seeing patterns in randomness: irrational superstition or adaptive behavior?
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Seeing Patterns in Randomness: Irrational Superstition or Adaptive Behavior?. Angela J. Yu University of California, San Diego March 9, 2010. “Irrational” Probabilistic Reasoning in Humans. “hot hand” 2AFC: sequential effects (rep/alt). (Gillovich, Vallon, & Tversky, 1985). - PowerPoint PPT PresentationTRANSCRIPT
Seeing Patterns in Randomness:
Irrational Superstition or
Adaptive Behavior?
Angela J. Yu
University of California, San Diego
March 9, 2010
“Irrational” Probabilistic Reasoning in Humans
1 2 2 2 2 2 1 1 2 1 2 1 …1 2 2 2 2 2Random stimulus sequence:
1 2 1 2
• “hot hand”
• 2AFC: sequential effects (rep/alt)
(Gillovich, Vallon, & Tversky, 1985)
(Soetens, Boer, & Hueting, 1985)
(Wilke & Barrett, 2009)
“Superstitious” Predictions
Subjects are “superstitious” when viewing randomized stimuli
O o o o o o O O o O o O O…
repetitions alternations
slow slowfast fast
Trials
• Subjects slower & more error-prone when local pattern is violated
• Patterns are by chance, not predictive of next stimulus
• Such “superstitious” behavior is apparently sub-optimal
“Graded” Superstition(Cho et al, 2002)(Soetens et al, 1985) [o o O O O]
RARR = or [O O o o o]
RT
ER
Hypothesis:
Sequential adjustments may be
adaptive for changing environments.
tt-1t-2t-3
Outline
• “Ideal predictor” in a fixed vs. changing world
• Exponential forgetting normative and descriptive
• Optimal Bayes or exponential filter?
• Neural implementation of prediction/learning
I. Fixed Belief Model (FBM)
A (0)R (1) R (1)
hiddenbias
observedstimuli
?
…?
II. Dynamic Belief Model (DBM)
A (0)R (1) R (1)
changingbias
observedstimuli
?
?
.3 .8.3
QuickTime™ and a decompressor
are needed to see this picture.
RA bias
What the FBM subject should believe about the bias of the coin,given a sequence of observations: R R A R R R
FBM Subject’s Response to Random Inputs
FBM Subject’s Response to Random InputsWhat the FBM subject should believe about the bias of the coin,
given a long sequence of observations: R R A R A A R A A R A…
QuickTime™ and a decompressor
are needed to see this picture.
RA bias
What the DBM subject should believe about the bias of the coin,given a long sequence of observations: R R A R A A R A A R A…
QuickTime™ and a decompressor
are needed to see this picture.
RA bias
DBM Subject’s Response to Random Inputs
Randomized Stimuli: FBM > DBM
Given a sequence of truly random data ( = .5) …
FBM: belief distrib. over
Simulated trials
Prob
abili
ty
DBM: belief distrib. over
Simulated trials
Prob
abili
ty
Driven by long-term average Driven by transient patterns
“Natural Environment”: DBM > FBM
In a changing world, where undergoes un-signaled changes …
FBM: posterior over
Simulated trials
Prob
abili
ty
Adapt poorly to changes Adapt rapidly to changes
DBM: posterior over
Simulated trials
Prob
abili
ty
Persistence of Sequential Effects
• Sequential effects persist in data• DBM produces R/A asymmetry• Subjects=DBM (changing world)
FBM
P(st
imul
us)
DBM
P(st
imul
us)
Human Data(data from Cho et al, 2002)
RT
Outline
• “Ideal predictor” in a fixed vs. changing world
• Exponential forgetting normative and descriptive
• Optimal Bayes or exponential filter?
• Neural implementation of prediction/learning
Bayesian Computations in Neurons?
Optimal PredictionWhat subjects need to compute
Too hard to represent, too hard to compute!
Generative ModelWhat subjects need to know
(Sugrue, Corrado, & Newsome, 2004)
Simpler Alternative for Neural Computation?Inspiration: exponential forgetting in tracking true changes
Exponential Forgetting in Behavior
Exponential discounting is a good descriptive model
Linear regression:R/A R/A
Human Data
Trials into the Past
Coe
ffic
ient
s
(re-analysis of Cho et al)
Linear regression:R/A R/A
Exponential discounting is a good normative model
DBM Prediction
Trials into the Past
Coe
ffic
ient
s
Exponential Forgetting Approximates DBM
Discount Rate vs. Assumed Rate of Change
…DBM
= .95
Simulated trials
Prob
abili
ty
= .77
Simulated trials
Trials into the Past
DBM Simulation
Coe
ffic
ient
s
Human Data
Trials into the Past
Coe
ffic
ient
s
= .57 = .57
Reverse-engineering Subjects’ Assumptions
= p(t=t-1)
= .57 = .77
changes once every four trials
2/3
Analytical Approximation
Quality of approximation vs.
.57
.77
nonlinear Bayesian computations 3-param model
1-param linear model
Outline
• “Ideal predictor” in a fixed vs. changing world
• Exponential forgetting normative and descriptive
• Optimal Bayes or exponential filter?
• Neural implementation of prediction/learning
Subjects’ RT vs. Model Stimulus Probability
Repetition Trials
R A R R R R …
Subjects’ RT vs. Model Stimulus Probability
Repetition Trials
R A R R R R …RT
Subjects’ RT vs. Model Stimulus Probability
Repetition Trials Alternation Trials
R A R R R R …RT
Subjects’ RT vs. Model Stimulus Probability
Repetition vs. Alternation Trials
Multiple-Timescale Interactions
Optimal discrimination(Wald, 1947) 2
1
• discrete time, SPRT
• continuous-time, DDM
DBM
(Yu, NIPS 2007)(Frazier & Yu, NIPS 2008)(Gold & Shadlen, Neuron 2002)
SPRT/DDM & Linear Effect of Prior on RT
Timesteps
RT hist
Bias: P(s1)
<RT>
Bias: P(s1) x
tanh x
0
SPRT/DDM & Linear Effect of Prior on RT
Empirical RT vs. Stim Probability
Bias: P(s1)
<RT>
Predicted RT vs. Stim Probability
Outline
• “Ideal predictor” in a fixed vs. changing world
• Exponential forgetting normative and descriptive
• Optimal Bayes or exponential filter?
• Neural implementation of prediction/learning
Neural Implementation of Prediction
Leaky-integrating neuron:
• Perceptual decision-making(Grice, 1972; Smith, 1995; Cook & Maunsell, 2002; Busmeyer & Townsend, 1993; McClelland, 1993; Bogacz et al, 2006; Yu, 2007; …)
• Trial-to-trial interactions(Kim & Myung, 1995; Dayan & Yu, 2003; Simen, Cohen & Holmes, 2006; Mozer, Kinoshita, & Shettel, 2007; …)
bias input recurrent
=1/2 (1-) 1/3 2/3
Neuromodulation & Dynamic Filters
Leaky-integrating neuron:
bias input recurrent
Norepinephrine (NE)(Hasselmo, Wyble, & Wallenstein 1996; Kobayashi, 2000)
Trials
NE: Unexpected Uncertainty(Yu & Dayan, Neuron, 2000)
Learning the Value of Humans (Behrens et al, 2007) and rats (Gallistel & Latham, 1999)
may encode meta-changes in the rate of change,
Bayesian Learning
00 1
.3 .9.3
…
…
…
…
Iteratively compute joint posterior
Marginal posterior over
Marginal posterior over
• Neurons don’t need to represent probabilities explicitly
• Just need to estimate
• Stochastic gradient descent (-rule)
Neural Parameter Learning?
learning rate error gradient
€
ˆ α n ← ˆ α n−1 + ε(xn − ˆ P t ) ˆ P t′
€
Pt′ = − 1
6 (1− β )−2 + 13 Qt−1
€
Qt−1 = x t−1 + βQt−2 + 2Pt−1 − 1−α1−β
€
Q1 = x1
Learning Results
Trials
Stochastic Gradient Descent
Trials
Bayesian Learning
Summary
H: “Superstition” reflects adaptation to changing world
Exponential “memory” near-optimal & fits behavior; linear RT
Neurobiology: leaky integration, stochastic -rule, neuromodulation
Random sequence and changing biases hard to distinguish
Questions: multiple outcomes? Explicit versus implicit prediction?
Unlearning Temporal Correlation is Slow
Marginal posterior over
Marginal posterior over
Trials
Prob
abili
tyPr
obab
ility
(see Bialek, 2005)
Insight from Brain’s “Mistakes”
Ex: visual illusions
(Adelson, 1995)
(Adelson, 1995)
lightnessdepth
context
Neural computation specialized for natural problems
Ex: visual illusions
Insight from Brain’s “Mistakes”
Discount Rate vs. Assumed Rate of ChangeIterative form of linear exponential
Exact inference is non-linear
Linear approximation
Empirical distribution
Bayesian Inference
Posterior
Generative Model(what subject “knows”)
1: repetition0: alternation
Optimal Prediction(Bayes’ Rule)
Bayesian Inference
Optimal Prediction(Bayes’ Rule)
Generative Model(what subject “knows”)
Power-Law Decay of MemoryHuman memory
Stationary process!
Hierarchical Chinese Restaurant Process
10 7 4 …(Teh, 2006)
Natural (language) statistics
(Anderson & Schooler, 1991)
Ties Across Time, Space, and ModalitySequential
effects
RT
Stroop
GREENSSHSS
Eriksen
time
modalityspace
(Yu, Dayan, Cohen, JEP: HPP 2008)(Liu, Yu, & Holmes, Neur Comp 2008)
Sequential Effects Perceptual Discrimination
Optimal discrimination(Wald, 1947) R
A
• discrete time, SPRT
• continuous-time, DDM
DBM
PFC
(Yu & Dayan, NIPS 2005)(Yu, NIPS 2007)
(Frazier & Yu, NIPS 2008)(Gold & Glimcher, Neuron 2002)
Monkey G
Coe
ffic
ient
s
Trials into past
= .72
Exponential Discounting for Changing Rewards
Monkey F
Coe
ffic
ient
s
Trials into past
= .63
(Sugrue, Corrado, & Newsome, 2004)
Monkey G
Coe
ffic
ient
s
Trials into past
= .72
Monkey F
Coe
ffic
ient
s
Trials into past
= .63
Human & Monkey Share Assumptions?
MonkeyHuman
≈!
= .68 = .80
Simulation Results
Trials
Learning via stochastic -rule
Monkeys’ Discount Rates in Choice Task(Sugrue, Corrado, & Newsome, 2004)
Monkey FC
oeff
icie
nts
Trials into past
= .63
.63
.68
Monkey G
Coe
ffic
ient
s
Trials into past
= .72
.72
.80
Human & Monkey Share Assumptions?
.72
.80
.63
.68
MonkeyHuman
≈!