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49 nips-2006-Causal inference in sensorimotor integration

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Author: Konrad P. Körding, Joshua B. Tenenbaum

Abstract: Many recent studies analyze how data from different modalities can be combined. Often this is modeled as a system that optimally combines several sources of information about the same variable. However, it has long been realized that this information combining depends on the interpretation of the data. Two cues that are perceived by different modalities can have different causal relationships: (1) They can both have the same cause, in this case we should fully integrate both cues into a joint estimate. (2) They can have distinct causes, in which case information should be processed independently. In many cases we will not know if there is one joint cause or two independent causes that are responsible for the cues. Here we model this situation as a Bayesian estimation problem. We are thus able to explain some experiments on visual auditory cue combination as well as some experiments on visual proprioceptive cue integration. Our analysis shows that the problem solved by people when they combine cues to produce a movement is much more complicated than is usually assumed, because they need to infer the causal structure that is underlying their sensory experience.

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Two cues that are perceived by different modalities can have different causal relationships: (1) They can both have the same cause, in this case we should fully integrate both cues into a joint estimate. [sent-8, score-0.681]

2 In many cases we will not know if there is one joint cause or two independent causes that are responsible for the cues. [sent-10, score-0.29]

3 We are thus able to explain some experiments on visual auditory cue combination as well as some experiments on visual proprioceptive cue integration. [sent-12, score-1.083]

4 Our analysis shows that the problem solved by people when they combine cues to produce a movement is much more complicated than is usually assumed, because they need to infer the causal structure that is underlying their sensory experience. [sent-13, score-0.644]

5 Traditionally, cue combination is formalized as a simple weighted combination of estimates coming from each modality (Fig 1A). [sent-17, score-0.435]

6 Research often focuses on exploring in which coordinate system the problem is being solved [2, 3] and how much weight is given to each variable as a function of the uncertainty in each modality and the prior[4, 5, 6, 7, 8]. [sent-21, score-0.33]

7 However, in many cases people can not be certain of the causal structure. [sent-25, score-0.367]

8 If two cues share a common cause (as in Fig 1B) they should clearly be combined. [sent-26, score-0.317]

9 In such cases people can not know which of the two models to use and have to estimate the causal structure of the problem along with the parameter values. [sent-28, score-0.442]

10 C) In many cases people will be unable to know if one common cause or two independent causes are responsible for the cues. [sent-34, score-0.477]

11 In that case people will have to estimate which causal structure is present from the properties of their sensory stimuli. [sent-35, score-0.455]

12 2 Cue combination: one common cause A large number of recent studies have interpreted the results from cue combination studies in a Bayesian framework[13]. [sent-38, score-0.412]

13 We discuss the case of visuoauditory integration as the statistical relations are identical in other cue combination cases. [sent-39, score-0.269]

14 It is acknowledged that if a signal is coming from a speciﬁc position the signal received by the nervous system in each modality will be noisy. [sent-41, score-0.534]

15 If the real position of a stimulus is xreal then the nervous system will not be able to directly know this variable but the visual modality will obtain a noisy estimate thereof x vis . [sent-42, score-1.247]

16 Typically it is assumed that in the process that the visually perceived position is a noisy version of the real position x vis = xreal +noise. [sent-43, score-0.99]

17 A statistical treatment thus results in p(xvis |xreal ) = N (xreal − xvis , σvis ) where σvis is the variance introduced by the visual modality and N (μ, σ) stands for a Gaussian distribution with mean μ and standard deviation σ. [sent-44, score-0.484]

18 If two cues are available, for example vision and audition then it is assumed that both cues x vis and xaud provide noisy observations of the relevant variable x real . [sent-45, score-0.855]

19 These papers assumed that we have two sources of information about one and the same variable and have shown that in psychophysical experiments people often show this kind of optimal integration and that the weights can be predicted from the variances [13, 14, 15, 4, 16]. [sent-50, score-0.336]

20 3 Combination of visual and auditory cues: uncertainty about the causal structure Here we consider the range of experiments where people hear a tone and simultaneously see a visual stimulus that may or may not come from the same position. [sent-53, score-1.224]

21 Subjects are asked to estimate which direction the tone is coming from and point to that direction – placing this experiment in the realm of sensorimotor integration. [sent-54, score-0.31]

22 Subjects are asked to estimate which direction the tone is coming from and do so with a motor response. [sent-55, score-0.341]

23 To optimally estimate where the tone is coming from people need to infer the causal structure (Fig 1 C) and decide if they should assume a single cause or two causes. [sent-56, score-0.717]

24 For different distances between the visual and the auditory stimulus they analyzed the strategies people use to estimate the position of the auditory stimuli (see ﬁgure 2). [sent-59, score-1.139]

25 It has long been realized that integration of different cues should only occur if the cues have the same cause [9, 10, 8, 19]. [sent-60, score-0.573]

26 2 Inference The probability that the two signals are from the same source will only weakly depend on the spatial prior but mostly depend on the distance Δ av = xaud − xvis between visually and auditory perceived positions. [sent-67, score-0.812]

28 Using Equation 4 we can then calculate the optimal solution which is: x = p(same|Δav )ˆsame + (1 − p(same|Δav ))ˆdifferent ˆ x x (6) We know the optimal estimates in the same case already from equation 3 and in the different case the optimal estimate exclusively relies on the auditory signal. [sent-69, score-0.322]

29 We furthermore assume that the position sensed by the sensory system is a noisy version of x observed = x + where is drawn from a ˆ Gaussian with zero mean and a standard deviation of σ motor . [sent-70, score-0.443]

30 In both auditory and visual trials the noise will have two sources, motor noise and sensory noise. [sent-78, score-0.638]

31 Even if people knew perfectly where a stimulus was coming from they would make small errors at pointing because their motor system is not perfect. [sent-79, score-0.467]

32 We assume that visual only trials are dominated by motor noise, stemming from motor errors and memory errors and that the noise in the visual trials is essentially exclusively motor noise (σ vis = 0. [sent-80, score-1.105]

33 5 deg and because variances are added linearly σ aud = 82 − 2. [sent-84, score-0.26]

34 A Bayes B Visual stimulus Gain α [%] Bayes unoptimized Auditory stimulus Gain>0 MAP C Gain<0 mean Combination Experiment Wallace et al Estimated one cause Estimated two causes One cause -5 Two causes 0 Position [deg] -5 Figure 2: Uncertainty if one or two causes are relevant. [sent-87, score-0.672]

35 A) The gain, the relative weight of vision for the estimation of the position of the auditory signal is shown. [sent-89, score-0.472]

36 It is plotted as a function of the spatial disparity, the distance between visual and auditory stimulus. [sent-90, score-0.44]

37 A gain value of α = 100% implies that subjects only use visual information. [sent-91, score-0.377]

38 A negative α means that on average subjects point away from the visual stimulus. [sent-92, score-0.306]

39 The visual stimulus is always at -5 deg (solid line) and the auditory stimulus is always straight ahead at 0deg(dotted line). [sent-95, score-0.596]

40 Visual perception is very low noise and the perceived position x vis is shown as red dots (each dot is one trial). [sent-96, score-0.578]

41 Auditory perception is noisy and the perceived auditory position x aud is shown as black dots. [sent-97, score-0.78]

42 In the white area where the subject perceive two causes, the average position of perceived auditory signals is further to the right. [sent-98, score-0.602]

43 From the speciﬁcations of the experiments we know that the distribution of auditory sources has a width of 20deg relative to the ﬁxation point and we assume that this width is known to the subjects from repeated trials. [sent-105, score-0.506]

44 The model predicts the counterintuitive ﬁnding that the trials where people inferred two causes exhibit negative gain. [sent-111, score-0.316]

45 On some trials noise in the auditory signal will make it appear as if the auditory signal is very close to the visual signal. [sent-118, score-0.8]

46 In this case the system will infer that both have the same source and part of the reported high gain for the fused cases will be because noise already perturbed the auditory signal towards the visual. [sent-119, score-0.424]

47 However, on some trials the auditory signal will be randomly perturbed away from the visual signal. [sent-120, score-0.517]

48 5 Maximum A Posteriori over causal structure In the derivations above we assumed that people are fully Bayesian, in the sense that they consider both possible structures for cue-integration and integrate over them to arrive at an optimal decision. [sent-125, score-0.367]

49 This difference is observed because the MAP model strongly underestimates the uncertainty in the single cause case and strongly overestimates the uncertainty in the dual cause case (Fig 2C). [sent-131, score-0.432]

50 The Bayesian model on the other hand always considers that it could be wrong, leading to more variance in the single cause case and less in the dual cause case. [sent-132, score-0.359]

51 This effect goes away if we assume that people underestimate the variance of the auditory source relative to the ﬁxation spot (data not shown). [sent-134, score-0.434]

52 In summary, the problem of crossmodal integration is much more complicated than it seems as it necessitates inference of the causal structure. [sent-137, score-0.294]

53 4 Combination of visual and proprioceptive cues Typical experiments in movement psychophysics where a virtual reality display is used to disturb the perceived position of the hand lead to an analogous problem. [sent-139, score-0.897]

54 In these experiments subjects proprioceptively feel their hand somewhere, but they cannot see their hand; at the same time, they visually perceive a cursor somewhere. [sent-140, score-0.714]

55 Subjects again can not be sure if the seen cursor position and the felt hand position are cues about the same variable (hand=cursor) or if each of them are independent and the experiment is just cheating them leading to the same causal structure inference problem described above. [sent-141, score-1.25]

56 In these experiments one hand is moving a cursor to either the position of a visually displayed (v) target or the position of the other hand (p). [sent-144, score-0.979]

57 People need to estimate two distinct variables: (1) the direction in which they are to move their arm, a visually perceived variable, the so-called movement vector (MV) and (2) a proprioceptively perceived variable, the conﬁguration of their joints (J). [sent-145, score-0.453]

58 Subjects obtain visual information about the position of the cursor and they obtain proprioceptive information from feeling the position of their hand. [sent-146, score-1.062]

59 Traditionally it would have been assumed that the seen cursor position and the proprioceptively felt hand position are cues caused by one single variable, the hand. [sent-147, score-1.062]

60 As a result, the position of the cursor uniquely deﬁnes the conﬁguration of the joints and vice versa. [sent-148, score-0.56]

61 As in the cue combination case above there should not be full cue combination but instead each variable (MV) and (J) should be estimated separately. [sent-149, score-0.398]

62 In this experiment a situation is produced where the visual position of a cursor is different from the actual position of the right hand. [sent-150, score-0.915]

63 The experimental studies then report the gain α, the linear weight α of vision on the estimate of (MV) and (J) in both the visual and the proprioceptive target conditions(ﬁgure 3A and B). [sent-154, score-0.564]

64 If people only inferred about one common cause then the weight of vision should always be the same, indicating that more than just one cause is assumed by the subjects. [sent-155, score-0.506]

65 In the sensorimotor integration case there is uncertainty about the alignment of the two coordinate systems. [sent-158, score-0.342]

66 When using information from one coordinate system for an estimation in a different coordinate system there is uncertainty about the alignment of the coordinate systems. [sent-160, score-0.392]

67 This means that when we use visual information to estimate the position of a joint in joint space our visual system appears to be more noisy and vice versa. [sent-161, score-0.666]

68 As we are only interested in estimates along one dimension and can model the uncertainty about the alignment of the coordinate systems as a one dimensional Gaussian with width σ trans . [sent-162, score-0.328]

69 When using information from one modality for estimations of a variable in the other coordinate system we need 2 2 2 to use σef f ective = σmodality + σtrans . [sent-163, score-0.26]

70 The two target conditions in the experiments, moving the cursor to a visual target (v) and moving the cursor to the position of the other hand (p) produce two different estimation problems. [sent-164, score-1.362]

71 When we try to move a cursor to a visually displayed target we must compute MV in visual space. [sent-165, score-0.71]

72 If to the contrary we try to move a cursor with one hand to the position of the other hand then we must calculate MV in joint space. [sent-166, score-0.722]

73 Altogether people are faced with 4 problems, they have to estimate (MV) and (J) in both the visual (v) condition and the proprioceptive (p) condition. [sent-168, score-0.572]

74 2 Probabilistic model As above we assume that visual and proprioceptive uncertainty lead to probability distributions in the respective space that are characterized by Gaussians of width σ vis and σprop . [sent-170, score-0.746]

75 Under these circumstances it is only important how likely on average people ﬁnd that the two percepts are uniﬁed (p unif ied = psame p(Δpv |same)). [sent-173, score-0.357]

76 We assume that when moving the cursor to a visual target the average squared deviation of the cursor and the target in visual space is minimized. [sent-174, score-1.345]

77 We assume that when moving the cursor to a proprioceptive target the average squared deviation of the cursor and the target in proprioceptive space is minimized. [sent-175, score-1.253]

78 Apart from this difference the whole derivation of the equations is identical to the one above for the auditory visual integration. [sent-176, score-0.44]

79 3 Tool use Above we assumed that cursor and hand either have the same cause (the position of the hand, or different causes and are therefore unrelated. [sent-179, score-0.861]

80 The cursor could be seen as a tool that is seen displaced relative to our hand. [sent-181, score-0.52]

81 As tools are typically short the probability is largest that the tip of a tool is at the position of the hand and this probability will decay with increasing distance between the hand and the position of the tool. [sent-183, score-0.62]

82 The distance between the tip of the tool and the hand is thus another random variable that is assumed to be Gaussian with width σtool (see ﬁg. [sent-184, score-0.3]

83 The cursor will be close to the position of the hand. [sent-194, score-0.56]

84 model has several parameters, important the uncertainties of proprioception and of the coordinate transformation compared to the visual uncertainty. [sent-197, score-0.32]

85 He estimated σ vis = 1cm,σprop = 3cm,σtrans = 5cm. [sent-206, score-0.295]

86 The image of an arm is rendered on top of the cursor position. [sent-210, score-0.452]

87 In our interpretation, the rendering of the arm makes the probability much higher that actually the position of the visual display is the position of the hand and p unif ied would be much higher. [sent-212, score-0.74]

88 5 Analysis if subjects view a cursor as a tool Another possible model that seemed very likely to us was assuming that the cursor should appear somewhere close to the hand modeling the cursor hand relationship as another Gaussian variable (Fig 3E). [sent-214, score-1.62]

89 We ﬁt the 3 parameters of this model, the uncertainty of proprioception and the coordinate transformation relative to the visual uncertainty as well as the width of the Gaussian describing the tool. [sent-215, score-0.474]

90 Sober and Sabes [5, 6] explain the ﬁnding that two variables are estimated by the ﬁnding that cortex exhibits two important streams of information processing, one for visual processing and the other for motor tasks [20]. [sent-222, score-0.307]

91 If people see a cursor close to their hand they do not assume that they actually see their hand. [sent-224, score-0.652]

92 The models that we introduced can be understood as special instantiations of a model where the cursor position relative to the hand is drawn from a general probability distribution. [sent-225, score-0.627]

93 5 Discussion An impressive range of recent studies show that people do not just estimate one variable in situations of cue combination [5, 6, 17, 18]. [sent-226, score-0.487]

94 Here we have shown that the statistical problem that people solve in such situations involves an inference about the causal structure. [sent-227, score-0.396]

95 The problem faced by the nervous system is similar to cognitive problems that occur in the context of causal induction. [sent-229, score-0.294]

96 Many experiments show that people and in particular infants interpret events in terms of cause and effect [11, 21, 22]. [sent-230, score-0.333]

97 Demonstration of cue recruitment: change in visual appearance by means of pavlovian conditioning. [sent-241, score-0.312]

98 A theory of causal learning in children: causal maps and bayes nets. [sent-312, score-0.36]

99 Optimal integration of texture and motion cues to depth. [sent-329, score-0.256]

100 Integration of proprioceptive and visual position-information: An experimentally supported model. [sent-338, score-0.34]

similar papers computed by tfidf model

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