Lecture 3: Broken Bayes -...
Transcript of Lecture 3: Broken Bayes -...
Lecture 3: Broken Bayes
CoSMo 2018 Minneapolis, MN
Larry Maloney
Whyareyousowonderful?(Well,maybenotsowonderful…)
anomalocaridid
Cambrianexplosion
MFLand&D-ENilson,(2002)AnimalEyes.Oxford.
Now
500millionyears
eyesskeletonsmovementplanning
David Blackwell John von Neumann
Oskar Morgenstern 1954
Abraham Wald
M. A. Girschick
Statistical Decision Theory
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
A X d ( x )
decision
W
Action
π(w) prior
Bayesian Decision Theory
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.
A Bayesian Game
:speededreaching
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
c
Action selection
OurperformanceisclosetoopMmalinthereachingandMmingtasksconsideredsofar.
Athoughtproblem:whatifreachingerrorwerenotisotropic(round)?
Action selection
Action selection
Action selection
Anisotropydoesn’taffecttheop2malaimpointinthissimpletask.
?
$100Whichtargetwouldyouliketotry?
Add a gain function…
$100
>
Whichtargetwouldyouliketotry?
$100
>
Whichtargetwouldyouliketotry?
Bayesian
Prior
Likelihood
Gain
Action
Probability
Posterior
Objective Distributions
Prior
Likelihood
UMlity
Action
Probability
Posterior
Internal Representations
von Neumann & Morgenstern
We measure these
Wemeasurethese
Bayesian Computation Representing motor uncertainty
True distribution
Subject’s representation
Probability density function pdf
Bayesian Computation Representing motor uncertainty
True distribution
Subject’s representation
Probability density function pdf
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
Gain Function
Bayesian Computation
Optimal Choice
Experiment
Gain Function Subject’s Choice
Same? Near Optimal
Bayesian computation
Near Optimal
Bayesian Computation Bayesian computation
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
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
Inferring people’s internal models of uncertainty based on their choices
Gain Function Subject’s Choice
The Inverse Problem Bayesian Computation The inverse problem
Choice Task
$100
>
Whichtargetwouldyouliketotry?
Bayesian Computation Bayesian computation
Choice Task
?
$100 Whichtargetwouldyouliketotry?
Bayesian Computation Choice task
Choice Task
~
$100 Whichtargetwouldyouliketotry?
Bayesian Computation Bayesian computation
>
<
~
>
<
~
Zhang, Daw, Maloney (2013) PLoS CB
Bayesian Computation Experiment
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
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
Procedure
Bayesian Computation Zhang et al (2013)
Touchthetargetwithin400msec
300trials
+
?
500ms
500ms
500ms
500ms
500ms
500ms
Whichiseasiertohit?1stor2nd?
?
Whichiseasiertohit?
Definitelythecircle
?
Whichiseasiertohit?
Definitelytherectangle
?
Whichiseasiertohit?
10possiblerectangles
RadiusofthecirclewasadjustedbyadapMveprocedures(staircase)
~
Results: True error distribution
Vertically elongated, bivariate Gaussian
xσ (cm)
yσ(cm)
0 0.2 0.4 0.60
0.2
0.4
0.6
xσ (cm)
yσ(cm)
0 0.2 0.4 0.60
0.2
0.4
0.6
Median 1.44y xσ σ =
Vertically elongated
?
Results: Subjects' internal model of their own error distribution
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
True
in
Subjects’ Model
σ σy x
σ σ′ ′y x
0 0.5 1 1.50
0.5
1
1.5
True
in
Subjects’ Model
σ σy x
σ σ′ ′y x
0 0.5 1 1.50
0.5
1
1.5
True
in
Subjects’ Model
σ σy x
σ σ′ ′y x
0 0.5 1 1.50
0.5
1
1.5
true distribution subject's model
Summary
Zhang, Daw, Maloney (2015) Nature Neuroscience
Experiment
A1-Dversionof ?
Experiment
Touchthetargetwithin400msec
300trials
(target illustrated not in real scale)
Experiment
Experiment
?
Whichiseasiertohit?
Definitely the Triple
Experiment
?
Whichiseasiertohit?
Definitely the Single
Experiment
?
Whichiseasiertohit?
Experiment
Results: True error distribution
Experiment
Experiment
How do we estimate the participants’ subjective pdf?
?
Experiment
Maximum Likelihood Fitting
The data are choices between pairs of targets à 1 (second one)
à 0 (first one)
etc
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.
Warm up: So lets try some “large families of pdfs”!
Unimodal histograms
x
( )f x
Non-Parametric Analysis
1 2 3 4 5 6 7 8 9 10
Important constraint: monotone decreasing from center
x
( )f x
x
( )f x
constraint: monotone decreasing from center
x
( )f x
constrained to be unimodal
So what do we find?
x
( )f x
−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
x
( )f x
Don’t get too excited. Remember: we constrained the distributions to be Non-increasing away from center.
“bumps”
U-mix
x
( )f x
(uniform mixtures)
Hypothesis
Parameters: number of steps location of steps
heights of steps
( )f x
x
( )f x
( )f x
( )f x
U1
U2
U3
U4
Nested-hypothesis tests (Mood, Graybill, & Boes, 1974)
U-mix Family
Model Fit: Possible pdf Models
t distributions
Mixture of N Gaussians
Linear decay
Mixture of Non-overlapping Uniforms (U-Mix)
And more
Model Fit: Possible pdf Models
t distributions
Mixture of N Gaussians
Linear decay
Mixture of Non-overlapping Uniforms (U-Mix)
And more
0
3
6
U1
N. o
f sub
ject
s
U2 U3 U4
−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
−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
S4 S4
Non-parametric analysis U-mix
Experiment
−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
Subjects’ u-mix