The aim of this paper is to provide perceptual scientists with a quantitative framework for modeling a variety of common perceptual behaviors, and to unify various perceptual inference tasks by exposing their common computational underpinnings. This paper derives a model Bayesian observer for perceptual contexts with linear Gaussian generative processes. I demonstrate the relationship between four fundamental perceptual situations by expressing their corresponding posterior distributions as consequences of the model's predictions under their respective assumptions

Battaglia, Peter W.

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Computer Science and Artificial Intelligence LaboratoryTechnical Reportm a s s a c h u s e t t s  i n s t i t u t e  o f  t e c h n o l o g y,  c a m b r i d g e ,  m a  0 213 9  u s a  —  w w w. c s a i l . m i t . e d uMIT-CSAIL-TR-2010-046 September 21, 2010Bayesian perceptual inference in linear Gaussian modelsPeter W. BattagliaBayesian perceptual inference in linear Gaussian modelsPeter W. BattagliaSeptember 21, 2010BCS and CSAIL, MITCambridge, MA 02139pbatt@mit.eduAbstractThe aim of this paper is to provide perceptual scientists with a quantitative frameworkfor modeling a variety of common perceptual behaviors, and to unify various perceptual in-ference tasks by exposing their common computational underpinnings. This paper derives amodel Bayesian observer for perceptual contexts with linear Gaussian generative processes.I demonstrate the relationship between four fundamental perceptual situations by expressingtheir corresponding posterior distributions as consequences of the model's predictions undertheir respective assumptions.1 IntroductionPerception is the process of inferring scene properties that cannot be directly observed from theobservable sensory evidence they generate. There is substantial regularity in the structure of scenesin the world, and sensations are generated by a consistent, yet noisy, process; modeling perceptionas Bayesian inference is attractive because it oﬀers principled, normative behavioral predictionsfor perceptual tasks in this type of structured, uncertain world. Scientists compare these predic-tions with experimental measurements to reveal an organism's internal computational proceduresresponsible for its perceptually-guided behaviors.However formulating and evaluating Bayesian perception models can be daunting due to theirmathematical complexities and algorithmic subtleties. This report aims to increase perceptionscientists' access to Bayesian modeling by presenting a powerful class of models (linear Gaussian)that naturally encompass an important range of perceptual phenomena. These models are easy toconstruct, and can make exact, interpretable predictions.Linear Gaussian models apply to perceptual situations in which the scene properties are apri-ori Gaussian-distributed, and generate sensory evidence by a linear process corrupted by Gaussiannoise. They can also apply to log-Gaussian prior and noise distributions, with log-linear (multi-plicative/divisive) sensory generative processes, and may be extended to more complicated genera-tive processes for which Taylor series expansions provide good approximations. In practice, linearGaussian models can be applied to many common sensation/perception situations, like spatiallocalization, temporal perception, size, lightness, and color constancy, and many others.1Section 2 presents the abstract linear Gaussian model and a deriviation of the posterior dis-tribution over unobserved variables given observed variables. Section 3 tailors the framework tomodeling four qualitatively-distinct, elementary perceptual situations [10]. Section 4 brieﬂy outlinesa decision-making framework compatible with the perceptual inference framework.2 Linear Gaussian ModelThis section presents the linear Gaussian model, and derives the posterior inference formulae. Thederivation is based on general linear algebra rules, properties of Gaussians, and is examined ingreater depth by [13].2.1 Derivation of posterior for linear Gaussian modelConsider latent random vector, L, that generates observable data vector, D, as:D = GL+ Ωwhere G is a matrix that represents the deterministic component of the observation transform, Ωrepresents zero-mean (0), additive observation noise, and N (X;Y,Z) is a normal distribution overX, with mean vector Y , and covariance matrix Z. Assume L and Ω have normal prior distributions,Pr(L) = N (L;µL,ΣL)Pr(Ω) = N (Ω;0,ΣΩ)By rules for linear transformations between normal random variables, the conditional and marginallikelihoods of the data are,Pr(D | L) = N (D;GL,ΣΩ)Pr(D) = N (D;GµL, GΣLGT + ΣΩ)Reformulating the conditional likelihood as an unnormalized distribution over L gives,Pr(D | L) = c · N(L;(GTΣ−1Ω G)−1GTΣ−1Ω D,(GTΣ−1Ω G)−1)where c is a constant.The joint distribution over L and D can be factored:Pr(L,D) = Pr(D | L)Pr(L)and Bayes' theorem deﬁnes the posterior:Pr(L | D) = Pr(D | L)Pr(L)Pr(D)=c · kPr(D)N (L;µpost,Σpost)2where k and Pr(D) are constants (µpost and Σpost are deﬁned next).Because Pr(L | D) and N (L;µpost,Σpost) are both densities, constant c·kPr(D) must equal 1. So,Pr(L | D) = N (L;µpost,Σpost)Σpost =(GTΣ−1Ω G+ Σ−1L)−1µpost = Σpost(GTΣ−1Ω D + Σ−1L µL)=(GTΣ−1Ω G+ Σ−1L)−1GTΣ−1Ω D +(GTΣ−1Ω G+ Σ−1L)−1Σ−1L µLIf the posterior over only a subset of the elements of L is desired, because the posterior is normalthe undesired latent elements can be easily marginalized out by deleting their corresponding rows(and columns) from µpost and Σpost.3 Application to perceptual inferenceThis section considers several elementary perceptual situations, characterized by Kersten et al.(2004) [10] (Figure 4), by deﬁning their generative processes under linear Gaussian assumptions,and their corresponding posterior inference rules. In each case, there are between 1 and 2 latent anddata elements, but this can be extended arbitrarily by adding elements to the L and D vectors, andtheir respective parameter vectors/matrices. After each application, several references are providedin which the authors explicitly or implicitly use some form of the model.3.1 Basic BayesConsider an observer who wishes to infer latent scene property L = l from observed sensory dataD = d, with G = g, Ω = ω, µL = µl, ΣL = σ2l , and ΣΩ = σ2ω.The posterior over L given D is:Pr(L | D) = N (L;µpost,Σpost)Σpost =(g2σ2ω+1σ2l)−1=σ2ωσ2lg2σ2l + σ2ωµpost =(g2σ2ω+1σ2l)−1(dgσ2ω+µlσ2l)=dgσ2l + µlσ2ωg2σ2l + σ2ωThis model was used by [19] for modeling motion perception.Using the common assumptions that g = 1 and σ2l →∞ results in a simple perceptual inferencerule,Pr(L | D) = N (L;µpost,Σpost) = N(l; d, σ2ω)This special case has been used implicitly by too many authors to name, in a very broad numberof perception studies.33.2 Cue combinationConsider an observer who wishes to infer latent scene property L = l from two pieces of observed sen-sory data D =[d1d2], with G =[g1g2], Ω =[ω1ω2], µL = µl, ΣL = σ2l , and ΣΩ =[σ2ω1 00 σ2ω2].The posterior over L given D is:Pr(L | D) = N (L;µpost,Σpost)Σpost =(g21σ2ω1+g22σ2ω2+1σ2l)−1=σ2ω1σ2ω2σ2lg21σ2ω2σ2l + g22σ2ω1σ2l + σ2ω1σ2ω2µpost =(g21σ2ω1+g22σ2ω2+1σ2l)−1(d1g21σ2ω1+d2g22σ2ω2+µlσ2l)=d1g21σ2ω2σ2lg21σ2ω2σ2l + g22σ2ω1σ2l + σ2ω1σ2ω2+d2g22σ2ω1σ2lg21σ2ω2σ2l + g22σ2ω1σ2l + σ2ω1σ2ω2+µlσ2ω1σ2ω2g21σ2ω2σ2l + g22σ2ω1σ2l + σ2ω1σ2ω2Using the common assumptions that g1 = g2 = 1 and σ2l → ∞ results in a simple perceptualinference rule,Pr(L | D) = N (L;µpost,Σpost) = N(l;d1σ2ω2 + d2σ2ω1σ2ω1 + σ2ω2,σ2ω1σ2ω2σ2ω1 + σ2ω2)This model was used by [9, 20, 7, 4, 2, 12, 8] and many more for modeling human cue integration.3.3 DiscountingConsider an observer who wishes to infer latent scene properties L =[l1l2], from observed sensorydata D = d, with G =[g1 g2], Ω = ω, µL =[µl1µl2], ΣL =[σ2l1 00 σ2l2], and ΣΩ = σ2ω.The posterior over L given D is:Pr(L | D) = N (L;µpost,Σpost)Σpost = g21σ2ω + 1σ2l1 g1g2σ2ωg1g2σ2ωg22σ2ω+ 1σ2l2−1µpost = g21σ2ω + 1σ2l1 g1g2σ2ωg1g2σ2ωg22σ2ω+ 1σ2l2−1  dg1σ2ω + µl1σ2l1dg2σ2ω+µl2σ2l2Using the common assumptions that g1 = 1 and σ2l1→∞,Pr(L | D) = N (L;µpost,Σpost) = N(L;[d− µl2g2µl2],[g22σ2l2+ σ2ω −g2σ2l2−g2σ2l2 σ2l2])4If only l1 is relevant to the observer (i.e. l2 is a nuisance variable), then,Pr(l1 | D) = N (l1;µpost,Σpost) = N(l1; d− µl2g2, g22σ2l2 + σ2ω)This model was implicitly used by [17, 16, 1], and many more, and can explain a variety ofhuman perceptual biases.3.4 Explaining awayConsider an observer who wishes to infer latent scene properties L =[l1l2], from observed sensorydata D =[d1d2], with G =[g1,1 g1,20 g2,2], Ω =[ω1ω2], µL =[µl1µl2], ΣL =[σ2l1 00 σ2l2], andΣΩ =[σ2ω1 00 σ2ω2].The posterior over L given D is:Pr(L | D) = N (L;µpost,Σpost)Σpost = g21,1σ2ω1 + 1σ2l1 g1,1g1,2σ2ω1g1,1g1,2σ2ω1g21,2σ2ω1+g22,2σ2ω2+ 1σ2l2−1µpost = g21,1σ2ω1 + 1σ2l1 g1,1g1,2σ2ω1g1,1g1,2σ2ω1g21,2σ2ω1+g22,2σ2ω2+ 1σ2l2−1  d1g1,1σ2ω1 + µl1σ2l1d1g1,2σ2ω1+d2g2,2σ2ω2+µl2σ2l2Using the common assumptions that g1,1 = g2,2 = 1, σ2l1→∞, and σ2l2 →∞,Pr(L | D) = N (L;µpost,Σpost) = N(L;[d1 − d2g1,2d2],[g21,2σ2ω2 + σ2ω1 −g1,2σ2ω2−g1,2σ2ω2 σ2ω2])If only l1 is relevant to the observer (i.e. l2 is a nuisance variable), then,Pr(l1 | D) = N (l1;µpost,Σpost) = N(l1; d1 − d2g1,2, g21,2σ2ω2 + σ2ω1)A slightly more general assumption is that g1,1 = g2,2 = 1, σ2l1→∞ (σ2l2 remains ﬁnite),Pr(L | D) = N (L;µpost,Σpost)Σpost =1σ2l2 + σ2ω2[g21,2σ2l2σ2ω2 + σ2l2σ2ω1 + σ2ω1σ2ω2 −g1,2σ2l2σ2ω2−g1,2σ2l2σ2ω2 σ2l2σ2ω2]µpost = d1 − d2g1,2σ2l2σ2l2+σ2ω2− µl2g1,2σ2ω2σ2l2+σ2ω2d2g1,2σ2l2σ2l2+σ2ω2+µl2g1,2σ2ω2σ2l2+σ2ω25If only l1 is relevant to the observer (i.e. l2 is a nuisance variable), then,Pr(l1 | D) = N (l1;µpost,Σpost) = N(l1; d1 −d2g1,2σ2l2+ µl2g1,2σ2ω2σ2l2 + σ2ω2,g21,2σ2l2σ2ω2 + σ2l2σ2ω1 + σ2ω1σ2ω2σ2l2 + σ2ω2)This model was used by [5, 3, 6, 11], and can be applied to the broad class of perceptualconstancy phenomena.4 Perceptual decision-makingThis section brieﬂy outlines Bayesian decision theory, and how to apply the inference rules fromSection 3 to make perceptual judgments. For a detailed treatment of Bayesian perceptual decision-making, see [14, 15].4.1 Bayesian decision theoryBayesian decision theory (BDT) prescribes optimal decision-making based on inferred posteriordistributions over the state of the world. BDT deﬁnes a risk function, R(α,D), that representsthe expected reward (negative loss) for diﬀerent combinations of data D and actions α. Therisk function is an expectation over the observer's rewards, characterized by the reward (negativeloss) function Λ(α,L), under the information inferred about the environment, characterized by theposterior Pr(L | D),R(α,D) =ˆLΛ(α,L)Pr(L | D)dLOptimal agents select actions with maximal expected reward by computing α˜ = arg maxα R(α,D).Sometimes random components, ν, are included to characterize the eﬀects of motor noise, decisionnoise, etc., αˆ = α˜+ ν.In normal psychophysical experiments, participants are typically asked to respond with anindication of the L state, so the space of α is either equivalent to the L space, or some straightforwardfunction of it, f(·). In some cases they are assumed to place low, identical reward across all incorrectanswers, and greater, identical reward across all correct answers:Λ(α,L) ={b ;α = f(L¯)c ;α 6= f(L¯)where L¯ is the true latent world state and b c.For standard discrimination tasks, f(·) may be:f(L) ={0 ; lr ≤ θ1 ; lr > θwhere lr is an element of the latent state relevant to the task and θ is some threshold value (e.g. lrrepresents absence/presence, 0/1, of an auditory tone and θ is 0).6For tasks that ask participants to produce a response, α, on some continuous axis, i.e. pointing,grasping, etc., f(·) may deﬁne a range, [−θ, θ], that constitutes success:f(L) ={1 ;−θ ≤ lr ≤ θ0 ; otherwiseIn work from the cognitive psychology domain and more recent perceptual studies [21], it hasbeen suggested that humans sample from their posterior distributions, rather than computing de-terministic functions of the distribution, to produce responses. Also, in some cases the perceptualinference process may appear to depend on the task [18].5 ConclusionThe beneﬁts of a Bayesian framework for modeling perceptually-guided behaviors is that it explainsthese behaviors as resulting from a principled inferential process, based on sensory input and theobserver's perceptual system's internal knowledge and beliefs, which is used rationally to acquirereward and avoid penalty. The linear Gaussian model and its example applications provide anaccessible and useful model that can be used to model a broad set of perceptual behaviors. It mayhelp explain various cue integration, bias, constancy phenomena.The problem of using this framework to design an experiment and analyze the resultant data isbeyond the scope of this article, but is an important element for leveraging this framework on typ-ical perceptual science problems. In brief, by treating the experimental stimuli as L (Section 2.1),and the recorded participant responses as αˆ (Section 4.1), the model parameters, (G,µL,ΣL,ΣΩν),which encode the experimenter's hypotheses and assumptions about the observer's perceptual rea-soning, can be estimated or inferred using standard statistical methods.AcknowledgementsPaul Schrater, Frank Jaekel, and Dan Kersten provided helpful direction and criticism. Funding:NIH Grant (NRSA) Number F32EY019228-02, and a UMN Doctoral Dissertation Fellowship.References[1] W.J. Adams, E.W. Graf, and M.O. Ernst. Experience can change the'light-from-above'prior.Nature Neuroscience, 7(10):10571058, 2004.[2] D. Alais and D. Burr. The ventriloquist eﬀect results from near-optimal bimodal integration.Current Biology, 14(3):257262, 2004.[3] P.W. Battaglia, M. Di Luca, M.O. Ernst, P.R. Schrater, T. Machulla, and D. Kersten.Within-and Cross-Modal Distance Information Disambiguate Visual Size-Change Perception.6(3):e1000697, 2010.[4] P.W. Battaglia, R.A. Jacobs, and R.N. Aslin. Bayesian integration of visual and auditorysignals for spatial localization. Journal of the Optical Society of America A, 20(7):13911397,2003.7[5] P.W. Battaglia, P.R. Schrater, and D.J. Kersten. Auxiliary object knowledge inﬂuencesvisually-guided interception behavior. In Proceedings of the 2nd symposium on Applied percep-tion in graphics and visualization, volume 95, pages 145152. ACM, 2005.[6] MG Bloj, D. Kersten, and AC Hurlbert. Perception of three-dimensional shape inﬂuencescolour perception through mutual illumination. Nature, 402(6764):877879, 1999.[7] M.O. Ernst and M.S. Banks. Humans integrate visual and haptic information in a statisticallyoptimal fashion. Nature, 415(6870):429433, 2002.[8] J.M. Hillis, S.J. Watt, M.S. Landy, and M.S. Banks. Slant from texture and disparity cues:Optimal cue combination. 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Bayesian perceptual inference in linear Gaussian models

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Bayesian perceptual inference in linear Gaussian models

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