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## Bayesian Perception

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**General Idea**Ernst and Banks, Nature, 2002**General Idea**• Bayesian formulation: Conditional Independence assumption**noise**v=w+n + w t=w+n + noise General Idea Generative model: w? Ernst and Banks, Nature, 2002**Bimodal**P(w|t,v)= aP(v|w)P(t|w) Visual P(v|w) Touch P(t|w) General Idea Probability Width**General Idea**Mean and variance**Visual**P(v|w) Touch P(t|w) General Idea Probability Width v t**General Idea**Mean and variance**Optimal Variance**Variance Fisher information sums for independent signals**Unimodal Tactile STD**Unimodal visual STD Measured bimodal STD General Idea Predicted by the Bayesian model 0.2 0.15 Threshold (STD) 0.1 0.05 0 0 67 133 200 Visual noise level (%) Note: unimodal estimates may not be optimal but the multimodal estimate is optimal Ernst and Banks, Nature, 2002**Adaptive Cue Integration**• Note: the reliability of the cue change on every trial • This implies that the weights of the linear combination have to be changed on every trial! • Or do they?**General Idea**• Perception is a statistical inference • The brain stores knowledge about P(I,V) where I is the set of natural images, and V are the perceptual variables (color, motion, object identity) • Given an image, the brain computes P(V|I)**General Idea**• Decisions are made by collapsing the distribution onto a single value: • or**Key Ideas**• The nervous systems represents probability distributions. i.e., it represents the uncertainty inherent to all stimuli. • The nervous system stores generative models, or forward models, of the world (P(I|V)), and prior knowlege about the state of the world (P(V)) • Biological neural networks can perform complex statistical inferences.**The Aperture Problem**Vertical velocity (deg/s) Horizontal velocity (deg/s)**The Aperture Problem**Vertical velocity (deg/s) Horizontal velocity (deg/s)**The Aperture Problem**Vertical velocity (deg/s) Horizontal velocity (deg/s)**The Aperture Problem**Vertical velocity (deg/s) Horizontal velocity (deg/s)**The Aperture Problem**Vertical velocity (deg/s) Horizontal velocity (deg/s)**Standard Models of Motion Perception**• IOC: interception of constraints • VA: Vector average • Feature tracking**Standard Models of Motion Perception**IOC VA Vertical velocity (deg/s) Horizontal velocity (deg/s)**Standard Models of Motion Perception**IOC VA Vertical velocity (deg/s) Horizontal velocity (deg/s)**Standard Models of Motion Perception**IOC VA Vertical velocity (deg/s) Horizontal velocity (deg/s)**Standard Models of Motion Perception**IOC VA Vertical velocity (deg/s) Horizontal velocity (deg/s)**Standard Models of Motion Perception**• Problem: perceived motion is close to either IOC or VA depending on stimulus duration, eccentricity, contrast and other factors.**Standard Models of Motion Perception**• Example: Rhombus Percept: IOC Percept: VA IOC IOC VA VA Vertical velocity (deg/s) Vertical velocity (deg/s) Horizontal velocity (deg/s) Horizontal velocity (deg/s)**Bayesian Model of Motion Perception**• Perceived motion correspond to the MAP estimate**50**0 Vertical Velocity -50 -50 0 50 Horizontal Velocity Prior • Human observers favor slow motions**50**Vertical Velocity 0 -50 -50 0 50 Horizontal Velocity Likelihood • Weiss and Adelson**Likelihood**Binary maskPresumably, this is set by segmentation cues**Bayesian Model of Motion Perception**• Perceived motion corresponds to the MAP estimate Only one free parameter**Motion through an Aperture**• Humans perceive the slowest motion. • More generally: we tend to perceive the most likely interpretation of an image**Motion through an Aperture**Likelihood 50 Vertical Velocity 0 -50 -50 0 50 ML Horizontal Velocity 50 50 Vertical Velocity Vertical Velocity MAP 0 0 -50 -50 Prior Posterior -50 0 50 -50 0 50 Horizontal Velocity Horizontal Velocity**Motion and Constrast**• Humans tend to underestimate velocity in low contrast situations**Motion and Contrast**Likelihood 50 Vertical Velocity 0 -50 High Contrast -50 0 50 ML Horizontal Velocity 50 50 Vertical Velocity Vertical Velocity MAP 0 0 -50 -50 Prior Posterior -50 0 50 -50 0 50 Horizontal Velocity Horizontal Velocity**Motion and Contrast**Likelihood 50 Vertical Velocity 0 -50 Low Contrast -50 0 50 ML Horizontal Velocity MAP 50 50 Vertical Velocity Vertical Velocity 0 0 -50 -50 Prior Posterior -50 0 50 -50 0 50 Horizontal Velocity Horizontal Velocity**Motion and Contrast**• Driving in the fog: in low contrast situations, the prior dominates**Moving Rhombus**Likelihood 50 50 Vertical Velocity Vertical Velocity 0 0 -50 -50 High Contrast -50 0 50 -50 0 50 IOC Horizontal Velocity Horizontal Velocity MAP 50 50 Vertical Velocity Vertical Velocity 0 0 -50 -50 -50 0 50 -50 0 50 Prior Posterior Horizontal Velocity Horizontal Velocity**Moving Rhombus**Likelihood 50 50 Vertical Velocity Vertical Velocity 0 0 -50 -50 Low Contrast -50 0 50 -50 0 50 IOC Horizontal Velocity Horizontal Velocity 50 50 MAP Vertical Velocity Vertical Velocity 0 0 -50 -50 -50 0 50 -50 0 50 Prior Posterior Horizontal Velocity Horizontal Velocity