrect decision. In other words, a correctly deciding cue swaps positions with its nearest neighbor upwards in the cue order, and an incorrectly deciding cue swaps positions with its nearest neighbor downwards. The tally swap rule is a hybrid of the simple swap rule and the tally rule. It keeps a tally of correct minus incorrect discriminations per cue so far (so memory load is high) but only locally swaps cues: When a cue makes a correct decision and its tally is greater than or equal to that of its upward neighbor, the two cues swap positions. When a cue makes an incorrect decision and its tally is smaller than or equal to that of its downward neighbor, the two cues also swap positions. We also evaluate two types of move-to-front rules. First, the Take The Last (TTL) rule moves the last discriminating cue (that is, whichever cue was found to discriminate for the current decision) to the front of the order. This is equivalent to the Take The Last heuristic [6], [7], which uses a memory of cues that discriminated in the past to determine cue search order for subsequent decisions. Second, TTLcorrect moves the last discriminating cue to the front of the order only if it correctly discriminated; otherwise, the cue order remains unchanged. This rule thus takes accuracy as well as frugality into account. Finally, we include an associative learning rule that uses the delta rule to update cue weights according to whether they make correct or incorrect discriminations, and then reorders all cues in decreasing order of this weight after each decision. This corresponds to a simple network with nine input units encoding the difference in cue value between the two objects (A and B) being decided on (i.e., ini = -1 if cuei(A) cuei(B), and 0 if cuei(A)=cuei(B) or cuei was not checked) and with one output unit whose target value encodes the correct decision (t = 1 if criterion(A)>criterion(B), otherwise -1), and with the weights between inputs and output updated according to wi = lr · (t - ini·wi) · ini with learning rate lr = 0.1. We expect this rule to behave similarly to Oliver’s rule initially (moving a cue to the front of the list by giving it the largest weight when weights are small) and to swap later on (moving cues only a short distance once weights are larger). 3 Si mul ati on Study of Si mpl e O r de r i ng Rul e s To test the performance of these order learning rules, we use the German cities data set [6], [7], consisting of the 83 largest-population German cities (those with more than 100,000 inhabitants), described on 9 cues that give some information about population size. Discrimination rate and validity of the cues are negatively correlated (r = -.47). We present results averaged over 10,000 learning trials for each rule, starting from random initial cue orders. Each trial consisted of 100 decisions between randomly selected decision pairs. For each decision, the current cue order was used to look up cues until a discriminating cue was found, which was used to make the decision (employing a onereason or lexicographic decision strategy). After each decision, the cue order was updated using the particular order-learning rule. We start by considering the cumulative accuracies (i.e., online or amortized performance—[12]) of the rules, defined as the total percentage of correct decisions made so far at any point in the learning process. The contrasting measure of offline accuracy—how well the current learned cue order would do if it were applied to the entire test set—will be subsequently reported (see Figure 1). For all but the move-to-front rules, cumulative accuracies soon rise above that of the Minimalist heuristic (proportion correct = .70) which looks up cues in random order and thus serves as a lower benchmark. However, at least throughout the first 100 decisions, cumulative accuracies stay well below the (offline) accuracy that would be achieved by using TTB for all decisions (proportion correct = .74), looking up cues in the true order of their ecological validities. Except for the move-to-front rules, whose cumulative accuracies are very close to Minimalist (mean proportion correct in 100 decisions: TTL: .701; TTL-correct: .704), all learning rules perform on a surprisingly similar level, with less than one percentage point difference in favor of the most demanding rule (i.e., delta rule: .719) compared to the least (i.e., simple swap: .711; for comparison: tally swap: .715; tally: .716; validity learning rule: .719). Offline accuracies are slightly higher, again with the exception of the move to front rules (TTL: .699; TTL-correct: .702; simple swap: .714; tally swap: .719; tally: .721; validity learning rule: .724; delta rule: .725; see Figure 1). In longer runs (10,000 decisions) the validity learning rule is able to converge on TTB’s accuracy, but the tally rule’s performance changes little (to .73). Figure 1: Mean offline accuracy of order learning rules Figure 2: Mean offline frugality of order learning rules All learning rules are, however, more frugal than TTB, and even more frugal than Minimalist, both in terms of online as well as offline frugality. Let us focus on their offline frugality (see Figure 2): On average, the rules look up fewer cues than Minimalist before reaching a decision. There is little difference between the associative rule, the tallying rules and the swapping rules (mean number of cues looked up in 100 decisions: delta rule: 3.20; validity learning rule: 3.21; tally: 3.01; tally swap: 3.04; simple swap: 3.13). Most frugal are the two move-to front rules (TTL-correct: 2.87; TTL: 2.83). Consistent with this finding, all of the learning rules lead to cue orders that show positive correlations with the discrimination rate cue order (reaching the following values after 100 decisions: validity learning rule: r = .18; tally: r = .29; tally swap: r = .24; simple swap: r = .18; TTL-correct: r = .48; TTL: r = .56). This means that cues that often lead to discriminations are more likely to end up in the first positions of the order. This is especially true for the move-to-front rules. In contrast, the cue orders resulting from all learning rules but the validity learning rule do not correlate or correlate negatively with the validity cue order, and even the correlations of the cue orders resulting from the validity learning rule after 100 decisions only reach an average r = .12. But why would the discrimination rates of cues exert more of a pull on cue order than validity, even when the validity learning rule is applied? As mentioned earlier, this is what we would expect for the move-to-front rules, but it was unexpected for the other rules. Part of the explanation comes from the fact that in the city data set we used for the simulations, validity and discrimination rate of cues are negatively correlated. Having a low discrimination rate means that a cue has little chance to be used and hence to demonstrate its high validity. Whatever learning rule is used, if such a cue is displaced downward to the lower end of the order by other cues, it may have few chances to escape to the higher ranks where it belongs. The problem is that when a decision pair is finally encountered for which that cue would lead to a correct decision, it is unlikely to be checked because other, more discriminating although less valid, cues are looked up before and already bring about a decision. Thus, because one-reason decision making is intertwined with the learning mechanism and so influences which cues can be learned about, what mainly makes a cue come early in the order is producing a high number of correct decisions and not so much a high ratio of correct discriminations to total discriminations regardless of base rates. This argument indicates that performance may differ in environments where cue validities and discrimination rates correlate positively. We tested the learning rules on one such data set (r=.52) of mammal species life expectancies, predicted from 9 cues. It also differs from the cities environment with a greater difference between TTB’s and Minimalist’s performance (6.5 vs. 4 percentage points). In terms of offline accuracy, the validity learning rule now indeed more closely approaches TTB’s accuracy after 100 decisions (.773 vs. .782)., The tally rule, in contrast, behaves very much as in the cities environment, reaching an accuracy of .752, halfway between TTB and Minimalist (accuracy =.716). Thus only some learning rules can profit from the positive correlation. 4 D i s c u s s i on Most of the simpler cue order learning rules we have proposed do not fall far behind a validity learning rule in accuracy, and although the move-to-front rules cannot beat the accuracy achieved if cues were selected randomly, they compensate for this failure by being highly frugal. Interestingly, the rules that do achieve higher accuracy than Minimalist also beat random cue selection in terms of frugality. On the other hand, all rules, even the delta rule and the validity learning rule, stay below TTB’s accuracy across a relatively high number of decisions. But often it is necessary to make good decisions without much experience. Therefore, learning rules should be preferred that quickly lead to orders with good performance. The relatively complex rules with relatively high memory requirement, i.e., the delta and the validity learning rule, but also the tally learning rule, more quickly rise in accuracy compared the rules with lower requirements. Especially the tally rule thus represents a good compromise between cost, correctness and psychological plausibility considerations. Remember that the rules based on tallies assume full memory of all correct minus incorrect decisions made by a cue so far. But this does not make the rule implausible, at least from a psychological perspective, even though computer scientists were reluctant to adopt such counting approaches because of their extra memory requirements. There is considerable evidence that people are actually very good at remembering the frequencies of events. Hasher and Zacks [13] conclude from a wide range of studies that frequencies are encoded in an automatic way, implying that people are sensitive to this information without intention or special effort. Estes [14] pointed out the role frequencies play in decision making as a shortcut for probabilities. Further, the tally rule and the tally swap rule are comparatively simple, not having to keep track of base rates or perform divisions as does the validity rule. From the other side, the simple swap and move to front rules may not be much simpler, because storing a cue order may be about as demanding as storing a set of tallies. We have run experiments (reported elsewhere) in which indeed the tally swap rule best accounts for people’s actual processes of ordering cues. Our goal in this paper was to explore how well simple cue-ordering rules could work in conjunction with lexicographic decision strategies. This is important because it is necessary to take into account the set-up costs of a heuristic in addition to its application costs when considering the mechanism’s overall simplicity. As the example of the validity search order of TTB shows, what is easy to apply may not necessarily be so easy to set up. But simple rules can also be at work in the construction of a heuristic’s building blocks. We have proposed such rules for the construction of one building block, the search order. Simple learning rules inspired by research in computer science can enable a one-reason decision heuristic to perform only slightly worse than if it had full knowledge of cue validities from the very beginning. Giving up the assumption of full a priori knowledge for the slight decrease in accuracy seems like a reasonable bargain: Through the addition of learning rules, one-reason decision heuristics might lose some of their appeal to decision theorists who were surprised by the performance of such simple mechanisms compared to more complex algorithms, but they gain psychological plausibility and so become more attractive as explanations for human decision behavior. References [1] Fishburn, P.C. (1974). Lexicographic orders, utilities and decision rules: A survey. Management Science, 20, 1442-1471. [2] Payne, J.W., Bettman, J.R., & Johnson, E.J. (1993). The adaptive decision maker. New York: Cambridge University Press. [3] Bröder, A. (2000). Assessing the empirical validity of the “Take-The-Best” heuristic as a model of human probabilistic inference. Journal of Experimental Psychology: Learning, Memory, and Cognition, 26 (5), 1332-1346. [4] Bröder, A. (2003). Decision making with the “adaptive toolbox”: Influence of environmental structure, intelligence, and working memory load. Journal of Experimental Psychology: Learning, Memory, & Cognition, 29, 611-625. [5] Gigerenzer, G., Todd, P.M., & The ABC Research Group (1999). Simple heuristics that make us smart. New York: Oxford University Press. [6] Gigerenzer, G., & Goldstein, D.G. (1996). Reasoning the fast and frugal way: Models of bounded rationality. Psychological Review, 103 (4), 650-669. [7] Gigerenzer, G., & Goldstein, D.G. (1999). Betting on one good reason: The Take The Best Heuristic. In G. Gigerenzer, P.M. Todd & The ABC Research Group, Simple heuristics that make us smart. New York: Oxford University Press. [8] Czerlinski, J., Gigerenzer, G., & Goldstein, D.G. (1999). How good are simple heuristics? In G. Gigerenzer, P.M. Todd & The ABC Research Group, Simple heuristics that make us smart. New York: Oxford University Press. [9] Newell, B.R., & Shanks, D.R. (2003). Take the best or look at the rest? Factors influencing ‘one-reason’ decision making. Journal of Experimental Psychology: Learning, Memory, and Cognition, 29, 53-65. [10] Juslin, P., & Persson, M. (2002). PROBabilities from EXemplars (PROBEX): a “lazy” algorithm for probabilistic inference from generic knowledge. Cognitive Science, 26, 563-607. [11] Rivest, R. (1976). On self-organizing sequential search heuristics. Communications of the ACM, 19(2), 63-67. [12] Bentley, J.L. & McGeoch, C.C. (1985). Amortized analyses of self-organizing sequential search heuristics. Communications of the ACM, 28(4), 404-411. [13] Hasher, L., & Zacks, R.T. (1984). Automatic Processing of fundamental information: The case of frequency of occurrence. American Psychologist, 39, 1372-1388. [14] Estes, W.K. (1976). The cognitive side of probability learning. Psychological Review, 83, 3764.

5 0.47810075 46 nips-2004-Constraining a Bayesian Model of Human Visual Speed Perception

Author: Alan Stocker, Eero P. Simoncelli

Abstract: It has been demonstrated that basic aspects of human visual motion perception are qualitatively consistent with a Bayesian estimation framework, where the prior probability distribution on velocity favors slow speeds. Here, we present a reﬁned probabilistic model that can account for the typical trial-to-trial variabilities observed in psychophysical speed perception experiments. We also show that data from such experiments can be used to constrain both the likelihood and prior functions of the model. Speciﬁcally, we measured matching speeds and thresholds in a two-alternative forced choice speed discrimination task. Parametric ﬁts to the data reveal that the likelihood function is well approximated by a LogNormal distribution with a characteristic contrast-dependent variance, and that the prior distribution on velocity exhibits signiﬁcantly heavier tails than a Gaussian, and approximately follows a power-law function. Humans do not perceive visual motion veridically. Various psychophysical experiments have shown that the perceived speed of visual stimuli is affected by stimulus contrast, with low contrast stimuli being perceived to move slower than high contrast ones [1, 2]. Computational models have been suggested that can qualitatively explain these perceptual effects. Commonly, they assume the perception of visual motion to be optimal either within a deterministic framework with a regularization constraint that biases the solution toward zero motion [3, 4], or within a probabilistic framework of Bayesian estimation with a prior that favors slow velocities [5, 6]. The solutions resulting from these two frameworks are similar (and in some cases identical), but the probabilistic framework provides a more principled formulation of the problem in terms of meaningful probabilistic components. Speciﬁcally, Bayesian approaches rely on a likelihood function that expresses the relationship between the noisy measurements and the quantity to be estimated, and a prior distribution that expresses the probability of encountering any particular value of that quantity. A probabilistic model can also provide a richer description, by deﬁning a full probability density over the set of possible “percepts”, rather than just a single value. Numerous analyses of psychophysical experiments have made use of such distributions within the framework of signal detection theory in order to model perceptual behavior [7]. Previous work has shown that an ideal Bayesian observer model based on Gaussian forms µ posterior low contrast probability density probability density high contrast likelihood prior a posterior likelihood prior v ˆ v ˆ visual speed µ b visual speed Figure 1: Bayesian model of visual speed perception. a) For a high contrast stimulus, the likelihood has a narrow width (a high signal-to-noise ratio) and the prior induces only a small shift µ of the mean v of the posterior. b) For a low contrast stimuli, the measurement ˆ is noisy, leading to a wider likelihood. The shift µ is much larger and the perceived speed lower than under condition (a). for both likelihood and prior is sufﬁcient to capture the basic qualitative features of global translational motion perception [5, 6]. But the behavior of the resulting model deviates systematically from human perceptual data, most importantly with regard to trial-to-trial variability and the precise form of interaction between contrast and perceived speed. A recent article achieved better ﬁts for the model under the assumption that human contrast perception saturates [8]. In order to advance the theory of Bayesian perception and provide signiﬁcant constraints on models of neural implementation, it seems essential to constrain quantitatively both the likelihood function and the prior probability distribution. In previous work, the proposed likelihood functions were derived from the brightness constancy constraint [5, 6] or other generative principles [9]. Also, previous approaches deﬁned the prior distribution based on general assumptions and computational convenience, typically choosing a Gaussian with zero mean, although a Laplacian prior has also been suggested [4]. In this paper, we develop a more general form of Bayesian model for speed perception that can account for trial-to-trial variability. We use psychophysical speed discrimination data in order to constrain both the likelihood and the prior function. 1 1.1 Probabilistic Model of Visual Speed Perception Ideal Bayesian Observer Assume that an observer wants to obtain an estimate for a variable v based on a measurement m that she/he performs. A Bayesian observer “knows” that the measurement device is not ideal and therefore, the measurement m is affected by noise. Hence, this observer combines the information gained by the measurement m with a priori knowledge about v. Doing so (and assuming that the prior knowledge is valid), the observer will – on average – perform better in estimating v than just trusting the measurements m. According to Bayes’ rule 1 p(v|m) = p(m|v)p(v) (1) α the probability of perceiving v given m (posterior) is the product of the likelihood of v for a particular measurements m and the a priori knowledge about the estimated variable v (prior). α is a normalization constant independent of v that ensures that the posterior is a proper probability distribution. ^ ^ P(v2 > v1) 1 + Pcum=0.5 0 a b Pcum=0.875 vmatch vthres v2 Figure 2: 2AFC speed discrimination experiment. a) Two patches of drifting gratings were displayed simultaneously (motion without movement). The subject was asked to ﬁxate the center cross and decide after the presentation which of the two gratings was moving faster. b) A typical psychometric curve obtained under such paradigm. The dots represent the empirical probability that the subject perceived stimulus2 moving faster than stimulus1. The speed of stimulus1 was ﬁxed while v2 is varied. The point of subjective equality, vmatch , is the value of v2 for which Pcum = 0.5. The threshold velocity vthresh is the velocity for which Pcum = 0.875. It is important to note that the measurement m is an internal variable of the observer and is not necessarily represented in the same space as v. The likelihood embodies both the mapping from v to m and the noise in this mapping. So far, we assume that there is a monotonic function f (v) : v → vm that maps v into the same space as m (m-space). Doing so allows us to analytically treat m and vm in the same space. We will later propose a suitable form of the mapping function f (v). An ideal Bayesian observer selects the estimate that minimizes the expected loss, given the posterior and a loss function. We assume a least-squares loss function. Then, the optimal estimate v is the mean of the posterior in Equation (1). It is easy to see why this model ˆ of a Bayesian observer is consistent with the fact that perceived speed decreases with contrast. The width of the likelihood varies inversely with the accuracy of the measurements performed by the observer, which presumably decreases with decreasing contrast due to a decreasing signal-to-noise ratio. As illustrated in Figure 1, the shift in perceived speed towards slow velocities grows with the width of the likelihood, and thus a Bayesian model can qualitatively explain the psychophysical results [1]. 1.2 Two Alternative Forced Choice Experiment We would like to examine perceived speeds under a wide range of conditions in order to constrain a Bayesian model. Unfortunately, perceived speed is an internal variable, and it is not obvious how to design an experiment that would allow subjects to express it directly 1 . Perceived speed can only be accessed indirectly by asking the subject to compare the speed of two stimuli. For a given trial, an ideal Bayesian observer in such a two-alternative forced choice (2AFC) experimental paradigm simply decides on the basis of the two trial estimates v1 (stimulus1) and v2 (stimulus2) which stimulus moves faster. Each estimate v is based ˆ ˆ ˆ on a particular measurement m. For a given stimulus with speed v, an ideal Bayesian observer will produce a distribution of estimates p(ˆ|v) because m is noisy. Over trials, v the observers behavior can be described by classical signal detection theory based on the distributions of the estimates, hence e.g. the probability of perceiving stimulus2 moving 1 Although see [10] for an example of determining and even changing the prior of a Bayesian model for a sensorimotor task, where the estimates are more directly accessible. faster than stimulus1 is given as the cumulative probability Pcum (ˆ2 > v1 ) = v ˆ ∞ 0 p(ˆ2 |v2 ) v v2 ˆ 0 p(ˆ1 |v1 ) dˆ1 dˆ2 v v v (2) Pcum describes the full psychometric curve. Figure 2b illustrates the measured psychometric curve and its ﬁt from such an experimental situation. 2 Experimental Methods We measured matching speeds (Pcum = 0.5) and thresholds (Pcum = 0.875) in a 2AFC speed discrimination task. Subjects were presented simultaneously with two circular patches of horizontally drifting sine-wave gratings for the duration of one second (Figure 2a). Patches were 3deg in diameter, and were displayed at 6deg eccentricity to either side of a ﬁxation cross. The stimuli had an identical spatial frequency of 1.5 cycle/deg. One stimulus was considered to be the reference stimulus having one of two different contrast values (c1 =[0.075 0.5]) and one of ﬁve different speed values (u1 =[1 2 4 8 12] deg/sec) while the second stimulus (test) had one of ﬁve different contrast values (c2 =[0.05 0.1 0.2 0.4 0.8]) and a varying speed that was determined by an interleaved staircase procedure. For each condition there were 96 trials. Conditions were randomly interleaved, including a random choice of stimulus identity (test vs. reference) and motion direction (right vs. left). Subjects were asked to ﬁxate during stimulus presentation and select the faster moving stimulus. The threshold experiment differed only in that auditory feedback was given to indicate the correctness of their decision. This did not change the outcome of the experiment but increased signiﬁcantly the quality of the data and thus reduced the number of trials needed. 3 Analysis With the data from the speed discrimination experiments we could in principal apply a parametric ﬁt using Equation (2) to derive the prior and the likelihood, but the optimization is difﬁcult, and the ﬁt might not be well constrained given the amount of data we have obtained. The problem becomes much more tractable given the following weak assumptions: • We consider the prior to be relatively smooth. • We assume that the measurement m is corrupted by additive Gaussian noise with a variance whose dependence on stimulus speed and contrast is separable. • We assume that there is a mapping function f (v) : v → vm that maps v into the space of m (m-space). In that space, the likelihood is convolutional i.e. the noise in the measurement directly deﬁnes the width of the likelihood. These assumptions allow us to relate the psychophysical data to our probabilistic model in a simple way. The following analysis is in the m-space. The point of subjective equality (Pcum = 0.5) is deﬁned as where the expected values of the speed estimates are equal. We write E vm,1 ˆ vm,1 − E µ1 = E vm,2 ˆ = vm,2 − E µ2 (3) where E µ is the expected shift of the perceived speed compared to the veridical speed. For the discrimination threshold experiment, above assumptions imply that the variance var vm of the speed estimates vm is equal for both stimuli. Then, (2) predicts that the ˆ ˆ discrimination threshold is proportional to the standard deviation, thus vm,2 − vm,1 = γ var vm ˆ (4) likelihood a b prior vm Figure 3: Piece-wise approximation We perform a parametric ﬁt by assuming the prior to be piece-wise linear and the likelihood to be LogNormal (Gaussian in the m-space). where γ is a constant that depends on the threshold criterion Pcum and the exact shape of p(ˆm |vm ). v 3.1 Estimating the prior and likelihood In order to extract the prior and the likelihood of our model from the data, we have to ﬁnd a generic local form of the prior and the likelihood and relate them to the mean and the variance of the speed estimates. As illustrated in Figure 3, we assume that the likelihood is Gaussian with a standard deviation σ(c, vm ). Furthermore, the prior is assumed to be wellapproximated by a ﬁrst-order Taylor series expansion over the velocity ranges covered by the likelihood. We parameterize this linear expansion of the prior as p(vm ) = avm + b. We now can derive a posterior for this local approximation of likelihood and prior and then deﬁne the perceived speed shift µ(m). The posterior can be written as 2 vm 1 1 p(m|vm )p(vm ) = [exp(− )(avm + b)] α α 2σ(c, vm )2 where α is the normalization constant ∞ b p(m|vm )p(vm )dvm = π2σ(c, vm )2 α= 2 −∞ p(vm |m) = (5) (6) We can compute µ(m) as the ﬁrst order moment of the posterior for a given m. Exploiting the symmetries around the origin, we ﬁnd ∞ a(m) µ(m) = σ(c, vm )2 vp(vm |m)dvm ≡ (7) b(m) −∞ The expected value of µ(m) is equal to the value of µ at the expected value of the measurement m (which is the stimulus velocity vm ), thus a(vm ) σ(c, vm )2 E µ = µ(m)|m=vm = (8) b(vm ) Similarly, we derive var vm . Because the estimator is deterministic, the variance of the ˆ estimate only depends on the variance of the measurement m. For a given stimulus, the variance of the estimate can be well approximated by ∂ˆm (m) v var vm = var m ( ˆ |m=vm )2 (9) ∂m ∂µ(m) |m=vm )2 ≈ var m = var m (1 − ∂m Under the assumption of a locally smooth prior, the perceived velocity shift remains locally constant. The variance of the perceived speed vm becomes equal to the variance of the ˆ measurement m, which is the variance of the likelihood (in the m-space), thus var vm = σ(c, vm )2 ˆ (10) With (3) and (4), above derivations provide a simple dependency of the psychophysical data to the local parameters of the likelihood and the prior. 3.2 Choosing a Logarithmic speed representation We now want to choose the appropriate mapping function f (v) that maps v to the m-space. We deﬁne the m-space as the space in which the likelihood is Gaussian with a speedindependent width. We have shown that discrimination threshold is proportional to the width of the likelihood (4), (10). Also, we know from the psychophysics literature that visual speed discrimination approximately follows a Weber-Fechner law [11, 12], thus that the discrimination threshold increases roughly proportional with speed and so would the likelihood. A logarithmic speed representation would be compatible with the data and our choice of the likelihood. Hence, we transform the linear speed-domain v into a normalized logarithmic domain according to v + v0 vm = f (v) = ln( ) (11) v0 where v0 is a small normalization constant. The normalization is chosen to account for the expected deviation of equal variance behavior at the low end. Surprisingly, it has been found that neurons in the Medial Temporal area (Area MT) of macaque monkeys have speed-tuning curves that are very well approximated by Gaussians of constant width in above normalized logarithmic space [13]. These neurons are known to play a central role in the representation of motion. It seems natural to assume that they are strongly involved in tasks such as our performed psychophysical experiments. 4 Results Figure 4 shows the contrast dependent shift of speed perception and the speed discrimination threshold data for two subjects. Data points connected with a dashed line represent the relative matching speed (v2 /v1 ) for a particular contrast value c2 of the test stimulus as a function of the speed of the reference stimulus. Error bars are the empirical standard deviation of ﬁts to bootstrapped samples of the data. Clearly, low contrast stimuli are perceived to move slower. The effect, however, varies across the tested speed range and tends to become smaller for higher speeds. The relative discrimination thresholds for two different contrasts as a function of speed show that the Weber-Fechner law holds only approximately. The data are in good agreement with other data from the psychophysics literature [1, 11, 8]. For each subject, data from both experiments were used to compute a parametric leastsquares ﬁt according to (3), (4), (7), and (10). In order to test the assumption of a LogNormal likelihood we allowed the standard deviation to be dependent on contrast and speed, thus σ(c, vm ) = g(c)h(vm ). We split the speed range into six bins (subject2: ﬁve) and parameterized h(vm ) and the ratio a/b accordingly. Similarly, we parameterized g(c) for the seven contrast values. The resulting ﬁts are superimposed as bold lines in Figure 4. Figure 5 shows the ﬁtted parametric values for g(c) and h(v) (plotted in the linear domain), and the reconstructed prior distribution p(v) transformed back to the linear domain. The approximately constant values for h(v) provide evidence that a LogNormal distribution is an appropriate functional description of the likelihood. The resulting values for g(c) suggest for the likelihood width a roughly exponential decaying dependency on contrast with strong saturation for higher contrasts. discrimination threshold (relative) reference stimulus contrast c1: 0.075 0.5 subject 1 normalized matching speed 1.5 contrast c2 1 0.5 1 10 0.075 0.5 0.79 0.5 0.4 0.3 0.2 0.1 0 10 1 contrast: 1 10 discrimination threshold (relative) normalized matching speed subject 2 1.5 contrast c2 1 0.5 10 1 a 0.5 0.4 0.3 0.2 0.1 10 1 1 b speed of reference stimulus [deg/sec] 10 stimulus speed [deg/sec] Figure 4: Speed discrimination data for two subjects. a) The relative matching speed of a test stimulus with different contrast levels (c2 =[0.05 0.1 0.2 0.4 0.8]) to achieve subjective equality with a reference stimulus (two different contrast values c1 ). b) The relative discrimination threshold for two stimuli with equal contrast (c1,2 =[0.075 0.5]). reconstructed prior subject 1 p(v) [unnormalized] 1 Gaussian Power-Law g(c) 1 h(v) 2 0.9 1.5 0.8 0.1 n=-1.41 0.7 1 0.6 0.01 0.5 0.5 0.4 0.3 1 p(v) [unnormalized] subject 2 10 0.1 1 1 1 1 10 1 10 2 0.9 n=-1.35 0.1 1.5 0.8 0.7 1 0.6 0.01 0.5 0.5 0.4 1 speed [deg/sec] 10 0.3 0 0.1 1 contrast speed [deg/sec] Figure 5: Reconstructed prior distribution and parameters of the likelihood function. The reconstructed prior for both subjects show much heavier tails than a Gaussian (dashed ﬁt), approximately following a power-law function with exponent n ≈ −1.4 (bold line). 5 Conclusions We have proposed a probabilistic framework based on a Bayesian ideal observer and standard signal detection theory. We have derived a likelihood function and prior distribution for the estimator, with a fairly conservative set of assumptions, constrained by psychophysical measurements of speed discrimination and matching. The width of the resulting likelihood is nearly constant in the logarithmic speed domain, and decreases approximately exponentially with contrast. The prior expresses a preference for slower speeds, and approximately follows a power-law distribution, thus has much heavier tails than a Gaussian. It would be interesting to compare the here derived prior distributions with measured true distributions of local image velocities that impinge on the retina. Although a number of authors have measured the spatio-temporal structure of natural images [14, e.g. ], it is clearly difﬁcult to extract therefrom the true prior distribution because of the feedback loop formed through movements of the body, head and eyes. Acknowledgments The authors thank all subjects for their participation in the psychophysical experiments. References [1] P. Thompson. Perceived rate of movement depends on contrast. Vision Research, 22:377–380, 1982. [2] L.S. Stone and P. Thompson. Human speed perception is contrast dependent. Vision Research, 32(8):1535–1549, 1992. [3] A. Yuille and N. Grzywacz. A computational theory for the perception of coherent visual motion. Nature, 333(5):71–74, May 1988. [4] Alan Stocker. Constraint Optimization Networks for Visual Motion Perception - Analysis and Synthesis. PhD thesis, Dept. of Physics, Swiss Federal Institute of Technology, Z¨ rich, Switzeru land, March 2002. [5] Eero Simoncelli. Distributed analysis and representation of visual motion. PhD thesis, MIT, Dept. of Electrical Engineering, Cambridge, MA, 1993. [6] Y. Weiss, E. Simoncelli, and E. Adelson. Motion illusions as optimal percept. Nature Neuroscience, 5(6):598–604, June 2002. [7] D.M. Green and J.A. Swets. Signal Detection Theory and Psychophysics. Wiley, New York, 1966. [8] F. H¨ rlimann, D. Kiper, and M. Carandini. Testing the Bayesian model of perceived speed. u Vision Research, 2002. [9] Y. Weiss and D.J. Fleet. Probabilistic Models of the Brain, chapter Velocity Likelihoods in Biological and Machine Vision, pages 77–96. Bradford, 2002. [10] K. Koerding and D. Wolpert. Bayesian integration in sensorimotor learning. 427(15):244–247, January 2004. Nature, [11] Leslie Welch. The perception of moving plaids reveals two motion-processing stages. Nature, 337:734–736, 1989. [12] S. McKee, G. Silvermann, and K. Nakayama. Precise velocity discrimintation despite random variations in temporal frequency and contrast. Vision Research, 26(4):609–619, 1986. [13] C.H. Anderson, H. Nover, and G.C. DeAngelis. Modeling the velocity tuning of macaque MT neurons. Journal of Vision/VSS abstract, 2003. [14] D.W. Dong and J.J. Atick. Statistics of natural time-varying images. Network: Computation in Neural Systems, 6:345–358, 1995.

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