Quantum dynamics I. Peter Kvam. Michigan State University Max Planck Institute for Human Development

Transcription

1 Quantum dynamics I Peter Kvam Michigan State University Max Planck Institute for Human Development Full-day workshop on quantum models of cognition 37 th Annual Meeting of the Cognitive Science Society

2 Dynamics Busemeyer, J. R., Wang, Z., & Townsend, J. T. (006). Quantum dynamics of human decision making. Journal of Mathematical Psychology, 50 (3), Kvam, P. D., Pleskac, T. J., Yu, S., & Busemeyer, J. R. (in press). Proceedings of the National Academy of Sciences.

3 Dynamics - Intro We talked a bit about superposition already

4 Superposition

5 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

6 Measurement

7 Rescaling

8 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

9 Rotation

10 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, 0,, 100%) Or for preferences on a Likert scale (1,, 3,, 9) We ll be using compatible states for multiple responses

11 Random walk / diffusion models Memory recognition Ratcliff (1978) Perceptual discrimination Link & Heath (1975) Usher & McClelland (001) Categorization Nosofsy & Palmeri (1997) Ashby (000) Preferential / risky decisions Busemeyer & Townsend (1993) Sensory processing Smith (1995) Rudd (1996) Multi-attribute decisions Roe, Busemeyer & Townsend (001) Diederich (1997) Neural activation Gold & Shadlen (007) Schall (003) Liu & Pleskac (011) Confidence judgments Pleskac & Busemeyer (010) Ratcliff & Starns (013)

12 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%

13 Diffusion path

14 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 φ x (t) Transitions during time t are given by the transition matrix P(t) φ x t = P t φ x (0)

15 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 ψ x (t) Transitions specified by the unitary matrix U(t) ψ x t = U t ψ x (0)

16 Specifying transition matrices Both models use a drift rate (μ) and a diffusion rate (σ ) 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 Qt ( t) e (0) The quantum walk uses a (Hermitian) Hamiltonian and the Schrödinger equation to specify U(t): d ( t) i H ( t) dt iht ( t) e (0)

17 Specifying transition matrices For the Markov model, the columns of the intensity matrix must sum to zero, i q ij = 0 For the quantum model, the Hamiltonian must be Hermitian, h ij = h ji *

18 Specifying the intensity matrix (Markov model) 1,, 1, 1, 1 jj j j j j q q q Example: 5-state model Q

19 Specifying the intensity matrix (Markov model) 1,, 1, 1, 1 jj j j j j q q q Example: 5-state model Q Drift

20 Specifying the intensity matrix (Markov model) 1,, 1, 1, 1 jj j j j j q q q Example: 5-state model Q Diffusion

21 Specifying the Hamiltonian (Quantum model) h i,i = i and h i,i+1 = h i,i-1 = Example: 5-state model H 1 / m / m / m / m / m m = # of states

22 Specifying the Hamiltonian (Quantum model) Drift Diffusion h i,i = i and h i,i+1 = h i,i-1 = Example: 5-state model H 1 / m / m / m / m / m m = # of states

23 Initial state Markov random walk Quantum walk Mixed State Superposition State Pr( conf x t) ( t). Pr( conf x t) ( t) x x

24 After transitions Markov random walk Quantum walk μ μ P t e P t Q ( t1t0 ( ) ) ( ) q q q 1 0 j, j j1, j j1, j ( ) / ( ) / ( 1 0 ( ) ih t t t e ) ( t ) h h 1 0 j, j j ( * ) # states h j1, j j1, j

25 Response times Markov model 0.01 Stopping Time Density 0.7 Stopping Time Distribution Probability Correct InCorrect Probability Correct InCorrect Time in msec Time in msec Busemeyer, Wang, & Townsend (006)

26 Response times quantum model 0.07 Stopping Time Density 1 Stopping Time Distribution Probability Correct InCorrect Probability Correct InCorrect Time in msec Time in msec Busemeyer, Wang, & Townsend (006)

27 Response time comparison Markov random walk outperformed quantum walk in Busemeyer, Wang, & Townsend (006) A partially coherent quantum random walk outperformed the diffusion model in Fuss & Navarro (013)

28 Lunch? We ll start up again at 1 p.m. Afternoon: 1- p.m. Quantum Dynamics II Peter Kvam -:30 p.m. Advanced tools for building quantum models I James Yearsley :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

29 Quantum Dynamics II Peter Kvam Michigan State University Max Planck Institute for Human Development Full-day workshop on quantum models of cognition 37 th Annual Meeting of the Cognitive Science Society

30 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, t 1 Pr A, t 1 + Pr C = x ~A, t 1 Pr(~A, t 1 )

31 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, t 1 Pr A, t 1 + Pr C = x ~A, t 1 Pr(~A, t 1 ) If a person chooses A at time t 1, it does not change their state Therefore, subsequent responses should not be affected (e.g. at time t ) We tested this using a -response task Choice (t 1 ) then confidence (t )

32 Markov prediction M y = Maps evidence states onto confidence M correct, M incorrect = Maps evidence onto correct / incorrect states L = sums the probabilities across states I = Identity matrix

33 Markov prediction Law of total probability holds Choice or no choice at t 1 should make no difference in marginal distributions of confidence ratings at t

34 Quantum prediction M y = Maps evidence states onto confidence M correct, M incorrect = Maps evidence onto correct / incorrect states L = sums the probabilities across states I = Identity matrix

35 Quantum prediction Law of total probability is violated A choice made at time t 1 should result in different marginal distributions of confidence when rated at time t

36 Random dot motion stimulus

37 Experiment Kvam, Pleskac, Yu, & Busemeyer (in press)

38 Model predictions Markov Random Walk w Quantum Random Walk w Mixed State Superposition State Pr( conf x t) ( t). Pr( conf x t) ( t) x x Kvam, Pleskac, Yu, & Busemeyer (in press)

39 Model predictions Markov Random Walk Quantum Random Walk Kvam, Pleskac, Yu, & Busemeyer (in press)

40 Model predictions Markov Random Walk Quantum Random Walk 1 Pr( correct t) ( t ) ( t ) ( t correct) n 1 n ( t1), ( t1) 1 ( t ) ( t ) n 1 n51 1 Pr( correct) ( t ) ( t ) x 1 x51 1 proj51 100( t1) proj50( t1) ( t 1 correct) 1 proj51 100( t1) proj50( t1) Kvam, Pleskac, Yu, & Busemeyer (in press)

41 Model predictions Markov Random Walk Quantum Random Walk ( 1 ( ) Q t t ) 1 t e ( t ) 1 ( t ) e ( t ) ih ( t t ) 1 Kvam, Pleskac, Yu, & Busemeyer (in press)

42 Model predictions Markov Random Walk Quantum Random Walk φ t no choice = φ t choice ψ t no choice ψ(t choice) Kvam, Pleskac, Yu, & Busemeyer (in press)

43 Experiment Kvam, Pleskac, Yu, & Busemeyer (in press)

44 Methods 4 conditions 4 levels of dot coherence ( / 4 / 8 / 16 %) 3 levels of second stage time (50 / 750 / 1500 ms) 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

45 Results Cumulative distribution of confidence ratings Clear difference between mean confidence in choice (M = Kvam, Pleskac, Yu, & Busemeyer (in press)

46 Results Kvam, Pleskac, Yu, & Busemeyer (in press)

47 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 ) Starting point variability (w) Second-stage decay (γ), attenuates drift after t 1 Fitting method: Grid approximation of likelihood function Sampled evenly across 1 x 1 x 51 x 1 grid of the 4 parameters Computed Pr(Data Model) at each point Computed Bayes Factor using uniform priors over the grid

48 Results model fits Kvam, Pleskac, Yu, & Busemeyer (in press)

49 Results model fits Kvam, Pleskac, Yu, & Busemeyer (in press)

50 Results model fits Kvam, Pleskac, Yu, & Busemeyer (in press)

51 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 (006) 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

52 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 Choose A > B A B Time 1 Time

53 Free choice paradigm

54 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 Choose A > B A B Time 1 Time Time-dependent bolstering

55 Dissonance - Suppression Early dissonance work also found the opposite effect, where preference for a chosen option decreased Suppression

56 Dissonance - Suppression Early dissonance work also found the opposite effect, where preference for a chosen option decreased Suppression Time-dependent suppression

57 Dissonance - Background Brehm, 1955; Festinger, 1957; Festinger, 1963 In theory, choice is the key factor A B Choose A > B A B Time 1 Time No A B A B Choice Time 1 Time Choice-dependent bolstering

58 Dissonance - Background Brehm, 1955; Festinger, 1957; Festinger, 1963 In theory, choice is the key factor Choose A > B A B A B Time 1 Time A B No Choice A B Time 1 Time Choice-dependent suppression

59 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

60 Task Stimulus onset Response 1 (Choice / Click) Response (Rate preference)

61 Task 1 st response Worth $11 Rating: * * * Average Meal: $17 Distance: 4.7 miles < > Worth $13 Rating: * * * Average Meal: $1 Distance: 0. miles

62 Task nd response Worth $11 Rating: * * * Average Meal: $17 Distance: 4.7 miles Worth $13 Rating: * * * Average Meal: $1 Distance: 0. miles

63 Task Stimulus onset Response 1 Choice / No-choice Response (Rate preference)

64 Decision field theory (DFT) framework Busemeyer & Townsend, 1993

65 Classical (Markov) DFT Busemeyer & Townsend, 1993; Busemeyer & Diederich, 00 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

66 Classical (Markov) DFT Busemeyer & Townsend, 1993; Busemeyer & Diederich, 00 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

67 Classical (Markov) DFT Busemeyer & Townsend, 1993; Busemeyer & Diederich, 00

68 Quantum DFT Based on the quantum random walk model (Busemeyer et al, 006; 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

69 Quantum DFT Prefer B Indifferent Prefer A

70 Quantum DFT Prefer B Pr( A) x xa Indifferent Prefer A

71 Quantum DFT Prefer B Pr( y A) xa y x Indifferent Prefer A

72 Quantum DFT Choice condition No choice condition States after choice / click (t 1 )

73 Quantum DFT

74 Quantum DFT Time-dependent bolstering

75 Quantum DFT Time-dependent suppression

76 Quantum DFT Choice-dependent suppression

77 Quantum DFT Choice-dependent bolstering

78 Model predictions (review) Classical DFT Should be no choice-depdent bolstering or suppression effects or time-dependent suppression No choice = choice Mean preference strength should change monotonically Quantum DFT Should be time-dependent and choice-dependent bolstering and suppression effects No choice choice 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

79 Task Stimulus onset Response 1 (Choice / Click) Response (Rate preference) 5 seconds 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

80 Results

81 Model fitting Each model used 4 parameters: Drift (µ): sets drift rate as a linear function of item difference, grouped by contrast level Diffusion (σ ): 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

82 Model fits z

83 Model fits

84 Model fits

85 Model fits

86 Model fits Log Bayes Factor (QDFT : DFT) = (very strong evidence for QDFT)

87 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

88 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

89 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)

90 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!