Download Quantum dynamics - Psychological Sciences

Quantum dynamics I
Peter Kvam
Michigan State University
Max Planck Institute for Human Development
Full-day workshop on quantum models of cognition
37th Annual Meeting of the Cognitive Science Society
Dynamics
Busemeyer, J. R., Wang, Z., & Townsend, J.
T. (2006). Quantum dynamics of human
decision making. Journal of Mathematical
Psychology, 50 (3), 220-241.
Kvam, P. D., Pleskac, T. J., Yu, S., &
Busemeyer, J. R. (in press). Proceedings of
the National Academy of Sciences.
Dynamics - Intro
• We talked a bit about superposition already
Superposition
Dynamics - Intro
• We talked a bit about superposition already
• Also, how to evaluate beliefs
– Taking a measurement of a quantum cognitive system
– This is one way of changing states
Measurement
Rescaling
Dynamics - Intro
• We talked a bit about superposition already
• Also, how to evaluate beliefs
– Taking a measurement of a quantum cognitive system
– This is one way of changing states
• Next, we’ll examine how beliefs change over
time with new information
– Rotations in n-dimensional Hilbert space
– Specified by unitary operators
Rotation
Unitary transformations
• Rotations are specified by unitary matrices
• Non-commutative with other operators
• Simple in geometric space, but more difficult to construct
for higher-dimensional spaces we’ll be using
– E.g. for response times
– Or for probability judgments (0, 10, 20, …, 100%)
– Or for preferences on a Likert scale (1, 2, 3, …, 9)
• We’ll be using compatible states for multiple responses
Random walk / diffusion models
•
Memory recognition
•
– Ratcliff (1978)
•
Perceptual discrimination
– Link & Heath (1975)
– Usher & McClelland (2001)
•
•
– Smith (1995)
– Rudd (1996)
•
Multi-attribute decisions
– Roe, Busemeyer & Townsend
(2001)
– Diederich (1997)
Categorization
– Nosofsy & Palmeri (1997)
– Ashby (2000)
Sensory processing
•
Neural activation
– Gold & Shadlen (2007)
– Schall (2003)
– Liu & Pleskac (2011)
Preferential / risky decisions
– Busemeyer & Townsend (1993)
•
Confidence judgments
– Pleskac & Busemeyer (2010)
– Ratcliff & Starns (2013)
Random walk / diffusion models
• A decision-maker has to choose between two options (‘left’
and ‘right’)
• Each new piece of information favors one or the other
– Increments or decrements current belief state
• Continues until a desired level of confidence is reached
– E.g. 30% / 70%
Diffusion path
Markov random walk
• A person is only in 1 state at any given time, but an external
observer is uncertain about which state
– Represented as a distribution of probabilities across states:
mixed state vector 𝝓𝒙 (𝒕)
– Transitions during time t are given by the transition matrix 𝑷(𝒕)
𝜙𝑥 𝑡 = 𝑃 𝑡 ∙ 𝜙𝑥 (0)
Quantum random walk
• A person can simultaneously entertain multiple levels of evidence /
beliefs at the same time
– Represented as a distribution of probability amplitudes across
states: superposition state vector 𝝍𝒙 (𝒕)
– Transitions specified by the unitary matrix 𝑼(𝒕)
𝜓𝑥 𝑡 = 𝑈 𝑡 ∙ 𝜓𝑥 (0)
Specifying transition matrices
• Both models use a drift rate (𝝁) and a diffusion rate (𝝈𝟐 )
– Drift: average rate of ‘true’ sampling
– Diffusion: average rate of random / noise sampling
• The Markov random walk uses an intensity matrix and the
Kolmogorov forward equation to specify 𝑷(𝒕):
d
 (t )  Q   (t )
dt
→  (t )  e   (0)
Q t
• The quantum walk uses a (Hermitian) Hamiltonian and the
Schrödinger equation to specify 𝑼(𝒕):
d
 (t )  i  H  (t )
dt
→  (t )  e
iH t
 (0)
Specifying transition matrices
• For the Markov model, the columns of the
intensity matrix must sum to zero,  i qij = 0
• For the quantum model, the Hamiltonian must
be Hermitian, hij = hji*
Specifying the intensity matrix
(Markov model)
q j 1, j
1  2  
1  2  
2
  2  , q j 1, j   2  , q jj   2
2  
2  

Example: 5-state model
 2
  2
 2
 
 2

Q 0

 0


 0

2
22

2
22



2
2
2

0
0

2
0
2
22


2
22

0

2
0
2
2
2
22

0

2

2
22



2
2
2


2

0 

0 


0 

2 
2 
2
 2
 
Specifying the intensity matrix
(Markov model)
q j 1, j
Drift
1   
1   
2
  2  , q j 1, j   2  , q jj   2
2  
2  

2
2
Example: 5-state model
 2
  2
 2
 
 2

Q 0

 0


 0

2
22

2
22



2
2
2

0
0

2
0
2
22


2
22

0

2
0
2
2
2
22

0

2

2
22



2
2
2


2

0 

0 


0 

2 
2 
2
 2
 
Specifying the intensity matrix
(Markov model)
Diffusion 2
1   
1  2  
2
q j 1, j   2  , q j 1, j   2  , q jj   2
2  
2  

Example: 5-state model
 2
  2
 2
 
 2

Q 0

 0


 0

2
22

2
22



2
2
2

0
0

2
0
2
22


2
22

0

2
0
2
2
2
22

0

2

2
22



2
2
2


2

0 

0 


0 

2 
2 
2
 2
 
Specifying the Hamiltonian
(Quantum model)
hi,i = i and hi,i+1 = hi,i-1 = 2
Example: 5-state model
1   / m
2
0
0
0 


2
2

2


/
m

0
0


2
2
H  0

3  / m

0 


2
2
0

4 / m

 0

 0
0
0
2
5   / m 
𝑚 = # of states
Specifying the Hamiltonian
(Quantum model)
Diffusion
Drift
hi,i = i and hi,i+1 = hi,i-1 = 2
Example: 5-state model
1   / m
2
0
0
0 


2
2

2


/
m

0
0


2
2
H  0

3  / m

0 


2
2
0

4 / m

 0

 0
0
0
2
5   / m 
𝑚 = # of states
Initial state
Markov random walk
Quantum walk
Mixed State
Superposition State
Pr(conf  x | t )  x (t ).
Pr(conf  x | t ) | x (t ) |2
After transitions
Markov random walk
Quantum walk
μ
P(t1 )  eQ (t1 t0 ) P(t0 )
q j , j   2
q j 1, j  ( 2   ) / 2
q j 1, j  ( 2   ) / 2
μ
 (t1 )  eiH (t t ) (t0 )
1
j
)
# states
 h j 1, j   2
h j , j  (  *
h j 1, j
0
Response times – Markov model
Stopping Time Density
Stopping Time Distribution
0.01
0.7
0.009
0.6
0.008
Correct
InCorrect
0.007
0.5
Correct
InCorrect
Probability
Probability
0.006
0.005
0.4
0.3
0.004
0.003
0.2
0.002
0.1
0.001
0
0
50
100
150
Time in msec
200
250
0
0
50
100
150
Time in msec
200
250
Busemeyer, Wang, & Townsend (2006)
Response times – quantum model
Stopping Time Density
Stopping Time Distribution
0.07
1
0.9
0.06
0.8
0.7
Correct
InCorrect
0.04
Correct
InCorrect
0.6
Probability
Probability
0.05
0.03
0.5
0.4
0.3
0.02
0.2
0.01
0.1
0
0
50
100
150
Time in msec
200
250
0
0
50
100
150
Time in msec
200
250
Busemeyer, Wang, & Townsend (2006)
Response time comparison
• Markov random walk outperformed quantum walk in
Busemeyer, Wang, & Townsend (2006)
• A partially coherent quantum random walk outperformed
the diffusion model in Fuss & Navarro (2013)
Lunch?
• We’ll start up again at 1 p.m.
• Afternoon:
– 1-2 p.m. Quantum Dynamics II – Peter Kvam
– 2-2:30 p.m. Advanced tools for building quantum models I –
James Yearsley
– 2:30-3 p.m. Coffee break
– 3-3:45 p.m. Advanced tools for building quantum models II –
James Yearsley
– 3:45-4 p.m. Discussion / questions – Jennifer, James, Peter
Quantum Dynamics II
Peter Kvam
Michigan State University
Max Planck Institute for Human Development
Full-day workshop on quantum models of cognition
37th Annual Meeting of the Cognitive Science Society
Multiple responses
• As in the earlier classical models, the Markov model
obeys the law of total probability
• This applies to sequential responses as well:
𝐏𝐫 𝑪 = 𝒙 = 𝐏𝐫 𝑪 = 𝒙 𝑨, 𝒕𝟏 ∗ 𝐏𝐫 𝑨, 𝒕𝟏 + 𝐏𝐫 𝑪 = 𝒙 ~𝑨, 𝒕𝟏 ∗ 𝐏𝐫(~𝑨, 𝒕𝟏 )
Multiple responses
• As in the earlier classical models, the Markov model obeys the law
of total probability
• This applies to sequential responses as well:
𝐏𝐫 𝑪 = 𝒙 = 𝐏𝐫 𝑪 = 𝒙 𝑨, 𝒕𝟏 ∗ 𝐏𝐫 𝑨, 𝒕𝟏 + 𝐏𝐫 𝑪 = 𝒙 ~𝑨, 𝒕𝟏 ∗ 𝐏𝐫(~𝑨, 𝒕𝟏 )
• If a person chooses A at time t1, it does not change their state
– Therefore, subsequent responses should not be affected (e.g. at time t2)
• We tested this using a 2-response task
– Choice (t1) then confidence (t2)
Markov prediction
My = Maps evidence states onto confidence
Mcorrect, Mincorrect = Maps evidence onto correct / incorrect states
L = sums the probabilities across states
I = Identity matrix
Markov prediction
• Law of total probability holds
• Choice or no choice at t1 should make no
difference in marginal distributions of confidence
ratings at t2
Quantum prediction
My = Maps evidence states onto confidence
Mcorrect, Mincorrect = Maps evidence onto correct / incorrect states
L = sums the probabilities across states
I = Identity matrix
Quantum prediction
• Law of total probability is violated
• A choice made at time t1 should result in
different marginal distributions of confidence
when rated at time t2
Random dot motion stimulus
Experiment
Kvam, Pleskac, Yu, & Busemeyer (in press)
Model predictions
Markov Random Walk
Quantum Random Walk
𝑤
𝑤
Mixed State
Superposition State
Pr(conf  x | t )  x (t ).
Pr(conf  x | t ) | x (t ) |2
Kvam, Pleskac, Yu, & Busemeyer (in press)
Model predictions
Markov Random Walk
Quantum Random Walk
Kvam, Pleskac, Yu, & Busemeyer (in press)
Model predictions
Markov Random Walk
100
1
Pr(correct | t )  50 (t1 )   n (t1 )
2
n 51
1
50 (t1 ), 51100 (t1 )
 (t1 | correct )  2
100
1
50 (t1 )   n (t1 )
2
n 51
Quantum Random Walk
100
1
2
2
Pr(correct )   50 (t1 )    x (t1 )
2
x 51
1
proj50 (t1 )
2
 (t1 | correct ) 
1
proj51100 (t1 ) 
proj50 (t1 )
2
proj51100 (t1 ) 
Kvam, Pleskac, Yu, & Busemeyer (in press)
Model predictions
Markov Random Walk
 (t2 )  e
 (t1 )
Q ( t2 t1 )
Quantum Random Walk
 (t2 )  eiH (t t ) (t1 )
2
1
Kvam, Pleskac, Yu, & Busemeyer (in press)
Model predictions
Markov Random Walk
𝜙 𝑡2 |𝑛𝑜 𝑐ℎ𝑜𝑖𝑐𝑒 = 𝜙 𝑡2 |𝑐ℎ𝑜𝑖𝑐𝑒
Quantum Random Walk
𝜓 𝑡2 |𝑛𝑜 𝑐ℎ𝑜𝑖𝑐𝑒 ≠ 𝜓(𝑡2 |𝑐ℎ𝑜𝑖𝑐𝑒)
Kvam, Pleskac, Yu, & Busemeyer (in press)
Experiment
Kvam, Pleskac, Yu, & Busemeyer (in press)
Methods
• 24 conditions
– 4 levels of dot coherence (2 / 4 / 8 / 16 %)
– 3 levels of second stage time (50 / 750 / 1500 ms)
– 2 main conditions (choice / no-choice)
• 9 Participants, Michigan State University students
– Each attended 6 total sessions
– Total ~3600 trials per participant
– Modeled individual level data
Results
• Cumulative distribution
of confidence ratings
• Clear difference
between mean
confidence in choice (M
=
Kvam, Pleskac, Yu, & Busemeyer (in press)
Results
Kvam, Pleskac, Yu, & Busemeyer (in press)
Model fitting
• Interference effect is clear in the data
– Qualitative evidence against the Markov model
• Compare quantitative model fits
– 4 parameters:
•
•
•
•
Drift multiplier (𝛿), gives drift as a linear function of coherence
Diffusion (𝜎 2 )
Starting point variability (𝑤)
Second-stage decay (𝛾), attenuates drift after t1
• Fitting method: Grid approximation of likelihood function
– Sampled evenly across 21 x 21 x 51 x 21 grid of the 4
parameters
– Computed Pr(Data | Model) at each point
– Computed Bayes Factor using uniform priors over the grid
Results – model fits
Kvam, Pleskac, Yu, & Busemeyer (in press)
Results – model fits
Kvam, Pleskac, Yu, & Busemeyer (in press)
Results – model fits
Kvam, Pleskac, Yu, & Busemeyer (in press)
Interference effects
• More extreme judgments in no-choice relative to
choice was unexpected
– Confirmation bias would suggest the opposite effect
– But there is some precedent:
• Sniezek et al (2006) – higher no choice than choice
confidence in general knowledge task
• Old dissonance work by Walster (1964) and Brehm &
Wicklund (1970)
• Crano & Messé (1970) suggest that effects in preference
should change direction over time
Dissonance - Background
Brehm, 1955; Festinger, 1957; Festinger, 1963
• Dissonance theory is based on the finding that decisions
between alternatives affect subsequent preferences
– Canonical finding is that people bring their preferences into
alignment with their decisions – “Bolstering”
A
B
Time 1
Choose
A>B
A
B
Time 2
Free choice paradigm
Dissonance - Background
Brehm, 1955; Festinger, 1957; Festinger, 1963
• Dissonance theory is based on the finding that decisions
between alternatives affect subsequent preferences
– Canonical finding is that people bring their preferences into
alignment with their decisions – “Bolstering”
A
B
Time 1
Choose
A>B
A
B
Time 2
Time-dependent bolstering
Dissonance - Suppression
• Early dissonance work also found the opposite effect, where
preference for a chosen option decreased – “Suppression”
Dissonance - Suppression
• Early dissonance work also found the opposite effect, where
preference for a chosen option decreased – “Suppression”
Time-dependent suppression
Dissonance - Background
Brehm, 1955; Festinger, 1957; Festinger, 1963
• In theory, choice is the key factor
A
B
Choose
A>B
Time 1
A
B
Time 1
A
B
Time 2
No
Choice
A
B
Time 2
Choice-dependent bolstering
Dissonance - Background
Brehm, 1955; Festinger, 1957; Festinger, 1963
• In theory, choice is the key factor
A
B
Choose
A>B
Time 1
A
B
Time 1
A
B
Time 2
No
Choice
A
B
Time 2
Choice-dependent suppression
Issues
• Why do these effects occur?
– Dissonance theory suggests that a choice creates a new
(conflicted) cognitive state
• Which effect occurs when?
– Effects depend on time and item type
– Time-dependent / choice-dependent, bolstering / suppression
• What models can account for these phenomena?
– Dissonance theory provides no quantitative predictions
– Extant models can predict time-dependent bolstering, but not
time-dependent suppression
– Extant process models do not predict choice-dependent
bolstering or suppression effects
Task
Stimulus
onset
Response 1
(Choice / Click)
Response 2
(Rate preference)
Task – 1st response
Worth $11
Rating: * * *
Average Meal: $17
Distance: 4.7 miles
<>
Worth $13
Rating: * * *
Average Meal: $12
Distance: 0.2 miles
Task – 2nd response
Worth $11
Rating: * * *
Average Meal: $17
Distance: 4.7 miles
Worth $13
Rating: * * *
Average Meal: $12
Distance: 0.2 miles
Task
Stimulus
onset
Response 1
Choice /
No-choice
Response 2
(Rate preference)
Decision field theory
(DFT) framework
Busemeyer & Townsend, 1993
Classical (Markov) DFT
Busemeyer & Townsend, 1993; Busemeyer & Diederich, 2002
• Preferences represented as a point
– A person can only give one response at any given time
– Mixed state vector
• Decisions and preferences are given by this point
– They do not change its position or system dynamics
Classical (Markov) DFT
Busemeyer & Townsend, 1993; Busemeyer & Diederich, 2002
• Preferences represented as a point
– A person can only give one response at any given time
– Mixed state vector
• Decisions and preferences are given by this point
– They do not change its position or system dynamics
– Predicts no choice-dependent effects
Classical (Markov) DFT
Busemeyer & Townsend, 1993; Busemeyer & Diederich, 2002
Quantum DFT
• Based on the quantum random walk model
– (Busemeyer et al, 2006; Kvam et al, in press)
• Preferences represented as a superposition across
preference levels
– At any given point, they may be entertaining many possible
preferences simultaneously
– Superposition state vector
• Preferences are constructed by collapsing this state onto
possible preference levels
Quantum DFT
Prefer B
Indifferent
Prefer A
Quantum DFT
Prefer B
Pr( A)    x
2
xA
Indifferent
Prefer A
Quantum DFT
Prefer B
Pr( y | A) 
y

xA
Indifferent
2
2
x
Prefer A
Quantum DFT
Choice condition
No choice condition
States after choice / click (t1)
Quantum DFT
Quantum DFT
Time-dependent bolstering
Quantum DFT
Time-dependent suppression
Quantum DFT
Choice-dependent suppression
Quantum DFT
Choice-dependent bolstering
Model predictions (review)
Classical DFT
• Should be no choice-depdent
bolstering or suppression
effects or time-dependent
suppression
– No choice = choice
Quantum DFT
• Should be time-dependent and
choice-dependent bolstering
and suppression effects
– No choice ≠ choice
• Mean preference strength
should change monotonically
• Mean preference strength
should oscillate over time
• Preference should reach an
asymptote and then stop
changing
• Preference will vary back and
forth, asymptote will be
achieved late if at all
Task
Stimulus
onset
Response 1
(Choice / Click)
5 seconds
Response 2
(Rate preference)
3 / 6 / 9 / 18 / 30 / 45
seconds
118 Participants, 48 trials each (8 each delay)
• Randomly assigned to choice / no choice
• Stimuli divided into low / medium / high contrast based on attributes
Results
Model fitting
• Each model used 4 parameters:
– Drift (µ): sets drift rate as a linear function of item difference,
grouped by contrast level
– Diffusion (σ2): sets noise in accumulation process
– Initial distribution width (w): sets prior beliefs
– Decay (d): sets the amount of information processing per unit
time after a decision is made (relative to pre-decision)
• Used a 31 x 31 x 31 x 10 grid approximation of
likelihood:
– Joint distributions of preference ratings for all conditions
– Choice probabilities based on item attributes + weights
– Uniform priors used to calculate Bayes Factor
Model fits
z
Model fits
Model fits
Model fits
Model fits
Log Bayes Factor (QDFT : DFT)
= 34.65 (very strong evidence for QDFT)
Theory performance
Accounts for…
Dissonance
Classical DFT
Quantum DFT

~
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~
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~
~
~
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

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
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Time-dependent bolstering
Time-dependent suppression
Choice-dependent bolstering
Choice-depedent suppression
Time course of preference
Memorylessness
Oscillations in preference
 Yes ~
With modifications
Conclusions – Dissonance study
• Dissonance is a verbal theory, doesn’t offer a
quantitative or dynamic account of preference
• Classical decision field theory doesn’t offer a
constructive characterization of decision-making
– Only preference formation via information gathering
• Quantum decision field theory addresses both issues
+ Predicts (surprising) oscillations in preference
+ Offers better fits to the empirical data than DFT
Remaining issues
• Confidence ratings are fit as nominal categories
– Errors in fit don’t take ordinal properties into account
– Example: Predicting a rating of 50 when it was actually 100 is
(fitness-wise) the same as predicting 95 when it was actually 100
• Therefore, unless fit is nearly perfect, predictions for mean
confidence will suffer
– By extension, so will fits to oscillations
• Where is the stable bolstering effect from dissonance studies?
– Delays may be too short
– Stimuli may be too objective (bolstering may rely on distortion)
Conclusions - Quantum Dynamics
• A quantum perspective on dynamic decision-making opens up
a wealth of new questions
– Challenges assumptions about belief representation,
measurement, and interaction with information accumulation
• Violations of the law of total probability: interference effects
• Quantum walks can out-perform Markov model (inference
tasks) as well as decision field theory (preference tasks)
– Predicts a effects (bolstering, suppression) which have
previously been absent from models of evidence accumulation
• But still more work to be done!