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Figure 2. Different causal structures and their respective optimal estimates of location of an auditory source. The red curve represents auditory likelihood distribution p(x,|s,) and the blue curve represents the visual likelihood p(xy|sv). Here, for the sake of simplicity, we assume that the prior is uniform (i.e. uninformative), and both auditory and visual likelihoods have a normal distribution. (a) If the audio and visual signals x4 and xy are assumed to have been caused by separate sources (C= 2), the signals should be kept separate. Thus, the best estimate of location of sound is only based on the auditory signal. The optimal estimate of location of sound S$, c-2 is therefore the mean of the auditory likelihood. (b) If the two signals are assumed to have been caused by the same object (C = 1), the two sensory signals should be combined by multiplying the two likelihoods that results in the brown distribution. Therefore, the optimal estimate of the location of sound §, c-1 is the mean of this combination distribution. (c) In general, the observer does not know the causal structure of events in the environment, and only has access to the sensory signals; therefore there is uncertainty about the cause of the signals. In this case, the optimal estimate of the location of sound, Sy, is a weighted average of the estimates corresponding to the two causal structures, the mean of the brown and the mean of the red distributions, each weighted by their respective probability Sa = p(C = 1|xXa,Xv)$aca1 + p(C = 2|xa, Xv) Sa.c=2-

Figure 2 Different causal structures and their respective optimal estimates of location of an auditory source. The red curve represents auditory likelihood distribution p(x,|s,) and the blue curve represents the visual likelihood p(xy|sv). Here, for the sake of simplicity, we assume that the prior is uniform (i.e. uninformative), and both auditory and visual likelihoods have a normal distribution. (a) If the audio and visual signals x4 and xy are assumed to have been caused by separate sources (C= 2), the signals should be kept separate. Thus, the best estimate of location of sound is only based on the auditory signal. The optimal estimate of location of sound S$, c-2 is therefore the mean of the auditory likelihood. (b) If the two signals are assumed to have been caused by the same object (C = 1), the two sensory signals should be combined by multiplying the two likelihoods that results in the brown distribution. Therefore, the optimal estimate of the location of sound §, c-1 is the mean of this combination distribution. (c) In general, the observer does not know the causal structure of events in the environment, and only has access to the sensory signals; therefore there is uncertainty about the cause of the signals. In this case, the optimal estimate of the location of sound, Sy, is a weighted average of the estimates corresponding to the two causal structures, the mean of the brown and the mean of the red distributions, each weighted by their respective probability Sa = p(C = 1|xXa,Xv)$aca1 + p(C = 2|xa, Xv) Sa.c=2-

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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Provided for non-commercial research and education use. Not for reproduction, distribution or commercial use.
This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Provided for non-commercial research and education use. Not for reproduction, distribution or commercial use.
Figure 3. Comparison of different models. The prior distributions for different models are shown in the leftmost two columns. For simplicity, for models with a 2 dimensional prior, we only focus on one dimension (e.g. location of sound), and show the one dimensional prior for a specific value of the second dimension (e.g. sy = 0) i.e. a slice from the two dimensional prior. This is shown in the second column. Likelihood functions are shown in the third column. Multiplying the priors and likelihoods creates the posterior distribution (e.g. p(sa|xa,xXv)) shown in the fourth column. A response (e.g. $4) is generated by taking the arg-max or mean of the posterio! distribution. The relationship between the physical source (e.g. sy) and the perceptual estimate (e.g. $4) is shown in the rightmost column. Red line represents model averaging for each model, blue is model selection (Box 1), dotted line is the response for the independent model (C = 2), dashed line for the full fusion (C = 1). All models except for those in (c), exhibit a nonlinear response behavior (both red and blue curves): the sources of information get combined so long as the discrepancy between ther is not large. In this regime, larger discrepancies result in a larger interaction. However, once the discrepancy becomes large, the sources no longer get combined, and the interaction goes back to zero. The models listed for each row are qualitatively similar, although small differences can exist. For example, Knill [34] uses a log-gaussian (as opposed to Gaussian) as one element of the mixed prior. Wozny et al. [27] use a three-dimensional Gaussian. (a) These are mixture models with the prior distributior composed of two components. The spike represents one hypothesis (common cause in [28] and [30]; and circle hypothesis in [34]) and the isotopic Gaussian represen another hypothesis (e.g. independent causes or ellipse shape; see text for explanation). The rightmost panel represents the response Sy, (e.g. perceived auditory location o: the perceived slant) for s,4 = 0 (representing a specific location of sound or the slant as conveyed by binocular disparity) as a function of the other source, sy (e.g. location o visual stimulus or the compression cue). As sy gets farther from sy, (i.e. zero) the response gets farther from s,, however when the discrepancy between sy and sy, get large the response goes back to s, again (in effect ignoring sy). (b) The prior is composed of two components, a Gaussian and a uniform distribution. The response behavior i qualitatively the same as that of models in (a). (c) The prior is a Gaussian ridge, together with the Gaussian likelihood, this results in a Gaussian posterior, and a linea response behavior. (d) The prior is a heavy-tailed distribution. This results in a nonlinear response behavior similar to those of (a) and (b). The prior has a mode at zerc speed. The rightmost panel shows response (perceived speed) as a function of true visual speed. As the visual sensation gets farther from the expected speed (zero), the error first increases, and then it decreases as the discrepancy between the two gets too large. (e) In this model, the prior is uniform, however one of the likelihood functions is a heavy-tailed distribution, shown in this case in the second column. This likelihood for the texture cue is combined with the Gaussian likelihood of another cue (binocula disparity) shown in the third column. The rightmost panel shows perceived slant as a function of texture signal, for a disparity cue consistent with zero slant. Comparison o the posterior and response in this model with those of (a), (b) and (d) shows that a heavy-tailed prior and heavy-tailed likelihood have the same effect on the posterior anc thisre tha racenanes hahaaine It should be noted that all of these models (Figure 3a,b,d,e) involve ‘complex’ priors (referring to mix- ture of priors or unstructured priors). This feature dis- tinguishes these models from models that use a single parametric (typically Gaussian) prior distribution. For example, Bresciani et al. [24] used a Gaussian distribution (Figure 3c) to model the difference between causes (p(sa, Sy) = N(s,4 — Sy, o)), and Wozny et al. [27] used a multi- variate Gaussian distribution as prior for three sources (p(Sa,Sv,Sr)). Because the product of Gaussians is itself a In the models of multisensory integration presented in Roach et al. [26] and Sato et al. [44] (Fig. 3b), the choice of causal structures is not between a single source or independent sources, but instead between correlated sources and independent sources. In heavy-tailed models (Figure 3d,e), the heavy-tailed distribution can be thought of as a mixture of two distributions, a narrow one and a wide one, very similar to the prior distributions shown in Figure 3a,b. In Stocker and Simoncelli [39], Lu et al. [40] and Dokka et al. [43] it is the prior that is heavy tailed, whereas in Natarajan et al. [42] and Girshick and Banks [41] it is the likelihood function that is heavy-tailed. How- ever, the effect on the posterior function and response
Figure 3. Comparison of different models. The prior distributions for different models are shown in the leftmost two columns. For simplicity, for models with a 2 dimensional prior, we only focus on one dimension (e.g. location of sound), and show the one dimensional prior for a specific value of the second dimension (e.g. sy = 0) i.e. a slice from the two dimensional prior. This is shown in the second column. Likelihood functions are shown in the third column. Multiplying the priors and likelihoods creates the posterior distribution (e.g. p(sa|xa,xXv)) shown in the fourth column. A response (e.g. $4) is generated by taking the arg-max or mean of the posterio! distribution. The relationship between the physical source (e.g. sy) and the perceptual estimate (e.g. $4) is shown in the rightmost column. Red line represents model averaging for each model, blue is model selection (Box 1), dotted line is the response for the independent model (C = 2), dashed line for the full fusion (C = 1). All models except for those in (c), exhibit a nonlinear response behavior (both red and blue curves): the sources of information get combined so long as the discrepancy between ther is not large. In this regime, larger discrepancies result in a larger interaction. However, once the discrepancy becomes large, the sources no longer get combined, and the interaction goes back to zero. The models listed for each row are qualitatively similar, although small differences can exist. For example, Knill [34] uses a log-gaussian (as opposed to Gaussian) as one element of the mixed prior. Wozny et al. [27] use a three-dimensional Gaussian. (a) These are mixture models with the prior distributior composed of two components. The spike represents one hypothesis (common cause in [28] and [30]; and circle hypothesis in [34]) and the isotopic Gaussian represen another hypothesis (e.g. independent causes or ellipse shape; see text for explanation). The rightmost panel represents the response Sy, (e.g. perceived auditory location o: the perceived slant) for s,4 = 0 (representing a specific location of sound or the slant as conveyed by binocular disparity) as a function of the other source, sy (e.g. location o visual stimulus or the compression cue). As sy gets farther from sy, (i.e. zero) the response gets farther from s,, however when the discrepancy between sy and sy, get large the response goes back to s, again (in effect ignoring sy). (b) The prior is composed of two components, a Gaussian and a uniform distribution. The response behavior i qualitatively the same as that of models in (a). (c) The prior is a Gaussian ridge, together with the Gaussian likelihood, this results in a Gaussian posterior, and a linea response behavior. (d) The prior is a heavy-tailed distribution. This results in a nonlinear response behavior similar to those of (a) and (b). The prior has a mode at zerc speed. The rightmost panel shows response (perceived speed) as a function of true visual speed. As the visual sensation gets farther from the expected speed (zero), the error first increases, and then it decreases as the discrepancy between the two gets too large. (e) In this model, the prior is uniform, however one of the likelihood functions is a heavy-tailed distribution, shown in this case in the second column. This likelihood for the texture cue is combined with the Gaussian likelihood of another cue (binocula disparity) shown in the third column. The rightmost panel shows perceived slant as a function of texture signal, for a disparity cue consistent with zero slant. Comparison o the posterior and response in this model with those of (a), (b) and (d) shows that a heavy-tailed prior and heavy-tailed likelihood have the same effect on the posterior anc thisre tha racenanes hahaaine It should be noted that all of these models (Figure 3a,b,d,e) involve ‘complex’ priors (referring to mix- ture of priors or unstructured priors). This feature dis- tinguishes these models from models that use a single parametric (typically Gaussian) prior distribution. For example, Bresciani et al. [24] used a Gaussian distribution (Figure 3c) to model the difference between causes (p(sa, Sy) = N(s,4 — Sy, o)), and Wozny et al. [27] used a multi- variate Gaussian distribution as prior for three sources (p(Sa,Sv,Sr)). Because the product of Gaussians is itself a In the models of multisensory integration presented in Roach et al. [26] and Sato et al. [44] (Fig. 3b), the choice of causal structures is not between a single source or independent sources, but instead between correlated sources and independent sources. In heavy-tailed models (Figure 3d,e), the heavy-tailed distribution can be thought of as a mixture of two distributions, a narrow one and a wide one, very similar to the prior distributions shown in Figure 3a,b. In Stocker and Simoncelli [39], Lu et al. [40] and Dokka et al. [43] it is the prior that is heavy tailed, whereas in Natarajan et al. [42] and Girshick and Banks [41] it is the likelihood function that is heavy-tailed. How- ever, the effect on the posterior function and response
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