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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 iH 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 22 2 22 2 2 2 0 0 2 0 2 22 2 22 0 2 0 2 2 2 22 0 2 2 22 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 22 2 22 2 2 2 0 0 2 0 2 22 2 22 0 2 0 2 2 2 22 0 2 2 22 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 22 2 22 2 2 2 0 0 2 0 2 22 2 22 0 2 0 2 2 2 22 0 2 2 22 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 ), 51100 (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 proj51100 (t1 ) proj50 (t1 ) 2 proj51100 (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 xA Indifferent Prefer A Quantum DFT Prefer B Pr( y | A) y xA 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 ~ ~ ~ ~ ~ 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!