nips nips2005 nips2005-34 knowledge-graph by maker-knowledge-mining

34 nips-2005-Bayesian Surprise Attracts Human Attention

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Author: Laurent Itti, Pierre F. Baldi

Abstract: The concept of surprise is central to sensory processing, adaptation, learning, and attention. Yet, no widely-accepted mathematical theory currently exists to quantitatively characterize surprise elicited by a stimulus or event, for observers that range from single neurons to complex natural or engineered systems. We describe a formal Bayesian deﬁnition of surprise that is the only consistent formulation under minimal axiomatic assumptions. Surprise quantiﬁes how data affects a natural or artiﬁcial observer, by measuring the difference between posterior and prior beliefs of the observer. Using this framework we measure the extent to which humans direct their gaze towards surprising items while watching television and video games. We ﬁnd that subjects are strongly attracted towards surprising locations, with 72% of all human gaze shifts directed towards locations more surprising than the average, a ﬁgure which rises to 84% when considering only gaze targets simultaneously selected by all subjects. The resulting theory of surprise is applicable across different spatio-temporal scales, modalities, and levels of abstraction. Life is full of surprises, ranging from a great christmas gift or a new magic trick, to wardrobe malfunctions, reckless drivers, terrorist attacks, and tsunami waves. Key to survival is our ability to rapidly attend to, identify, and learn from surprising events, to decide on present and future courses of action [1]. Yet, little theoretical and computational understanding exists of the very essence of surprise, as evidenced by the absence from our everyday vocabulary of a quantitative unit of surprise: Qualities such as the “wow factor” have remained vague and elusive to mathematical analysis. Informal correlates of surprise exist at nearly all stages of neural processing. In sensory neuroscience, it has been suggested that only the unexpected at one stage is transmitted to the next stage [2]. Hence, sensory cortex may have evolved to adapt to, to predict, and to quiet down the expected statistical regularities of the world [3, 4, 5, 6], focusing instead on events that are unpredictable or surprising. Electrophysiological evidence for this early sensory emphasis onto surprising stimuli exists from studies of adaptation in visual [7, 8, 4, 9], olfactory [10, 11], and auditory cortices [12], subcortical structures like the LGN [13], and even retinal ganglion cells [14, 15] and cochlear hair cells [16]: neural response greatly attenuates with repeated or prolonged exposure to an initially novel stimulus. Surprise and novelty are also central to learning and memory formation [1], to the point that surprise is believed to be a necessary trigger for associative learning [17, 18], as supported by mounting evidence for a role of the hippocampus as a novelty detector [19, 20, 21]. Finally, seeking novelty is a well-identiﬁed human character trait, with possible association with the dopamine D4 receptor gene [22, 23, 24]. In the Bayesian framework, we develop the only consistent theory of surprise, in terms of the difference between the posterior and prior distributions of beliefs of an observer over the available class of models or hypotheses about the world. We show that this deﬁnition derived from ﬁrst principles presents key advantages over more ad-hoc formulations, typically relying on detecting outlier stimuli. Armed with this new framework, we provide direct experimental evidence that surprise best characterizes what attracts human gaze in large amounts of natural video stimuli. We here extend a recent pilot study [25], adding more comprehensive theory, large-scale human data collection, and additional analysis. 1 Theory Bayesian Deﬁnition of Surprise. We propose that surprise is a general concept, which can be derived from ﬁrst principles and formalized across spatio-temporal scales, sensory modalities, and, more generally, data types and data sources. Two elements are essential for a principled deﬁnition of surprise. First, surprise can exist only in the presence of uncertainty, which can arise from intrinsic stochasticity, missing information, or limited computing resources. A world that is purely deterministic and predictable in real-time for a given observer contains no surprises. Second, surprise can only be deﬁned in a relative, subjective, manner and is related to the expectations of the observer, be it a single synapse, neuronal circuit, organism, or computer device. The same data may carry different amount of surprise for different observers, or even for the same observer taken at different times. In probability and decision theory it can be shown that the only consistent and optimal way for modeling and reasoning about uncertainty is provided by the Bayesian theory of probability [26, 27, 28]. Furthermore, in the Bayesian framework, probabilities correspond to subjective degrees of beliefs in hypotheses or models which are updated, as data is acquired, using Bayes’ theorem as the fundamental tool for transforming prior belief distributions into posterior belief distributions. Therefore, within the same optimal framework, the only consistent deﬁnition of surprise must involve: (1) probabilistic concepts to cope with uncertainty; and (2) prior and posterior distributions to capture subjective expectations. Consistently with this Bayesian approach, the background information of an observer is captured by his/her/its prior probability distribution {P (M )}M ∈M over the hypotheses or models M in a model space M. Given this prior distribution of beliefs, the fundamental effect of a new data observation D on the observer is to change the prior distribution {P (M )}M ∈M into the posterior distribution {P (M |D)}M ∈M via Bayes theorem, whereby P (D|M ) ∀M ∈ M, P (M |D) = P (M ). (1) P (D) In this framework, the new data observation D carries no surprise if it leaves the observer beliefs unaffected, that is, if the posterior is identical to the prior; conversely, D is surprising if the posterior distribution resulting from observing D signiﬁcantly differs from the prior distribution. Therefore we formally measure surprise elicited by data as some distance measure between the posterior and prior distributions. This is best done using the relative entropy or Kullback-Leibler (KL) divergence [29]. Thus, surprise is deﬁned by the average of the log-odd ratio: P (M |D) S(D, M) = KL(P (M |D), P (M )) = P (M |D) log dM (2) P (M ) M taken with respect to the posterior distribution over the model class M. Note that KL is not symmetric but has well-known theoretical advantages, including invariance with respect to Figure 1: Computing surprise in early sensory neurons. (a) Prior data observations, tuning preferences, and top-down inﬂuences contribute to shaping a set of “prior beliefs” a neuron may have over a class of internal models or hypotheses about the world. For instance, M may be a set of Poisson processes parameterized by the rate λ, with {P (M )}M ∈M = {P (λ)}λ∈I +∗ the prior distribution R of beliefs about which Poisson models well describe the world as sensed by the neuron. New data D updates the prior into the posterior using Bayes’ theorem. Surprise quantiﬁes the difference between the posterior and prior distributions over the model class M. The remaining panels detail how surprise differs from conventional model ﬁtting and outlier-based novelty. (b) In standard iterative Bayesian model ﬁtting, at every iteration N , incoming data DN is used to update the prior {P (M |D1 , D2 , ..., DN −1 )}M ∈M into the posterior {P (M |D1 , D2 , ..., DN )}M ∈M . Freezing this learning at a given iteration, one then picks the currently best model, usually using either a maximum likelihood criterion, or a maximum a posteriori one (yielding MM AP shown). (c) This best model is used for a number of tasks at the current iteration, including outlier-based novelty detection. New data is then considered novel at that instant if it has low likelihood for the best model b a (e.g., DN is more novel than DN ). This focus onto the single best model presents obvious limitations, especially in situations where other models are nearly as good (e.g., M∗ in panel (b) is entirely ignored during standard novelty computation). One palliative solution is to consider mixture models, or simply P (D), but this just amounts to shifting the problem into a different model class. (d) Surprise directly addresses this problem by simultaneously considering all models and by measuring how data changes the observer’s distribution of beliefs from {P (M |D1 , D2 , ..., DN −1 )}M ∈M to {P (M |D1 , D2 , ..., DN )}M ∈M over the entire model class M (orange shaded area). reparameterizations. A unit of surprise — a “wow” — may then be deﬁned for a single model M as the amount of surprise corresponding to a two-fold variation between P (M |D) and P (M ), i.e., as log P (M |D)/P (M ) (with log taken in base 2), with the total number of wows experienced for all models obtained through the integration in eq. 2. Surprise and outlier detection. Outlier detection based on the likelihood P (D|M best ) of D given a single best model Mbest is at best an approximation to surprise and, in some cases, is misleading. Consider, for instance, a case where D has very small probability both for a model or hypothesis M and for a single alternative hypothesis M. Although D is a strong outlier, it carries very little information regarding whether M or M is the better model, and therefore very little surprise. Thus an outlier detection method would strongly focus attentional resources onto D, although D is a false positive, in the sense that it carries no useful information for discriminating between the two alternative hypotheses M and M. Figure 1 further illustrates this disconnect between outlier detection and surprise. 2 Human experiments To test the surprise hypothesis — that surprise attracts human attention and gaze in natural scenes — we recorded eye movements from eight na¨ve observers (three females and ı ﬁve males, ages 23-32, normal or corrected-to-normal vision). Each watched a subset from 50 videoclips totaling over 25 minutes of playtime (46,489 video frames, 640 × 480, 60.27 Hz, mean screen luminance 30 cd/m2 , room 4 cd/m2 , viewing distance 80cm, ﬁeld of view 28◦ × 21◦ ). Clips comprised outdoors daytime and nighttime scenes of crowded environments, video games, and television broadcast including news, sports, and commercials. Right-eye position was tracked with a 240 Hz video-based device (ISCAN RK-464), with methods as previously [30]. Two hundred calibrated eye movement traces (10,192 saccades) were analyzed, corresponding to four distinct observers for each of the 50 clips. Figure 2 shows sample scanpaths for one videoclip. To characterize image regions selected by participants, we process videoclips through computational metrics that output a topographic dynamic master response map, assigning in real-time a response value to every input location. A good master map would highlight, more than expected by chance, locations gazed to by observers. To score each metric we hence sample, at onset of every human saccade, master map activity around the saccade’s future endpoint, and around a uniformly random endpoint (random sampling was repeated 100 times to evaluate variability). We quantify differences between histograms of master Figure 2: (a) Sample eye movement traces from four observers (squares denote saccade endpoints). (b) Our data exhibits high inter-individual overlap, shown here with the locations where one human saccade endpoint was nearby (≈ 5◦ ) one (white squares), two (cyan squares), or all three (black squares) other humans. (c) A metric where the master map was created from the three eye movement traces other than that being tested yields an upper-bound KL score, computed by comparing the histograms of metric values at human (narrow blue bars) and random (wider green bars) saccade targets. Indeed, this metric’s map was very sparse (many random saccades landing on locations with nearzero response), yet humans preferentially saccaded towards the three active hotspots corresponding to the eye positions of three other humans (many human saccades landing on locations with near-unity responses). map samples collected from human and random saccades using again the Kullback-Leibler (KL) distance: metrics which better predict human scanpaths exhibit higher distances from random as, typically, observers non-uniformly gaze towards a minority of regions with highest metric responses while avoiding a majority of regions with low metric responses. This approach presents several advantages over simpler scoring schemes [31, 32], including agnosticity to putative mechanisms for generating saccades and the fact that applying any continuous nonlinearity to master map values would not affect scoring. Experimental results. We test six computational metrics, encompassing and extending the state-of-the-art found in previous studies. The ﬁrst three quantify static image properties (local intensity variance in 16 × 16 image patches [31]; local oriented edge density as measured with Gabor ﬁlters [33]; and local Shannon entropy in 16 × 16 image patches [34]). The remaining three metrics are more sensitive to dynamic events (local motion [33]; outlier-based saliency [33]; and surprise [25]). For all metrics, we ﬁnd that humans are signiﬁcantly attracted by image regions with higher metric responses. However, the static metrics typically respond vigorously at numerous visual locations (Figure 3), hence they are poorly speciﬁc and yield relatively low KL scores between humans and random. The metrics sensitive to motion, outliers, and surprising events, in comparison, yield sparser maps and higher KL scores. The surprise metric of interest here quantiﬁes low-level surprise in image patches over space and time, and at this point does not account for high-level or cognitive beliefs of our human observers. Rather, it assumes a family of simple models for image patches, each processed through 72 early feature detectors sensitive to color, orientation, motion, etc., and computes surprise from shifts in the distribution of beliefs about which models better describe the patches (see [25] and [35] for details). We ﬁnd that the surprise metric signiﬁcantly outperforms all other computational metrics (p < 10−100 or better on t-tests for equality of KL scores), scoring nearly 20% better than the second-best metric (saliency) and 60% better than the best static metric (entropy). Surprising stimuli often substantially differ from simple feature outliers; for example, a continually blinking light on a static background elicits sustained ﬂicker due to its locally outlier temporal dynamics but is only surprising for a moment. Similarly, a shower of randomly-colored pixels continually excites all low-level feature detectors but rapidly becomes unsurprising. Strongest attractors of human attention. Clearly, in our and previous eye-tracking experiments, in some situations potentially interesting targets were more numerous than in others. With many possible targets, different observers may orient towards different locations, making it more difﬁcult for a single metric to accurately predict all observers. Hence we consider (Figure 4) subsets of human saccades where at least two, three, or all four observers simultaneously agreed on a gaze target. Observers could have agreed based on bottom-up factors (e.g., only one location had interesting visual appearance at that time), top-down factors (e.g., only one object was of current cognitive interest), or both (e.g., a single cognitively interesting object was present which also had distinctive appearance). Irrespectively of the cause for agreement, it indicates consolidated belief that a location was attractive. While the KL scores of all metrics improved when progressively focusing onto only those locations, dynamic metrics improved more steeply, indicating that stimuli which more reliably attracted all observers carried more motion, saliency, and surprise. Surprise remained signiﬁcantly the best metric to characterize these agreed-upon attractors of human gaze (p < 10−100 or better on t-tests for equality of KL scores). Overall, surprise explained the greatest fraction of human saccades, indicating that humans are signiﬁcantly attracted towards surprising locations in video displays. Over 72% of all human saccades were targeted to locations predicted to be more surprising than on average. When only considering saccades where two, three, or four observers agreed on a common gaze target, this ﬁgure rose to 76%, 80%, and 84%, respectively. Figure 3: (a) Sample video frames, with corresponding human saccades and predictions from the entropy, surprise, and human-derived metrics. Entropy maps, like intensity variance and orientation maps, exhibited many locations with high responses, hence had low speciﬁcity and were poorly discriminative. In contrast, motion, saliency, and surprise maps were much sparser and more speciﬁc, with surprise signiﬁcantly more often on target. For three example frames (ﬁrst column), saccades from one subject are shown (arrows) with corresponding apertures over which master map activity at the saccade endpoint was sampled (circles). (b) KL scores for these metrics indicate signiﬁcantly different performance levels, and a strict ranking of variance < orientation < entropy < motion < saliency < surprise < human-derived. KL scores were computed by comparing the number of human saccades landing onto each given range of master map values (narrow blue bars) to the number of random saccades hitting the same range (wider green bars). A score of zero would indicate equality between the human and random histograms, i.e., humans did not tend to hit various master map values any differently from expected by chance, or, the master map could not predict human saccades better than random saccades. Among the six computational metrics tested in total, surprise performed best, in that surprising locations were relatively few yet reliably gazed to by humans. Figure 4: KL scores when considering only saccades where at least one (all 10,192 saccades), two (7,948 saccades), three (5,565 saccades), or all four (2,951 saccades) humans agreed on a common gaze location, for the static (a) and dynamic metrics (b). Static metrics improved substantially when progressively focusing onto saccades with stronger inter-observer agreement (average slope 0.56 ± 0.37 percent KL score units per 1,000 pruned saccades). Hence, when humans agreed on a location, they also tended to be more reliably predicted by the metrics. Furthermore, dynamic metrics improved 4.5 times more steeply (slope 2.44 ± 0.37), suggesting a stronger role of dynamic events in attracting human attention. Surprising events were signiﬁcantly the strongest (t-tests for equality of KL scores between surprise and other metrics, p < 10−100 ). 3 Discussion While previous research has shown with either static scenes or dynamic synthetic stimuli that humans preferentially ﬁxate regions of high entropy [34], contrast [31], saliency [32], ﬂicker [36], or motion [37], our data provides direct experimental evidence that humans ﬁxate surprising locations even more reliably. These conclusions were made possible by developing new tools to quantify what attracts human gaze over space and time in dynamic natural scenes. Surprise explained best where humans look when considering all saccades, and even more so when restricting the analysis to only those saccades for which human observers tended to agree. Surprise hence represents an inexpensive, easily computable approximation to human attentional allocation. In the absence of quantitative tools to measure surprise, most experimental and modeling work to date has adopted the approximation that novel events are surprising, and has focused on experimental scenarios which are simple enough to ensure an overlap between informal notions of novelty and surprise: for example, a stimulus is novel during testing if it has not been seen during training [9]. Our deﬁnition opens new avenues for more sophisticated experiments, where surprise elicited by different stimuli can be precisely compared and calibrated, yielding predictions at the single-unit as well as behavioral levels. The deﬁnition of surprise — as the distance between the posterior and prior distributions of beliefs over models — is entirely general and readily applicable to the analysis of auditory, olfactory, gustatory, or somatosensory data. While here we have focused on behavior rather than detailed biophysical implementation, it is worth noting that detecting surprise in neural spike trains does not require semantic understanding of the data carried by the spike trains, and thus could provide guiding signals during self-organization and development of sensory areas. At higher processing levels, top-down cues and task demands are known to combine with stimulus novelty in capturing attention and triggering learning [1, 38], ideas which may now be formalized and quantiﬁed in terms of priors, posteriors, and surprise. Surprise, indeed, inherently depends on uncertainty and on prior beliefs. Hence surprise theory can further be tested and utilized in experiments where the prior is biased, for ex- ample by top-down instructions or prior exposures to stimuli [38]. In addition, simple surprise-based behavioral measures such as the eye-tracking one used here may prove useful for early diagnostic of human conditions including autism and attention-deﬁcit hyperactive disorder, as well as for quantitative comparison between humans and animals which may have lower or different priors, including monkeys, frogs, and ﬂies. Beyond sensory biology, computable surprise could guide the development of data mining and compression systems (giving more bits to surprising regions of interest), to ﬁnd surprising agents in crowds, surprising sentences in books or speeches, surprising sequences in genomes, surprising medical symptoms, surprising odors in airport luggage racks, surprising documents on the world-wide-web, or to design surprising advertisements. Acknowledgments: Supported by HFSP, NSF and NGA (L.I.), NIH and NSF (P.B.). 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Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract The concept of surprise is central to sensory processing, adaptation, learning, and attention. [sent-4, score-0.682]

2 Yet, no widely-accepted mathematical theory currently exists to quantitatively characterize surprise elicited by a stimulus or event, for observers that range from single neurons to complex natural or engineered systems. [sent-5, score-0.805]

3 We describe a formal Bayesian deﬁnition of surprise that is the only consistent formulation under minimal axiomatic assumptions. [sent-6, score-0.618]

4 Surprise quantiﬁes how data affects a natural or artiﬁcial observer, by measuring the difference between posterior and prior beliefs of the observer. [sent-7, score-0.233]

5 Using this framework we measure the extent to which humans direct their gaze towards surprising items while watching television and video games. [sent-8, score-0.554]

6 We ﬁnd that subjects are strongly attracted towards surprising locations, with 72% of all human gaze shifts directed towards locations more surprising than the average, a ﬁgure which rises to 84% when considering only gaze targets simultaneously selected by all subjects. [sent-9, score-1.005]

7 The resulting theory of surprise is applicable across different spatio-temporal scales, modalities, and levels of abstraction. [sent-10, score-0.618]

8 Key to survival is our ability to rapidly attend to, identify, and learn from surprising events, to decide on present and future courses of action [1]. [sent-12, score-0.174]

9 Informal correlates of surprise exist at nearly all stages of neural processing. [sent-14, score-0.618]

10 In sensory neuroscience, it has been suggested that only the unexpected at one stage is transmitted to the next stage [2]. [sent-15, score-0.064]

11 Hence, sensory cortex may have evolved to adapt to, to predict, and to quiet down the expected statistical regularities of the world [3, 4, 5, 6], focusing instead on events that are unpredictable or surprising. [sent-16, score-0.144]

12 Surprise and novelty are also central to learning and memory formation [1], to the point that surprise is believed to be a necessary trigger for associative learning [17, 18], as supported by mounting evidence for a role of the hippocampus as a novelty detector [19, 20, 21]. [sent-18, score-0.794]

13 Finally, seeking novelty is a well-identiﬁed human character trait, with possible association with the dopamine D4 receptor gene [22, 23, 24]. [sent-19, score-0.209]

14 In the Bayesian framework, we develop the only consistent theory of surprise, in terms of the difference between the posterior and prior distributions of beliefs of an observer over the available class of models or hypotheses about the world. [sent-20, score-0.374]

15 We show that this deﬁnition derived from ﬁrst principles presents key advantages over more ad-hoc formulations, typically relying on detecting outlier stimuli. [sent-21, score-0.1]

16 Armed with this new framework, we provide direct experimental evidence that surprise best characterizes what attracts human gaze in large amounts of natural video stimuli. [sent-22, score-1.025]

17 We here extend a recent pilot study [25], adding more comprehensive theory, large-scale human data collection, and additional analysis. [sent-23, score-0.121]

18 We propose that surprise is a general concept, which can be derived from ﬁrst principles and formalized across spatio-temporal scales, sensory modalities, and, more generally, data types and data sources. [sent-25, score-0.707]

19 First, surprise can exist only in the presence of uncertainty, which can arise from intrinsic stochasticity, missing information, or limited computing resources. [sent-27, score-0.618]

20 A world that is purely deterministic and predictable in real-time for a given observer contains no surprises. [sent-28, score-0.099]

21 Second, surprise can only be deﬁned in a relative, subjective, manner and is related to the expectations of the observer, be it a single synapse, neuronal circuit, organism, or computer device. [sent-29, score-0.618]

22 The same data may carry different amount of surprise for different observers, or even for the same observer taken at different times. [sent-30, score-0.717]

23 Furthermore, in the Bayesian framework, probabilities correspond to subjective degrees of beliefs in hypotheses or models which are updated, as data is acquired, using Bayes’ theorem as the fundamental tool for transforming prior belief distributions into posterior belief distributions. [sent-32, score-0.307]

24 Therefore, within the same optimal framework, the only consistent deﬁnition of surprise must involve: (1) probabilistic concepts to cope with uncertainty; and (2) prior and posterior distributions to capture subjective expectations. [sent-33, score-0.762]

25 Consistently with this Bayesian approach, the background information of an observer is captured by his/her/its prior probability distribution {P (M )}M ∈M over the hypotheses or models M in a model space M. [sent-34, score-0.201]

26 Given this prior distribution of beliefs, the fundamental effect of a new data observation D on the observer is to change the prior distribution {P (M )}M ∈M into the posterior distribution {P (M |D)}M ∈M via Bayes theorem, whereby P (D|M ) ∀M ∈ M, P (M |D) = P (M ). [sent-35, score-0.271]

27 Therefore we formally measure surprise elicited by data as some distance measure between the posterior and prior distributions. [sent-37, score-0.773]

28 This is best done using the relative entropy or Kullback-Leibler (KL) divergence [29]. [sent-38, score-0.074]

29 Thus, surprise is deﬁned by the average of the log-odd ratio: P (M |D) S(D, M) = KL(P (M |D), P (M )) = P (M |D) log dM (2) P (M ) M taken with respect to the posterior distribution over the model class M. [sent-39, score-0.67]

30 Note that KL is not symmetric but has well-known theoretical advantages, including invariance with respect to Figure 1: Computing surprise in early sensory neurons. [sent-40, score-0.706]

31 (a) Prior data observations, tuning preferences, and top-down inﬂuences contribute to shaping a set of “prior beliefs” a neuron may have over a class of internal models or hypotheses about the world. [sent-41, score-0.042]

32 For instance, M may be a set of Poisson processes parameterized by the rate λ, with {P (M )}M ∈M = {P (λ)}λ∈I +∗ the prior distribution R of beliefs about which Poisson models well describe the world as sensed by the neuron. [sent-42, score-0.181]

33 New data D updates the prior into the posterior using Bayes’ theorem. [sent-43, score-0.112]

34 Surprise quantiﬁes the difference between the posterior and prior distributions over the model class M. [sent-44, score-0.112]

35 The remaining panels detail how surprise differs from conventional model ﬁtting and outlier-based novelty. [sent-45, score-0.618]

36 (b) In standard iterative Bayesian model ﬁtting, at every iteration N , incoming data DN is used to update the prior {P (M |D1 , D2 , . [sent-46, score-0.06]

37 , DN −1 )}M ∈M into the posterior {P (M |D1 , D2 , . [sent-49, score-0.052]

38 Freezing this learning at a given iteration, one then picks the currently best model, usually using either a maximum likelihood criterion, or a maximum a posteriori one (yielding MM AP shown). [sent-53, score-0.03]

39 (c) This best model is used for a number of tasks at the current iteration, including outlier-based novelty detection. [sent-54, score-0.118]

40 New data is then considered novel at that instant if it has low likelihood for the best model b a (e. [sent-55, score-0.054]

41 This focus onto the single best model presents obvious limitations, especially in situations where other models are nearly as good (e. [sent-58, score-0.064]

42 , M∗ in panel (b) is entirely ignored during standard novelty computation). [sent-60, score-0.088]

43 (d) Surprise directly addresses this problem by simultaneously considering all models and by measuring how data changes the observer’s distribution of beliefs from {P (M |D1 , D2 , . [sent-62, score-0.144]

44 A unit of surprise — a “wow” — may then be deﬁned for a single model M as the amount of surprise corresponding to a two-fold variation between P (M |D) and P (M ), i. [sent-70, score-1.236]

45 Outlier detection based on the likelihood P (D|M best ) of D given a single best model Mbest is at best an approximation to surprise and, in some cases, is misleading. [sent-75, score-0.708]

46 Although D is a strong outlier, it carries very little information regarding whether M or M is the better model, and therefore very little surprise. [sent-77, score-0.033]

47 Thus an outlier detection method would strongly focus attentional resources onto D, although D is a false positive, in the sense that it carries no useful information for discriminating between the two alternative hypotheses M and M. [sent-78, score-0.232]

48 Figure 1 further illustrates this disconnect between outlier detection and surprise. [sent-79, score-0.1]

49 2 Human experiments To test the surprise hypothesis — that surprise attracts human attention and gaze in natural scenes — we recorded eye movements from eight na¨ve observers (three females and ı ﬁve males, ages 23-32, normal or corrected-to-normal vision). [sent-80, score-1.755]

50 Each watched a subset from 50 videoclips totaling over 25 minutes of playtime (46,489 video frames, 640 × 480, 60. [sent-81, score-0.088]

51 Clips comprised outdoors daytime and nighttime scenes of crowded environments, video games, and television broadcast including news, sports, and commercials. [sent-83, score-0.112]

52 Two hundred calibrated eye movement traces (10,192 saccades) were analyzed, corresponding to four distinct observers for each of the 50 clips. [sent-85, score-0.248]

53 To characterize image regions selected by participants, we process videoclips through computational metrics that output a topographic dynamic master response map, assigning in real-time a response value to every input location. [sent-87, score-0.405]

54 A good master map would highlight, more than expected by chance, locations gazed to by observers. [sent-88, score-0.283]

55 To score each metric we hence sample, at onset of every human saccade, master map activity around the saccade’s future endpoint, and around a uniformly random endpoint (random sampling was repeated 100 times to evaluate variability). [sent-89, score-0.478]

56 We quantify differences between histograms of master Figure 2: (a) Sample eye movement traces from four observers (squares denote saccade endpoints). [sent-90, score-0.482]

57 (b) Our data exhibits high inter-individual overlap, shown here with the locations where one human saccade endpoint was nearby (≈ 5◦ ) one (white squares), two (cyan squares), or all three (black squares) other humans. [sent-91, score-0.358]

58 (c) A metric where the master map was created from the three eye movement traces other than that being tested yields an upper-bound KL score, computed by comparing the histograms of metric values at human (narrow blue bars) and random (wider green bars) saccade targets. [sent-92, score-0.701]

59 This approach presents several advantages over simpler scoring schemes [31, 32], including agnosticity to putative mechanisms for generating saccades and the fact that applying any continuous nonlinearity to master map values would not affect scoring. [sent-95, score-0.528]

60 The ﬁrst three quantify static image properties (local intensity variance in 16 × 16 image patches [31]; local oriented edge density as measured with Gabor ﬁlters [33]; and local Shannon entropy in 16 × 16 image patches [34]). [sent-98, score-0.297]

61 The remaining three metrics are more sensitive to dynamic events (local motion [33]; outlier-based saliency [33]; and surprise [25]). [sent-99, score-0.976]

62 For all metrics, we ﬁnd that humans are signiﬁcantly attracted by image regions with higher metric responses. [sent-100, score-0.303]

63 However, the static metrics typically respond vigorously at numerous visual locations (Figure 3), hence they are poorly speciﬁc and yield relatively low KL scores between humans and random. [sent-101, score-0.483]

64 The metrics sensitive to motion, outliers, and surprising events, in comparison, yield sparser maps and higher KL scores. [sent-102, score-0.385]

65 The surprise metric of interest here quantiﬁes low-level surprise in image patches over space and time, and at this point does not account for high-level or cognitive beliefs of our human observers. [sent-103, score-1.644]

66 Rather, it assumes a family of simple models for image patches, each processed through 72 early feature detectors sensitive to color, orientation, motion, etc. [sent-104, score-0.047]

67 , and computes surprise from shifts in the distribution of beliefs about which models better describe the patches (see [25] and [35] for details). [sent-105, score-0.784]

68 We ﬁnd that the surprise metric signiﬁcantly outperforms all other computational metrics (p < 10−100 or better on t-tests for equality of KL scores), scoring nearly 20% better than the second-best metric (saliency) and 60% better than the best static metric (entropy). [sent-106, score-1.203]

69 Surprising stimuli often substantially differ from simple feature outliers; for example, a continually blinking light on a static background elicits sustained ﬂicker due to its locally outlier temporal dynamics but is only surprising for a moment. [sent-107, score-0.414]

70 With many possible targets, different observers may orient towards different locations, making it more difﬁcult for a single metric to accurately predict all observers. [sent-111, score-0.256]

71 Hence we consider (Figure 4) subsets of human saccades where at least two, three, or all four observers simultaneously agreed on a gaze target. [sent-112, score-0.842]

72 Observers could have agreed based on bottom-up factors (e. [sent-113, score-0.09]

73 While the KL scores of all metrics improved when progressively focusing onto only those locations, dynamic metrics improved more steeply, indicating that stimuli which more reliably attracted all observers carried more motion, saliency, and surprise. [sent-121, score-0.707]

74 Surprise remained signiﬁcantly the best metric to characterize these agreed-upon attractors of human gaze (p < 10−100 or better on t-tests for equality of KL scores). [sent-122, score-0.453]

75 Overall, surprise explained the greatest fraction of human saccades, indicating that humans are signiﬁcantly attracted towards surprising locations in video displays. [sent-123, score-1.232]

76 Over 72% of all human saccades were targeted to locations predicted to be more surprising than on average. [sent-124, score-0.736]

77 When only considering saccades where two, three, or four observers agreed on a common gaze target, this ﬁgure rose to 76%, 80%, and 84%, respectively. [sent-125, score-0.744]

78 Figure 3: (a) Sample video frames, with corresponding human saccades and predictions from the entropy, surprise, and human-derived metrics. [sent-126, score-0.534]

79 Entropy maps, like intensity variance and orientation maps, exhibited many locations with high responses, hence had low speciﬁcity and were poorly discriminative. [sent-127, score-0.08]

80 In contrast, motion, saliency, and surprise maps were much sparser and more speciﬁc, with surprise signiﬁcantly more often on target. [sent-128, score-1.288]

81 For three example frames (ﬁrst column), saccades from one subject are shown (arrows) with corresponding apertures over which master map activity at the saccade endpoint was sampled (circles). [sent-129, score-0.713]

82 (b) KL scores for these metrics indicate signiﬁcantly different performance levels, and a strict ranking of variance < orientation < entropy < motion < saliency < surprise < human-derived. [sent-130, score-0.994]

83 KL scores were computed by comparing the number of human saccades landing onto each given range of master map values (narrow blue bars) to the number of random saccades hitting the same range (wider green bars). [sent-131, score-1.149]

84 A score of zero would indicate equality between the human and random histograms, i. [sent-132, score-0.18]

85 , humans did not tend to hit various master map values any differently from expected by chance, or, the master map could not predict human saccades better than random saccades. [sent-134, score-0.926]

86 Among the six computational metrics tested in total, surprise performed best, in that surprising locations were relatively few yet reliably gazed to by humans. [sent-135, score-1.09]

87 Figure 4: KL scores when considering only saccades where at least one (all 10,192 saccades), two (7,948 saccades), three (5,565 saccades), or all four (2,951 saccades) humans agreed on a common gaze location, for the static (a) and dynamic metrics (b). [sent-136, score-1.061]

88 Static metrics improved substantially when progressively focusing onto saccades with stronger inter-observer agreement (average slope 0. [sent-137, score-0.583]

89 Hence, when humans agreed on a location, they also tended to be more reliably predicted by the metrics. [sent-140, score-0.254]

90 37), suggesting a stronger role of dynamic events in attracting human attention. [sent-144, score-0.209]

91 Surprising events were signiﬁcantly the strongest (t-tests for equality of KL scores between surprise and other metrics, p < 10−100 ). [sent-145, score-0.784]

92 These conclusions were made possible by developing new tools to quantify what attracts human gaze over space and time in dynamic natural scenes. [sent-147, score-0.384]

93 Surprise explained best where humans look when considering all saccades, and even more so when restricting the analysis to only those saccades for which human observers tended to agree. [sent-148, score-0.799]

94 Surprise hence represents an inexpensive, easily computable approximation to human attentional allocation. [sent-149, score-0.169]

95 Our deﬁnition opens new avenues for more sophisticated experiments, where surprise elicited by different stimuli can be precisely compared and calibrated, yielding predictions at the single-unit as well as behavioral levels. [sent-151, score-0.7]

96 The deﬁnition of surprise — as the distance between the posterior and prior distributions of beliefs over models — is entirely general and readily applicable to the analysis of auditory, olfactory, gustatory, or somatosensory data. [sent-152, score-0.851]

97 At higher processing levels, top-down cues and task demands are known to combine with stimulus novelty in capturing attention and triggering learning [1, 38], ideas which may now be formalized and quantiﬁed in terms of priors, posteriors, and surprise. [sent-154, score-0.134]

98 Surprise, indeed, inherently depends on uncertainty and on prior beliefs. [sent-155, score-0.06]

99 Hence surprise theory can further be tested and utilized in experiments where the prior is biased, for ex- ample by top-down instructions or prior exposures to stimuli [38]. [sent-156, score-0.777]

100 All source code for all metrics is freely available at http://iLab. [sent-332, score-0.159]

similar papers computed by tfidf model

tfidf for this paper:

wordName wordTfidf (topN-words)

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Several Bayesian approaches have been developed for changepoint problems involving single or multiple changepoints [3,5]. We apply a Markov Chain Monte Carlo (MCMC) analysis to approximate the joint posterior distribution over changepoint assignments x while integrating out θ. Gibbs sampling will be used to sample from this posterior marginal distribution. The samples can then be used to predict the next outcome in the sequence. 3.1 I n f e r e nc e f o r c h a n g e p o i n t a s s i g n m e n t s . To apply Gibbs sampling, we evaluate the conditional probability of assigning a changepoint at time i, given all other changepoint assignments and the current α value. By integrating out θ, the conditional probability is P ( xi | x−i , y, α ) = ∫ P ( xi ,θ , α | x− i , y ) (1) θ where x− i represents all switch point assignments except xi. This can be simplified by considering the location of the most recent changepoint preceding and following time i and the outcomes occurring between these locations. Let niL be the number of time steps from the last changepoint up to and including the current time step i such that xi − nL =1 and xi − nL + j =0 for 0 < niL . Similarly, let niR be the number of time steps that i i follow time step i up to the next changepoint such that xi + n R =1 and xi + nR − j =0 for i R i 0 < n . Let y = L i ∑ i − niL < k ≤ i i yk and y = ∑ k < k ≤i + n R yk . The update equation for the R i i changepoint assignment can then be simplified to P ( xi = m | x−i ) ∝ ( ) ( ( ) D Γ 1 + y L + y R Γ 1 + Kn L + Kn R − y L − y R ⎧ i, j i, j i i i, j i, j ⎪ (1 − α ) ∏ L R Γ 2 + Kni + Kni ⎪ j =1 ⎪ ⎨ L L L R R R ⎪ D Γ 1 + yi, j Γ 1 + Kni − yi, j Γ 1 + yi, j Γ 1 + Kni − yi, j α∏ ⎪ Γ 2 + KniL Γ 2 + KniR ⎪ j =1 ⎩ ( ) ( ( ) ( ) ( ) ) ( ) m=0 ) (2) m =1 We initialize the Gibbs sampler by sampling each xt from a Bernoulli(α) distribution. All changepoint assignments are then updated sequentially by the Gibbs sampling equation above. The sampler is run for M iterations after which one set of changepoint assignments is saved. The Gibbs sampler is then restarted multiple times until S samples have been collected. Although we could have included an update equation for α, in this analysis we treat α as a known constant. This will be useful when characterizing the differences between human observers in terms of differences in α. 3.2 P r e d i c ti v e i n f er e n ce The next latent parameter value θt+1 and outcome yt+1 can be predicted on the basis of observed outcomes that occurred after the last inferred changepoint: θ t+1, j = t ∑ i =t* +1 yt+1, j = round (θt +1, j K ) yi, j / K , (3) where t* is the location of the most recent change point. By considering multiple Gibbs samples, we get a distribution over outcomes yt+1. We base the model predictions on the mean of this distribution. 3.3 I l l u s t r a t i o n o f m o d e l p er f o r m a n c e Figure 1 illustrates the performance of the model on a one dimensional sequence (D=1) generated from the changepoint model with T=160, α=0.05, and K=10. The Gibbs sampler was run for M=30 iterations and S=200 samples were collected. The top panel shows the actual changepoints (triangles) and the distribution of changepoint assignments averaged over samples. The bottom panel shows the observed data y (thin lines) as well as the θ values in the generative model (rescaled between 0 and 10). At locations with large changes between observations, the marginal changepoint probability is quite high. At other locations, the true change in the mean is very small, and the model is less likely to put in a changepoint. The lower right panel shows the distribution over predicted θt+1 values. xt 1 0.5 0 yt 10 1 5 θt+1 0.5 0 20 40 60 80 100 120 140 160 0 Figure 1. Results of model simulation. 4 Prediction experiment We tested performance of 9 human observers in the prediction task. The observers included the authors, a visitor, and one student who were aware of the statistical nature of the task as well as naïve students. The observers were seated in front of an LCD touch screen displaying a two-dimensional grid of 11 x 11 buttons. The changepoint model was used to generate a sequence of T=1500 stimuli for two binomial variables y1 and y2 (D=2, K=10). The change probability α was set to 0.1. The two variables y1 and y2 specified the two-dimensional button location. The same sequence was used for all observers. On each trial, the observer touched a button on the grid displayed on the touch screen. Following each button press, the button corresponding to the next {y1,y2} outcome in the sequence was highlighted. Observers were instructed to press the button that best predicted the next location of the highlighted button. The 1500 trials were divided into three blocks of 500 trials. Breaks were allowed between blocks. The whole experiment lasted between 15 and 30 minutes. Figure 2 shows the first 50 trials from the third block of the experiment. The top and bottom panels show the actual outcomes for the y1 and y2 button grid coordinates as well as the predictions for two observers (SB and MY). The figure shows that at trial 15, the y1 and y2 coordinates show a large shift followed by an immediate shift in observer’s MY predictions (on trial 16). Observer SB waits until trial 17 to make a shift. 10 5 0 outcomes SB predictions MY predictions 10 5 0 0 5 10 15 20 25 Trial 30 35 40 45 50 Figure 2. Trial by trial predictions from two observers. 4.1 T a s k er r o r We assessed prediction performance by comparing the prediction with the actual outcome in the sequence. Task error was measured by normalized city-block distance T 1 (4) task error= ∑ yt ,1 − ytO,1 + yt ,2 − ytO,2 (T − 1) t =2 where yO represents the observer’s prediction. Note that the very first trial is excluded from this calculation. Even though more suitable probabilistic measures for prediction error could have been adopted, we wanted to allow comparison of observer’s performance with both probabilistic and non-probabilistic models. Task error ranged from 2.8 (for participant MY) to 3.3 (for ML). We also assessed the performance of five models – their task errors ranged from 2.78 to 3.20. The Bayesian models (Section 3) had the lowest task errors, just below 2.8. This fits with our definition of the Bayesian models as “ideal observer” models – their task error is lower than any other model’s and any human observer’s task error. The fast and frugal models (Section 5) had task errors ranging from 2.85 to 3.20. 5 Modeling R esults We will refer to the models with the following letter codes: B=Bayesian Model, LB=limited Bayesian model, FF1..3=fast and frugal models 1..3. We assessed model fit by comparing the model’s prediction against the human observers’ predictions, again using a normalized city-block distance model error= T 1 ∑ ytM − ytO,1 + ytM − ytO,2 ,1 ,2 (T − 1) t=2 (5) where yM represents the model’s prediction. The model error for each individual observer is shown in Figure 3. It is important to note that because each model is associated with a set of free parameters, the parameters optimized for task error and model error are different. For Figure 3, the parameters were optimized to minimize Equation (5) for each individual observer, showing the extent to which these models can capture the performance of individual observers, not necessarily providing the best task performance. B LB FF1 FF2 MY MS MM EJ FF3 Model Error 2 1.5 1 0.5 0 PH NP DN SB ML 1 Figure 3. Model error for each individual observer. 5.1 B ay e s i a n p re d i ct i o n m o d e l s At each trial t, the model was provided with the sequence of all previous outcomes. The Gibbs sampling and inference procedures from Eq. (2) and (3) were applied with M=30 iterations and S=200 samples. The change probability α was a free parameter. In the full Bayesian model, the whole sequence of observations up to the current trial is available for prediction, leading to a memory requirement of up to T=1500 trials – a psychologically unreasonable assumption. We therefore also simulated a limited Bayesian model (LB) where the observed sequence was truncated to the last 10 outcomes. The LB model showed almost no decrement in task performance compared to the full Bayesian model. Figure 3 also shows that it fit human data quite well. 5.2 I n d i v i d u a l D i f f er e nc e s The right-hand panel of Figure 4 plots each observer’s task error as a function of the mean city-block distance between their subsequent button presses. This shows a clear U-shaped function. Observers with very variable predictions (e.g., ML and DN) had large average changes between successive button pushes, and also had large task error: These observers were too “twitchy”. Observers with very small average button changes (e.g., SB and NP) were not twitchy enough, and also had large task error. Observers in the middle had the lowest task error (e.g., MS and MY). The left-hand panel of Figure 4 shows the same data, but with the x-axis based on the Bayesian model fits. Instead of using mean button change distance to index twitchiness (as in 1 Error bars indicate bootstrapped 95% confidence intervals. the right-hand panel), the left-hand panel uses the estimated α parameters from the Bayesian model. A similar U-shaped pattern is observed: individuals with too large or too small α estimates have large task errors. 3.3 DN 3.2 Task Error ML SB 3.2 NP 3.1 Task Error 3.3 PH EJ 3 MM MS MY 2.9 2.8 10 -4 10 -3 10 -2 DN NP 3.1 3 PH EJ MM MS 2.9 B ML SB MY 2.8 10 -1 10 0 0.5 1 α 1.5 2 Mean Button Change 2.5 3 Figure 4. Task error vs. “twitchiness”. Left-hand panel indexes twitchiness using estimated α parameters from Bayesian model fits. Right-hand panel uses mean distance between successive predictions. 5.3 F a s t - a n d - F r u g a l ( F F ) p r e d ic t i o n m o d e l s These models perform the prediction task using simple heuristics that are cognitively plausible. The FF models keep a short memory of previous stimulus values and make predictions using the same two-step process as the Bayesian model. First, a decision is made as to whether the latent parameter θ has changed. Second, remembered stimulus values that occurred after the most recently detected changepoint are used to generate the next prediction. A simple heuristic is used to detect changepoints: If the distance between the most recent observation and prediction is greater than some threshold amount, a change is inferred. We defined the distance between a prediction (p) and an observation (y) as the difference between the log-likelihoods of y assuming θ=p and θ=y. Thus, if fB(.|θ, K) is the binomial density with parameters θ and K, the distance between observation y and prediction p is defined as d(y,p)=log(fB(y|y,K))-log(fB(y|p,K)). A changepoint on time step t+1 is inferred whenever d(yt,pt)>C. The parameter C governs the twitchiness of the model predictions. If C is large, only very dramatic changepoints will be detected, and the model will be too conservative. If C is small, the model will be too twitchy, and will detect changepoints on the basis of small random fluctuations. Predictions are based on the most recent M observations, which are kept in memory, unless a changepoint has been detected in which case only those observations occurring after the changepoint are used for prediction. The prediction for time step t+1 is simply the mean of these observations, say p. Human observers were reticent to make predictions very close to the boundaries. This was modeled by allowing the FF model to change its prediction for the next time step, yt+1, towards the mean prediction (0.5). This change reflects a two-way bet. If the probability of a change occurring is α, the best guess will be 0.5 if that change occurs, or the mean p if the change does not occur. Thus, the prediction made is actually yt+1=1/2 α+(1-α)p. Note that we do not allow perfect knowledge of the probability of a changepoint, α. Instead, an estimated value of α is used based on the number of changepoints detected in the data series up to time t. The FF model nests two simpler FF models that are psychologically interesting. If the twitchiness threshold parameter C becomes arbitrarily large, the model never detects a change and instead becomes a continuous running average model. Predictions from this model are simply a boxcar smooth of the data. Alternatively, if we assume no memory the model must based each prediction on only the previous stimulus (i.e., M=1). Above, in Figure 3, we labeled the complete FF model as FF1, the boxcar model as FF2 and the memoryless model FF3. Figure 3 showed that the complete FF model (FF1) fit the data from all observers significantly better than either the boxcar model (FF2) or the memoryless model (FF3). Exceptions were observers PH, DN and ML, for whom all three FF model fit equally well. This result suggests that our observers were (mostly) doing more than just keeping a running average of the data, or using only the most recent observation. The FF1 model fit the data about as well as the Bayesian models for all observers except MY and MS. Note that, in general, the FF1 and Bayesian model fits are very good: the average city block distance between the human data and the model prediction is around 0.75 (out of 10) buttons on both the x- and y-axes. 6 C onclusion We used an online prediction task to study changepoint detection. Human observers had to predict the next observation in stochastic sequences containing random changepoints. We showed that some observers are too “twitchy”: They perform poorly on the prediction task because they see changes where only random fluctuation exists. Other observers are not twitchy enough, and they perform poorly because they fail to see small changes. We developed a Bayesian changepoint detection model that performed the task optimally, and also provided a good fit to human data when sub-optimal parameter settings were used. Finally, we developed a fast-and-frugal model that showed how participants may be able to perform well at the task using minimal information and simple decision heuristics. Acknowledgments We thank Eric-Jan Wagenmakers and Mike Yi for useful discussions related to this work. This work was supported in part by a grant from the US Air Force Office of Scientific Research (AFOSR grant number FA9550-04-1-0317). R e f er e n ce s [1] Gilovich, T., Vallone, R. and Tversky, A. (1985). The hot hand in basketball: on the misperception of random sequences. Cognitive Psychology17, 295-314. [2] Albright, S.C. (1993a). A statistical analysis of hitting streaks in baseball. Journal of the American Statistical Association, 88, 1175-1183. [3] Stephens, D.A. (1994). Bayesian retrospective multiple changepoint identification. Applied Statistics 43(1), 159-178. [4] Carlin, B.P., Gelfand, A.E., & Smith, A.F.M. (1992). Hierarchical Bayesian analysis of changepoint problems. Applied Statistics 41(2), 389-405. [5] Green, P.J. (1995). Reversible jump Markov chain Monte Carlo computation and Bayesian model determination. Biometrika 82(4), 711-732. [6] Gigerenzer, G., & Goldstein, D.G. (1996). Reasoning the fast and frugal way: Models of bounded rationality. Psychological Review, 103, 650-669.

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A variety of computational principles have been suggested as explanations for adaptation. Many of these are based on the concept of maximizing the sensory information an observer can obtain about a stimulus despite limited sensory resources [4, 5, 6]. More mechanistically, adaptation can be interpreted as the attempt of the sensory system to adjusts its (limited) dynamic range such that it is maximally informative with respect to the statistics of the stimulus. A typical example is observed in the retina, which manages to encode light intensities that vary over nine orders of magnitude using ganglion cells whose dynamic range covers only two orders of magnitude. This is achieved by adapting to the local mean as well as higher order statistics of the visual input over short time-scales [7]. ∗ corresponding author. If a Bayesian framework is to provide a valid computational explanation of perceptual processes, then it needs to account for the behavior of a perceptual system, regardless of its adaptation state. In general, adaptation in a sensory estimation task seems to have two fundamental effects on subsequent perception: • Repulsion: The estimate of parameters of subsequent stimuli are repelled by those of the adaptor stimulus, i.e. the perceived values for the stimulus variable that is subject to the estimation task are more distant from the adaptor value after adaptation. This repulsive effect has been reported for perception of visual speed (e.g. [8, 9]), direction-of-motion [10], and orientation [11]. • Increased sensitivity: Adaptation increases the observer’s discrimination ability around the adaptor (e.g. for visual speed [12, 13]), however it also seems to decrease it further away from the adaptor as shown in the case of direction-of-motion discrimination [14]. In this paper, we show that these two perceptual effects can be explained within a Bayesian estimation framework of perception. Note that our description is at an abstract functional level - we do not attempt to provide a computational model for the underlying mechanisms responsible for adaptation, and this clearly separates this paper from other work which might seem at ﬁrst glance similar [e.g., 15]. 2 Adaptive Bayesian estimator framework Suppose that an observer wants to estimate a property of a stimulus denoted by the variable θ, based on a measurement m. In general, the measurement can be vector-valued, and is corrupted by both internal and external noise. Hence, combining the noisy information gained by the measurement m with a priori knowledge about θ is advantageous. According to Bayes’ rule 1 p(θ|m) = p(m|θ)p(θ) . (1) α That is, the probability of stimulus value θ given m (posterior) is the product of the likelihood p(m|θ) of the particular measurement and the prior p(θ). The normalization constant α serves to ensure that the posterior is a proper probability distribution. Under the assumpˆ tion of a squared-error loss function, the optimal estimate θ(m) is the mean of the posterior, thus ∞ ˆ θ(m) = θ p(θ|m) dθ . (2) 0 ˆ Note that θ(m) describes an estimate for a single measurement m. As discussed in [16], the measurement will vary stochastically over the course of many exposures to the same stimulus, and thus the estimator will also vary. We return to this issue in Section 3.2. Figure 1a illustrates a Bayesian estimator, in which the shape of the (arbitrary) prior distribution leads on average to a shift of the estimate toward a lower value of θ than the true stimulus value θstim . The likelihood and the prior are the fundamental constituents of the Bayesian estimator model. Our goal is to describe how adaptation alters these constituents so as to account for the perceptual effects of repulsion and increased sensitivity. Adaptation does not change the prior ... An intuitively sensible hypothesis is that adaptation changes the prior distribution. Since the prior is meant to reﬂect the knowledge the observer has about the distribution of occurrences of the variable θ in the world, repeated viewing of stimuli with the same parameter a b probability probability attraction ! posterior likelihood prior modified prior Ã θ θ ˆ θ' θ θadapt Figure 1: Hypothetical model in which adaptation alters the prior distribution. a) Unadapted Bayesian estimation conﬁguration in which the prior leads to a shift of the estimate ˆ θ, relative to the stimulus parameter θstim . Both the likelihood function and the prior distriˆ bution contribute to the exact value of the estimate θ (mean of the posterior). b) Adaptation acts by increasing the prior distribution around the value, θadapt , of the adapting stimulus ˆ parameter. Consequently, an subsequent estimate θ of the same stimulus parameter value θstim is attracted toward the adaptor. This is the opposite of observed perceptual effects, and we thus conclude that adjustments of the prior in a Bayesian model do not account for adaptation. value θadapt should presumably increase the prior probability in the vicinity of θadapt . Figure 1b schematically illustrates the effect of such a change in the prior distribution. The estimated (perceived) value of the parameter under the adapted condition is attracted to the adapting parameter value. In order to account for observed perceptual repulsion effects, the prior would have to decrease at the location of the adapting parameter, a behavior that seems fundamentally inconsistent with the notion of a prior distribution. ... but increases the reliability of the measurements Since a change in the prior distribution is not consistent with repulsion, we are led to the conclusion that adaptation must change the likelihood function. But why, and how should this occur? In order to answer this question, we reconsider the functional purpose of adaptation. We assume that adaptation acts to allocate more resources to the representation of the parameter values in the vicinity of the adaptor [4], resulting in a local increase in the signal-to-noise ratio (SNR). This can be accomplished, for example, by dynamically adjusting the operational range to the statistics of the input. This kind of increased operational gain around the adaptor has been effectively demonstrated in the process of retinal adaptation [17]. In the context of our Bayesian estimator framework, and restricting to the simple case of a scalar-valued measurement, adaptation results in a narrower conditional probability density p(m|θ) in the immediate vicinity of the adaptor, thus an increase in the reliability of the measurement m. This is offset by a broadening of the conditional probability density p(m|θ) in the region beyond the adaptor vicinity (we assume that total resources are conserved, and thus an increase around the adaptor must necessarily lead to a decrease elsewhere). Figure 2 illustrates the effect of this local increase in signal-to-noise ratio on the likeli- unadapted adapted θadapt p(m2| θ )' 1/SNR θ θ θ1 θ2 θ1 p(m2|θ) θ2 θ m2 p(m1| θ )' m1 m m p(m1|θ) θ θ θ θadapt p(m| θ2)' p(m|θ2) likelihoods p(m|θ1) p(m| θ1)' p(m|θadapt )' conditionals Figure 2: Measurement noise, conditionals and likelihoods. The two-dimensional conditional density, p(m|θ), is shown as a grayscale image for both the unadapted and adapted cases. We assume here that adaptation increases the reliability (SNR) of the measurement around the parameter value of the adaptor. This is balanced by a decrease in SNR of the measurement further away from the adaptor. Because the likelihood is a function of θ (horizontal slices, shown plotted at right), this results in an asymmetric change in the likelihood that is in agreement with a repulsive effect on the estimate. a b ^ ∆θ ^ ∆θ [deg] + 0 60 30 0 -30 - θ θ adapt -60 -180 -90 90 θadapt 180 θ [deg] Figure 3: Repulsion: Model predictions vs. human psychophysics. a) Difference in perceived direction in the pre- and post-adaptation condition, as predicted by the model. Postadaptive percepts of motion direction are repelled away from the direction of the adaptor. b) Typical human subject data show a qualitatively similar repulsive effect. Data (and ﬁt) are replotted from [10]. hood function. The two gray-scale images represent the conditional probability densities, p(m|θ), in the unadapted and the adapted state. They are formed by assuming additive noise on the measurement m of constant variance (unadapted) or with a variance that decreases symmetrically in the vicinity of the adaptor parameter value θadapt , and grows slightly in the region beyond. In the unadapted state, the likelihood is convolutional and the shape and variance are equivalent to the distribution of measurement noise. However, in the adapted state, because the likelihood is a function of θ (horizontal slice through the conditional surface) it is no longer convolutional around the adaptor. As a result, the mean is pushed away from the adaptor, as illustrated in the two graphs on the right. Assuming that the prior distribution is fairly smooth, this repulsion effect is transferred to the posterior distribution, and thus to the estimate. 3 Simulation Results We have qualitatively demonstrated that an increase in the measurement reliability around the adaptor is consistent with the repulsive effects commonly seen as a result of perceptual adaptation. In this section, we simulate an adapted Bayesian observer by assuming a simple model for the changes in signal-to-noise ratio due to adaptation. We address both repulsion and changes in discrimination threshold. In particular, we compare our model predictions with previously published data from psychophysical experiments examining human perception of motion direction. 3.1 Repulsion In the unadapted state, we assume the measurement noise to be additive and normally distributed, and constant over the whole measurement space. Thus, assuming that m and θ live in the same space, the likelihood is a Gaussian of constant width. In the adapted state, we assume a simple functional description for the variance of the measurement noise around the adapter. Speciﬁcally, we use a constant plus a difference of two Gaussians, a b relative discrimination threshold relative discrimination threshold 1.8 1 θ θadapt 1.6 1.4 1.2 1 0.8 -40 -20 θ adapt 20 40 θ [deg] Figure 4: Discrimination thresholds: Model predictions vs. human psychophysics. a) The model predicts that thresholds for direction discrimination are reduced at the adaptor. It also predicts two side-lobes of increased threshold at further distance from the adaptor. b) Data of human psychophysics are in qualitative agreement with the model. Data are replotted from [14] (see also [11]). each having equal area, with one twice as broad as the other (see Fig. 2). Finally, for simplicity, we assume a ﬂat prior, but any reasonable smooth prior would lead to results that are qualitatively similar. Then, according to (2) we compute the predicted estimate of motion direction in both the unadapted and the adapted case. Figure 3a shows the predicted difference between the pre- and post-adaptive average estimate of direction, as a function of the stimulus direction, θstim . The adaptor is indicated with an arrow. The repulsive effect is clearly visible. For comparison, Figure 3b shows human subject data replotted from [10]. The perceived motion direction of a grating was estimated, under both adapted and unadapted conditions, using a two-alternative-forced-choice experimental paradigm. The plot shows the change in perceived direction as a function of test stimulus direction relative to that of the adaptor. Comparison of the two panels of Figure 3 indicate that despite the highly simpliﬁed construction of the model, the prediction is quite good, and even includes the small but consistent repulsive effects observed 180 degrees from the adaptor. 3.2 Changes in discrimination threshold Adaptation also changes the ability of human observers to discriminate between the direction of two different moving stimuli. In order to model discrimination thresholds, we need to consider a Bayesian framework that can account not only for the mean of the estimate but also its variability. We have recently developed such a framework, and used it to quantitatively constrain the likelihood and the prior from psychophysical data [16]. This framework accounts for the effect of the measurement noise on the variability of the ˆ ˆ estimate θ. Speciﬁcally, it provides a characterization of the distribution p(θ|θstim ) of the estimate for a given stimulus direction in terms of its expected value and its variance as a function of the measurement noise. As in [16] we write ˆ ∂ θ(m) 2 ˆ var θ|θstim = var m ( ) |m=θstim . (3) ∂m Assuming that discrimination threshold is proportional to the standard deviation, ˆ var θ|θstim , we can now predict how discrimination thresholds should change after adaptation. Figure 4a shows the predicted change in discrimination thresholds relative to the unadapted condition for the same model parameters as in the repulsion example (Figure 3a). Thresholds are slightly reduced at the adaptor, but increase symmetrically for directions further away from the adaptor. For comparison, Figure 4b shows the relative change in discrimination thresholds for a typical human subject [14]. Again, the behavior of the human observer is qualitatively well predicted. 4 Discussion We have shown that adaptation can be incorporated into a Bayesian estimation framework for human sensory perception. Adaptation seems unlikely to manifest itself as a change in the internal representation of prior distributions, as this would lead to perceptual bias effects that are opposite to those observed in human subjects. Instead, we argue that adaptation leads to an increase in reliability of the measurement in the vicinity of the adapting stimulus parameter. We show that this change in the measurement reliability results in changes of the likelihood function, and that an estimator that utilizes this likelihood function will exhibit the commonly-observed adaptation effects of repulsion and changes in discrimination threshold. We further conﬁrm our model by making quantitative predictions and comparing them with known psychophysical data in the case of human perception of motion direction. Many open questions remain. The results demonstrated here indicate that a resource allocation explanation is consistent with the functional effects of adaptation, but it seems unlikely that theory alone can lead to a unique quantitative prediction of the detailed form of these effects. Speciﬁcally, the constraints imposed by biological implementation are likely to play a role in determining the changes in measurement noise as a function of adaptor parameter value, and it will be important to characterize and interpret neural response changes in the context of our framework. Also, although we have argued that changes in the prior seem inconsistent with adaptation effects, it may be that such changes do occur but are offset by the likelihood effect, or occur only on much longer timescales. Last, if one considers sensory perception as the result of a cascade of successive processing stages (with both feedforward and feedback connections), it becomes necessary to expand the Bayesian description to describe this cascade [e.g., 18, 19]. For example, it may be possible to interpret this cascade as a sequence of Bayesian estimators, in which the measurement of each stage consists of the estimate computed at the previous stage. Adaptation could potentially occur in each of these processing stages, and it is of fundamental interest to understand how such a cascade can perform useful stable computations despite the fact that each of its elements is constantly readjusting its response properties. References [1] K. K¨ rding and D. Wolpert. Bayesian integration in sensorimotor learning. o 427(15):244–247, January 2004. Nature, [2] D C Knill and W Richards, editors. Perception as Bayesian Inference. Cambridge University Press, 1996. [3] Y. Weiss, E. Simoncelli, and E. Adelson. Motion illusions as optimal percept. Nature Neuroscience, 5(6):598–604, June 2002. [4] H.B. Barlow. Vision: Coding and Efﬁciency, chapter A theory about the functional role and synaptic mechanism of visual after-effects, pages 363–375. Cambridge University Press., 1990. [5] M.J. Wainwright. Visual adaptation as optimal information transmission. Vision Research, 39:3960–3974, 1999. [6] N. Brenner, W. Bialek, and R. de Ruyter van Steveninck. Adaptive rescaling maximizes information transmission. Neuron, 26:695–702, June 2000. [7] S.M. Smirnakis, M.J. Berry, D.K. Warland, W. Bialek, and M. Meister. Adaptation of retinal processing to image contrast and spatial scale. Nature, 386:69–73, March 1997. [8] P. Thompson. Velocity after-effects: the effects of adaptation to moving stimuli on the perception of subsequently seen moving stimuli. Vision Research, 21:337–345, 1980. [9] A.T. Smith. Velocity coding: evidence from perceived velocity shifts. Vision Research, 25(12):1969–1976, 1985. [10] P. Schrater and E. Simoncelli. Local velocity representation: evidence from motion adaptation. Vision Research, 38:3899–3912, 1998. [11] C.W. Clifford. Perceptual adaptation: motion parallels orientation. Trends in Cognitive Sciences, 6(3):136–143, March 2002. [12] C. Clifford and P. Wenderoth. Adaptation to temporal modulaton can enhance differential speed sensitivity. Vision Research, 39:4324–4332, 1999. [13] A. Kristjansson. Increased sensitivity to speed changes during adaptation to ﬁrst-order, but not to second-order motion. Vision Research, 41:1825–1832, 2001. [14] R.E. Phinney, C. Bowd, and R. Patterson. Direction-selective coding of stereoscopic (cyclopean) motion. Vision Research, 37(7):865–869, 1997. [15] N.M. Grzywacz and R.M. Balboa. A Bayesian framework for sensory adaptation. Neural Computation, 14:543–559, 2002. [16] A.A. Stocker and E.P. Simoncelli. Constraining a Bayesian model of human visual speed perception. In Lawrence K. Saul, Yair Weiss, and L´ on Bottou, editors, Advances in Neural Infore mation Processing Systems NIPS 17, pages 1361–1368, Cambridge, MA, 2005. MIT Press. [17] D. Tranchina, J. Gordon, and R.M. Shapley. Retinal light adaptation – evidence for a feedback mechanism. Nature, 310:314–316, July 1984. ´ [18] S. Deneve. Bayesian inference in spiking neurons. In Lawrence K. Saul, Yair Weiss, and L eon Bottou, editors, Adv. Neural Information Processing Systems (NIPS*04), vol 17, Cambridge, MA, 2005. MIT Press. [19] R. Rao. Hierarchical Bayesian inference in networks of spiking neurons. In Lawrence K. Saul, Yair Weiss, and L´ on Bottou, editors, Adv. Neural Information Processing Systems (NIPS*04), e vol 17, Cambridge, MA, 2005. MIT Press.

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