1. Bayesian Perception

2. 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)

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

4. 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 (e.g. P(I|V)).
Biological neural networks can perform complex statistical inferences.

5. A simple problem
Estimating direction of motion from a noisy population code

6. Population Code
Tuning Curves
Pattern of activity (A)

7. Maximum Likelihood

8. Maximum Likelihood
The maximum likelihood estimate is the value of q maximizing the likelihood P(A|q). Therefore, we seek such that:
is unbiased and efficient.
Likelihood function
Noise distribution

10. MT V1

11. Preferred Direction MT V1
Preferred Direction

12. Linear Networks
Networks in which the activity at time t+1 is a linear function of the activity at the previous time step.

13. Linear Networks
Equivalent to population vector

14. Nonlinear Networks
Networks in which the activity at time t+1 is a nonlinear function of the activity at the previous time step.

15. Preferred Direction MT V1
Preferred Direction

16. Maximum Likelihood

17. Standard Deviation of

18. Standard Deviation of

19. Weight Pattern
Amplitude
Difference in preferred direction

20. Performance Over Time

22. General Result
Networks of nonlinear units with bell shaped tuning curves and a line attractor (stable smooth hills) are equivalent to a maximum likelihood estimator regardless of the exact form of the nonlinear activation function.

23. General Result Pro: Maximum likelihood estimation
Biological implementation (the attractors dynamics is akin to a generative model )
Con:
No explicit representations of probability distributions
No use of priors

24. Motion Perception

25. The Aperture Problem
The aperture in itself introduces uncertainty

26. The Aperture Problem

27. The Aperture Problem

28. The Aperture Problem

29. The Aperture Problem

30. The Aperture Problem

31. The Aperture Problem

32. The Aperture Problem

33. The Aperture Problem

34. The Aperture Problem

35. The Aperture Problem

36. The Aperture Problem

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

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

39. The Aperture Problem

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

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

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

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

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

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

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

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

48. 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)

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

50. Prior Human observers favor slow motions Rotating wheel
-50 50
Horizontal Velocity
Vertical Velocity
Rotating wheel
Switching dot patterns

51. Likelihood Weiss and Adelson -50 50 Horizontal Velocity50
Horizontal Velocity
Vertical Velocity

52. Likelihood

53. Likelihood

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

55. Motion through an Aperture
Humans perceive the slowest motion

56. Motion through an Aperture
Likelihood 50
Vertical Velocity
-50
-50 50 ML
Horizontal Velocity 50 50
Vertical Velocity
Vertical Velocity
MAP
-50
-50
Prior
Posterior
-50 50
-50 50
Horizontal Velocity
Horizontal Velocity

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

58. Motion and Contrast High Contrast Likelihood ML MAP Prior Posterior 50
Vertical Velocity
High
Contrast
-50
-50 50 ML
Horizontal Velocity 50 50
Vertical Velocity
Vertical Velocity
MAP
-50
-50
Prior
Posterior
-50 50
-50 50
Horizontal Velocity
Horizontal Velocity

59. Motion and Contrast Low Contrast Likelihood ML MAP Prior Posterior 50
Vertical Velocity
Low
Contrast
-50
-50 50 ML
Horizontal Velocity
MAP 50 50
Vertical Velocity
Vertical Velocity
-50
-50
Prior
Posterior
-50 50
-50 50
Horizontal Velocity
Horizontal Velocity

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

61. Moving Rhombus High Contrast Likelihood IOC MAP Prior Posterior 50 50
Vertical Velocity
Vertical Velocity
High
Contrast
-50
-50
-50 50
-50 50
IOC
Horizontal Velocity
Horizontal Velocity 50 50
MAP
Vertical Velocity
Vertical Velocity
-50
-50
Prior
-50 50
-50 50
Posterior
Horizontal Velocity
Horizontal Velocity

62. Moving Rhombus Low Contrast Likelihood IOC MAP Prior Posterior 50 50
Vertical Velocity
Vertical Velocity
-50
-50
Low
Contrast
-50 50
-50 50
Horizontal Velocity
Horizontal Velocity
IOC 50 50
MAP
Vertical Velocity
Vertical Velocity
-50
-50
-50 50
-50 50
Prior
Posterior
Horizontal Velocity
Horizontal Velocity

63. Moving Rhombus

64. Moving Rhombus 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)

65. Barberpole Illusion

66. Plaid Motion: Type I and II

67. Plaids and Contrast

68. Plaids and Time
Viewing time reduces uncertainty

69. Ellipses
Fat vs narrow ellipses

70. Ellipses
Adding unambiguous motion

71. Biological Implementation
Neurons might be representing probability distributions
How?

72. Biological Implementation
Encoding model

73. Biological Implementation
Decoding
Linear decoder: deconvolution

74. Biological Implementation
Decoding: nonlinear
Represent P(V|W) as a discretized histogram and use EM to evaluate the parameters