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Bayesian Perception

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  1. Bayesian Perception

  2. General Idea Ernst and Banks, Nature, 2002

  3. General Idea • Bayesian formulation: Conditional Independence assumption

  4. noise v=w+n + w t=w+n + noise General Idea Generative model: w? Ernst and Banks, Nature, 2002

  5. Bimodal P(w|t,v)= aP(v|w)P(t|w) Visual P(v|w) Touch P(t|w) General Idea Probability Width

  6. General Idea

  7. General Idea Mean and variance

  8. Visual P(v|w) Touch P(t|w) General Idea Probability Width v t

  9. General Idea Mean and variance

  10. Optimal Variance Variance Fisher information sums for independent signals

  11. 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

  12. 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?

  13. 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)

  14. General Idea • Decisions are made by collapsing the distribution onto a single value: • or

  15. 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.

  16. Motion Perception

  17. The Aperture Problem

  18. The Aperture Problem

  19. The Aperture Problem

  20. The Aperture Problem Vertical velocity (deg/s) Horizontal velocity (deg/s)

  21. The Aperture Problem Vertical velocity (deg/s) Horizontal velocity (deg/s)

  22. The Aperture Problem

  23. The Aperture Problem Vertical velocity (deg/s) Horizontal velocity (deg/s)

  24. The Aperture Problem Vertical velocity (deg/s) Horizontal velocity (deg/s)

  25. The Aperture Problem Vertical velocity (deg/s) Horizontal velocity (deg/s)

  26. Standard Models of Motion Perception • IOC: interception of constraints • VA: Vector average • Feature tracking

  27. Standard Models of Motion Perception IOC VA Vertical velocity (deg/s) Horizontal velocity (deg/s)

  28. Standard Models of Motion Perception IOC VA Vertical velocity (deg/s) Horizontal velocity (deg/s)

  29. Standard Models of Motion Perception IOC VA Vertical velocity (deg/s) Horizontal velocity (deg/s)

  30. Standard Models of Motion Perception IOC VA Vertical velocity (deg/s) Horizontal velocity (deg/s)

  31. Standard Models of Motion Perception • Problem: perceived motion is close to either IOC or VA depending on stimulus duration, eccentricity, contrast and other factors.

  32. 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)

  33. Moving Rhombus

  34. Bayesian Model of Motion Perception • Perceived motion correspond to the MAP estimate

  35. 50 0 Vertical Velocity -50 -50 0 50 Horizontal Velocity Prior • Human observers favor slow motions

  36. 50 Vertical Velocity 0 -50 -50 0 50 Horizontal Velocity Likelihood • Weiss and Adelson

  37. Likelihood

  38. Likelihood Binary maskPresumably, this is set by segmentation cues

  39. Posterior

  40. Bayesian Model of Motion Perception • Perceived motion corresponds to the MAP estimate Only one free parameter

  41. Likelihood

  42. Motion through an Aperture • Humans perceive the slowest motion. • More generally: we tend to perceive the most likely interpretation of an image

  43. 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

  44. Motion and Constrast • Humans tend to underestimate velocity in low contrast situations

  45. 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

  46. 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

  47. Motion and Contrast • Driving in the fog: in low contrast situations, the prior dominates

  48. 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

  49. 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

  50. Moving Rhombus