Making Decisions: Modeling perceptual decisions

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Making Decisions: Modeling perceptual decisions. Outline. Graphical models for belief example completed Discrete, continuous and mixed belief distributions Vision as an inference problem Modeling Perceptual beliefs: Priors and likelihoods

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### Making Decisions: Modeling perceptual decisions

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Outline
• Graphical models for belief example completed
• Discrete, continuous and mixed belief distributions
• Vision as an inference problem
• Modeling Perceptual beliefs: Priors and likelihoods
• Modeling Perceptual decisions: Actions, outcomes, and utility spaces for perceptual decisions

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Graphical Models: Modeling complex inferences

Nodes store conditional

Probability Tables

A

Arrows represent conditioning

B

C

This model represents the decomposition:

P(A,B,C) = P(B|A) P(C|A) P(A)

What would the diagram for P(B|A,C) P(A|C) P( C) look like?

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Sprinkler Problem

You see wet grass. Did it rain?

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Sprinkler Problem cont’d

4 variables: C = Cloudy, R=rain; S=sprinkler; W=wet grass

What’s the probability of rain?Need P(R|W)

Given: P( C) = [ 0.5 0.5]; P(S|C) ; P(R|C); P(W|R,S)

Do the math:

Get the joint:

P(R,W,S,C) = P(W|R,S) P(R|C) P(S|C) P( C)

Marginalize:

P(R,W) = SSSC P(R,W,S,C)

Divide:

P(R|W) = P(R,W)/P(W) = P(R,W)/ SR P(R,W)

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Continuous data, discrete beliefs

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Continuous data, Multi-class Beliefs

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Width

position

z

x

y

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

The problem of perception

Image

Measurement

Decisions/Actions

Scene Inference

Edges

Classify

Identify

Learn/Update

Plan Action

Control Action

Motion

Color

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Vision as Statistical Inference

Rigid Rotation

Or Shape Change?

What Moves,

Square or Light Source?

Kersten et al, 1996

?

?

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Modeling Perceptual Beliefs
• Observations O in the form of image measurements-
• Cone and rod outputs
• Luminance edges
• Texture
• Etc
• World states s include
• Intrinsic attributes of objects (reflectance, shape, material)
• Relations between objects (position, orientation, configuration)

Belief equation for perception

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Image data

y

x

O

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Possible world states consistent with O

z

y

x

s

Many 3D shapes can map to the same 2D image

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

z

y

x

Images are produced by physical processes that can be modeled via a rendering equation:

Modeling rendering probabilistically:

Likelihood: p( I | scene)

e

.

g

.

for no rendering noise

p( I | scene) = (I-f(scene))

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Prior further narrows selection

z

z

y

y

x

x

Select most probable

Priors weight likelihood selections

p(s)=p

1

is biggest

p(s)=p

3

p(s)=p

2

p(s)=p

3

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Graphical Models for Perceptual Inference

Object

Description

Variables

Lighting

Variables

Viewing

variables

Q

V

Priors

L

O1

O2

ON

Likelihoods

All of these nodes can be subdivided into individual

Variables for more compact representation of the probability

distribution

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

There is an ideal world…
• Matching environment to assumptions
• Prior on light source direction

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Prior on light source motion

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Combining Likelihood over different observations

What data is relevant?

How to combine?

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Modeling Perceptual Decisions
• One possible strategy for human perception is to try to get the right answer. How do we model this with decision theory?

Penalize responses with an error Utility function:

Given sensory data O= {o1, o2 , …, on } and some world property S perception is trying to infer, compute the decision:

Value of response

Response is best

value

Where:

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

Rigid RotationOr Shape Change?

Actions:

a = 1 Choose rigid

a = 0 Choose not rigid

Utility

Rigidity judgment

Let s is a random variable representing rigidity

s=1 it is rigid

s=0 not rigid

Let O1, O2, … , On be descriptions

of the ellipse for frames 1,2, … , n

Need likelihood

P( O1, O2, … , On | s )

Need Prior

P( s )

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005

“You must choose, but Choose Wisely”

Given previous formulation, can we minimize the number of errors we make?

• Given:

responses ai, categories si, current category s, data o

• To Minimize error:
• Decide aiif P(ai | o) > P(ak | o) for alli≠k

P( o | si) P(si ) > P(o | sk ) P(sk )

P( o | si)/ P(o | sk ) > P(sk ) / P(si )

P( o | si)/ P(o | sk ) > T

Optimal classifications always involve hard boundaries

PSY 5018H: Math Models Hum Behavior, Prof. Paul Schrater, Spring 2005