Bayesian Perception

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Bayesian Perception
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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(VI)

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General Idea

• Decisions are made by collapsing the distribution
  onto a single value
• or

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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(IV)).
• Biological neural networks can perform complex
  statistical inferences.

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A simple problem

• Estimating direction of motion from a noisy
  population code

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Population Code
Tuning Curves
Pattern of activity (A)
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Maximum Likelihood
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Maximum Likelihood

• The maximum likelihood estimate is the value of
  q maximizing the likelihood P(Aq). Therefore, we
  seek such that
• is unbiased and efficient.

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MT
V1
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Preferred Direction
MT
V1
Preferred Direction
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Linear Networks

• Networks in which the activity at time t1 is a
  linear function of the activity at the previous
  time step.

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Linear Networks
Equivalent to population vector
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Nonlinear Networks

• Networks in which the activity at time t1 is a
  nonlinear function of the activity at the
  previous time step.

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Preferred Direction
MT
V1
Preferred Direction
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Maximum Likelihood
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Standard Deviation of
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Standard Deviation of
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Weight Pattern
Amplitude
Difference in preferred direction
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Performance Over Time
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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.

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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

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Motion Perception
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
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The Aperture Problem
Vertical velocity (deg/s)
Horizontal velocity (deg/s)
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The Aperture Problem
Vertical velocity (deg/s)
Horizontal velocity (deg/s)
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The Aperture Problem
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The Aperture Problem
Vertical velocity (deg/s)
Horizontal velocity (deg/s)
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The Aperture Problem
Vertical velocity (deg/s)
Horizontal velocity (deg/s)
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Standard Models of Motion Perception

• IOC interception of constraints
• VA Vector average
• Feature tracking

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

• Problem perceived motion is close to either IOC
  or VA depending on stimulus duration,
  eccentricity, contrast and other factors.

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Standard Models of Motion Perception

• Example Rhombus

Percept VA
Percept IOC
IOC
IOC
VA
VA
Vertical velocity (deg/s)
Vertical velocity (deg/s)
Horizontal velocity (deg/s)
Horizontal velocity (deg/s)
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Bayesian Model of Motion Perception

• Perceived motion correspond to the MAP estimate

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Prior

• Human observers favor slow motions

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Likelihood

• Weiss and Adelson

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Likelihood
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Likelihood
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Bayesian Model of Motion Perception

• Perceived motion correspond to the MAP estimate

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Motion through an Aperture

• Humans perceive the slowest motion

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Motion through an Aperture
Likelihood
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Vertical Velocity
0
-50
-50
0
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ML
Horizontal Velocity
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50
Vertical Velocity
Vertical Velocity
MAP
0
0
-50
-50
Prior
Posterior
-50
0
50
-50
0
50
Horizontal Velocity
Horizontal Velocity
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Motion and Constrast

• Humans tend to underestimate velocity in low
  contrast situations

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Motion and Contrast
Likelihood
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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
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Motion and Contrast
Likelihood
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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
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Motion and Contrast

• Driving in the fog in low contrast situations,
  the prior dominates

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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
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Moving Rhombus
Likelihood
50
50
0
Vertical Velocity
0
Vertical Velocity
-50
-50
Low Contrast
-50
0
50
-50
0
50
IOC
Horizontal Velocity
Horizontal Velocity
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50
MAP
Vertical Velocity
Vertical Velocity
0
0
-50
-50
-50
0
50
-50
0
50
Prior
Posterior
Horizontal Velocity
Horizontal Velocity
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Moving Rhombus
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Moving Rhombus

• Example Rhombus

Percept VA
Percept IOC
IOC
IOC
VA
VA
Vertical velocity (deg/s)
Vertical velocity (deg/s)
Horizontal velocity (deg/s)
Horizontal velocity (deg/s)
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Barberpole Illusion
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Plaid Motion Type I and II
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Plaids and Contrast
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Plaids and Time

• Viewing time reduces uncertainty

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Ellipses

• Fat vs narrow ellipses

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Ellipses

• Adding unambiguous motion

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Biological Implementation

• Neurons might be representing probability
  distributions
• How?

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Biological Implementation

• Encoding model

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Biological Implementation

• Decoding
• Linear decoder deconvolution

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Biological Implementation

• Decoding nonlinear
• Represent P(VW) as a discretized histogram and
  use EM to evaluate the parameters