Lecture 3: Broken Bayes -...

81
Lecture 3: Broken Bayes CoSMo 2018 Minneapolis, MN Larry Maloney

Transcript of Lecture 3: Broken Bayes -...

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Lecture 3: Broken Bayes

CoSMo 2018 Minneapolis, MN

Larry Maloney

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Whyareyousowonderful?(Well,maybenotsowonderful…)

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anomalocaridid

Cambrianexplosion

MFLand&D-ENilson,(2002)AnimalEyes.Oxford.

Now

500millionyears

eyesskeletonsmovementplanning

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David Blackwell John von Neumann

Oskar Morgenstern 1954

Abraham Wald

M. A. Girschick

Statistical Decision Theory

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W = w1, w2, ... ,wm{ }A = a1,a2, ...,ap{ }X = x1, x2..., xn{ }

possible states of the world

possible sensory events

possible actions

Three Elements of SDT

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A X d ( x )

decision

W

Action

π(w) prior

Bayesian Decision Theory

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D: X à A d : Xà A

A Bayesian Problem

G( a , w ) Gain π(w) Prior L ( w | x ) Likelihood Linking hypothesis

Maximize expected gain by choice of d(.)

Despite changes in gain function, likelihood, prior.

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A Bayesian Game

:speededreaching

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Expected value as function of mean movement end point (x,y):

0

<-30 -15

30

points per trial

x (mm)

y (m

m)

! = 4.83 mm

-10 -5 0 5 10 15 20

-10

-5

-0

5

10

target: 100 penalty: -500

15

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c

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Action selection

OurperformanceisclosetoopMmalinthereachingandMmingtasksconsideredsofar.

Athoughtproblem:whatifreachingerrorwerenotisotropic(round)?

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Action selection

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Action selection

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Action selection

Anisotropydoesn’taffecttheop2malaimpointinthissimpletask.

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?

$100Whichtargetwouldyouliketotry?

Add a gain function…

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$100

>

Whichtargetwouldyouliketotry?

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$100

>

Whichtargetwouldyouliketotry?

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Bayesian

Prior

Likelihood

Gain

Action

Probability

Posterior

Objective Distributions

Prior

Likelihood

UMlity

Action

Probability

Posterior

Internal Representations

von Neumann & Morgenstern

We measure these

Wemeasurethese

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Bayesian Computation Representing motor uncertainty

True distribution

Subject’s representation

Probability density function pdf

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Bayesian Computation Representing motor uncertainty

True distribution

Subject’s representation

Probability density function pdf

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Trommershäuser,Maloney,&Landy(2003),SpatVisTrommershäuser,Maloney,&Landy(2003),JOptSocAmA

Körding&Wolpert(2004),Nature

Najemnik&Geisler(2005),Nature

Trommershäuser,Gepshtein,Maloney,Landy,&Banks(2005),JNeurosci

Trommershäuser,Ma]s,Landy,&Maloney(2006),ExpBrainRes

Trommershäuser,Landy,&Maloney(2006),PsychSciBa`aglia&Schrater(2007),JNeurosci

Dean,Wu,&Maloney(2007),JOVHudson,Maloney,&Landy(2008),PLoSCompBiol

Faisal&Wolpert(2009),JNeurophysiol

Wei&Körding(2010),FrontComputNeurosci······

People are Good at Motor Decisions Movement planning: near optimal

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Gain Function

Bayesian Computation

Optimal Choice

Experiment

Gain Function Subject’s Choice

Same? Near Optimal

Bayesian computation

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Near Optimal

Bayesian Computation Bayesian computation

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Gain Function

Bayesian Computation

Optimal Choice

Experiment

Gain Function Subject’s Choice

Same? Near Optimal

Gaussian

Uniform

Zhang, Daw, & Maloney, 2013

Bayesian Computation Bayesian computation

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Gain Function

Bayesian Computation

Optimal Choice

Experiment

Gain Function Subject’s Choice

Same? Near Optimal

Uniform

Zhang, Daw, & Maloney, 2013

Many tasks may simply be insensitive to systematic deviations in the internal model of uncertainty

Gaussian

Bayesian Computation Bayesian computation

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Inferring people’s internal models of uncertainty based on their choices

Gain Function Subject’s Choice

The Inverse Problem Bayesian Computation The inverse problem

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Choice Task

$100

>

Whichtargetwouldyouliketotry?

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Bayesian Computation Bayesian computation

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Choice Task

?

$100 Whichtargetwouldyouliketotry?

Bayesian Computation Choice task

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Choice Task

~

$100 Whichtargetwouldyouliketotry?

Bayesian Computation Bayesian computation

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>

<

~

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>

<

~

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Zhang, Daw, Maloney (2013) PLoS CB

Bayesian Computation Experiment

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Measuring subjects’ choices between targets of varying shapes and sizes allows us to infer their internal model of motor uncertainty

? Riesz-Fischer theorem, see Maloney & Mamassian, 2009

Bayesian computation

Page 35: Lecture 3: Broken Bayes - CompNeuroscicompneurosci.com/wiki/images/8/81/CoSMo2018_Module03_Lecture_Larry.pdfLecture 3: Broken Bayes CoSMo 2018 Minneapolis, MN Larry Maloney . Why are

Measuring subjects’ choices between targets of varying shapes and sizes allows us to infer their internal model of motor uncertainty

? Riesz-Fischer theorem, see Maloney & Mamassian, 2009

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Procedure

Bayesian Computation Zhang et al (2013)

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Touchthetargetwithin400msec

300trials

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+

?

500ms

500ms

500ms

500ms

500ms

500ms

Whichiseasiertohit?1stor2nd?

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?

Whichiseasiertohit?

Definitelythecircle

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?

Whichiseasiertohit?

Definitelytherectangle

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?

Whichiseasiertohit?

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10possiblerectangles

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RadiusofthecirclewasadjustedbyadapMveprocedures(staircase)

~

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Results: True error distribution

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Vertically elongated, bivariate Gaussian

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xσ (cm)

yσ(cm)

0 0.2 0.4 0.60

0.2

0.4

0.6

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xσ (cm)

yσ(cm)

0 0.2 0.4 0.60

0.2

0.4

0.6

Median 1.44y xσ σ =

Vertically elongated

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?

Results: Subjects' internal model of their own error distribution

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Each subject’s internal model was fitted to the subject’s choices as a bivariate Gaussian distribution with two free parameters:

σ ′x σ ′

yand

Maximum likelihood estimates

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True

in

Subjects’ Model

σ σy x

σ σ′ ′y x

0 0.5 1 1.50

0.5

1

1.5

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True

in

Subjects’ Model

σ σy x

σ σ′ ′y x

0 0.5 1 1.50

0.5

1

1.5

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True

in

Subjects’ Model

σ σy x

σ σ′ ′y x

0 0.5 1 1.50

0.5

1

1.5

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true distribution subject's model

Summary

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Zhang, Daw, Maloney (2015) Nature Neuroscience

Experiment

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A1-Dversionof ?

Experiment

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Touchthetargetwithin400msec

300trials

(target illustrated not in real scale)

Experiment

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Experiment

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?

Whichiseasiertohit?

Definitely the Triple

Experiment

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?

Whichiseasiertohit?

Definitely the Single

Experiment

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?

Whichiseasiertohit?

Experiment

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Results: True error distribution

Experiment

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Experiment

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How do we estimate the participants’ subjective pdf?

?

Experiment

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Maximum Likelihood Fitting

The data are choices between pairs of targets à 1 (second one)

à 0 (first one)

etc

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Maximum Likelihood Fitting

The data are choices between pairs of targets à 1 (second one)

à 0 (first one)

etc For any motor pdf we can simulate the choices of a model participant who carries out the experiment. What is the probability that this model participant’s choices will match the responses of a given subject? Search through a “large” family of pdfs to find the one that has the highest likelihood of producing the subject’s data.

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Warm up: So lets try some “large families of pdfs”!

Unimodal histograms

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x

( )f x

Non-Parametric Analysis

1 2 3 4 5 6 7 8 9 10

Important constraint: monotone decreasing from center

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x

( )f x

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x

( )f x

constraint: monotone decreasing from center

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x

( )f x

constrained to be unimodal

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So what do we find?

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x

( )f x

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−1 0 10

1

2

−1 0 10

1

2

3

−1 0 10

1

2

3

−1 0 10

1

2

−1 0 10

1

2

−1 0 10

1

2

−1 0 10

1

2

−1 0 10

1

2

−1 0 10

1

2

x

f x( )S4 S5 S6

S1 S2 S3

S7 S8 S9

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x

( )f x

Don’t get too excited. Remember: we constrained the distributions to be Non-increasing away from center.

“bumps”

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U-mix

x

( )f x

(uniform mixtures)

Hypothesis

Parameters: number of steps location of steps

heights of steps

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( )f x

x

( )f x

( )f x

( )f x

U1

U2

U3

U4

Nested-hypothesis tests (Mood, Graybill, & Boes, 1974)

U-mix Family

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Model Fit: Possible pdf Models

t distributions

Mixture of N Gaussians

Linear decay

Mixture of Non-overlapping Uniforms (U-Mix)

And more

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Model Fit: Possible pdf Models

t distributions

Mixture of N Gaussians

Linear decay

Mixture of Non-overlapping Uniforms (U-Mix)

And more

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Non-parametric analysis U-mix

Experiment

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Subjects’ u-mix