Bayesian Perception
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 (e.g. P(I|V)). Biological neural networks can perform complex statistical inferences.
A simple problem Estimating direction of motion from a noisy population code
Population Code Tuning Curves Pattern of activity (A)
Maximum Likelihood
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
MT V1
Preferred Direction MT V1 Preferred Direction
Linear Networks Networks in which the activity at time t+1 is a linear function of the activity at the previous time step.
Linear Networks Equivalent to population vector
Nonlinear Networks Networks in which the activity at time t+1 is a nonlinear function of the activity at the previous time step.
Preferred Direction MT V1 Preferred Direction
Maximum Likelihood
Standard Deviation of
Standard Deviation of
Weight Pattern Amplitude Difference in preferred direction
Performance Over Time
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.
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
Motion Perception
The Aperture Problem The aperture in itself introduces uncertainty
The Aperture Problem
The Aperture Problem
The Aperture Problem
The Aperture Problem
The Aperture Problem
The Aperture Problem
The Aperture Problem
The Aperture Problem
The Aperture Problem
The Aperture Problem
The Aperture Problem
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
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
Prior Human observers favor slow motions Rotating wheel -50 50 Horizontal Velocity Vertical Velocity Rotating wheel Switching dot patterns
Likelihood Weiss and Adelson -50 50 Horizontal Velocity 50 Horizontal Velocity Vertical Velocity
Likelihood
Likelihood
Bayesian Model of Motion Perception Perceived motion correspond to the MAP estimate
Motion through an Aperture Humans perceive the slowest motion
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
Motion and Constrast Humans tend to underestimate velocity in low contrast situations
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
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
Motion and Contrast Driving in the fog: in low contrast situations, the prior dominates
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
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
Moving Rhombus
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)
Barberpole Illusion
Plaid Motion: Type I and II
Plaids and Contrast
Plaids and Time Viewing time reduces uncertainty
Ellipses Fat vs narrow ellipses
Ellipses Adding unambiguous motion
Biological Implementation Neurons might be representing probability distributions How?
Biological Implementation Encoding model
Biological Implementation Decoding Linear decoder: deconvolution
Biological Implementation Decoding: nonlinear Represent P(V|W) as a discretized histogram and use EM to evaluate the parameters