+ All Categories
Home > Small Business & Entrepreneurship > Ldb Convergenze Parallele_trueblood_03

Ldb Convergenze Parallele_trueblood_03

Date post: 28-Aug-2014
Category:
Upload: laboratoridalbasso
View: 55 times
Download: 0 times
Share this document with a friend
Description:
 
Popular Tags:
46
An Introduction to Quantum Models of Cognition and Decision-making Jennifer S. Trueblood University of California, Irvine Thursday, September 5, 13
Transcript
Page 1: Ldb Convergenze Parallele_trueblood_03

An Introduction to Quantum Models of Cognition and

Decision-making

Jennifer S. TruebloodUniversity of California, Irvine

Thursday, September 5, 13

Page 2: Ldb Convergenze Parallele_trueblood_03

Motivation

• Judgments often deviate from classical probability theory

• Conjunction & disjunction fallacies, base rate fallacies, subadditivity, etc.

• Heuristics and biases approach lacks a coherent theoretical explanation for judgment effects

• Quantum probability theory has the potential to explain judgments from a theoretical standpoint

Thursday, September 5, 13

Page 3: Ldb Convergenze Parallele_trueblood_03

Probability Theories

• Classic (Kolmogorov 1933) Theory

• Boolean logic: follows the extension rule A ⊆ B, p(A) < p(B)

• Quantum (von Neumann, 1932) Theory

• A generalization of classical theory

• Drops unicity (a single sample space)

• Can violate the extension rule

Thursday, September 5, 13

Page 4: Ldb Convergenze Parallele_trueblood_03

Six Reasons for a Quantum Approach to Cognition and Decision

1. Judgments are based on indefinite states

2. Judgments create rather than record

3. Judgments disturb each other, introducing uncertainty

4. Judgments do not always obey classic logic

5. Judgments do not obey the principle of unicity

6. Cognitive phenomena may not be decomposable

Thursday, September 5, 13

Page 5: Ldb Convergenze Parallele_trueblood_03

Reason 1: Judgments are based on indefinite states

• Stochastic model: a particle producing a definite sample path through a state space

• Quantum model: a wave moving across time over the state space

• superposition state: an indefinite state capturing ambiguity and/or confusion

Thursday, September 5, 13

Page 6: Ldb Convergenze Parallele_trueblood_03

Reason 2: Judgments create rather than record

• Classic model: answers are “read outs” from stored states

• Quantum model: answers are constructed from the interaction of an indefinite state and the question we ask

• Example: ambiguity of emotional state

• “Are you excited?”

• “Are you sad?”

Thursday, September 5, 13

Page 7: Ldb Convergenze Parallele_trueblood_03

Reason 3: Judgments disturb each other

• Classical models cannot capture the effects of measurement disturbances

• Quantum theory can allow for one question to disturb the answer to another.

Thursday, September 5, 13

Page 8: Ldb Convergenze Parallele_trueblood_03

Reason 4: Judgments do not always obey classic logic

• Defendant is guilty or innocent; defendant is good or bad

• Classical logic obeys the distributive axiom:

• Guilty ⋀ (Good ⋁ Bad) = (Guilty ⋀ Good) ⋁ (Guilty ⋀ Bad)

• The law of total probability in classical probability theory is derived from the distributive axiom

• p(Guilty) = p(Good)p(Guilty | Good) + p(Bad)p(Guilty | Bad)

• Quantum logic does not always obey the distributive axiom and thus can violate the law of total probability

Thursday, September 5, 13

Page 9: Ldb Convergenze Parallele_trueblood_03

Reason 5: Judgments do not obey the principle of unicity

• Classical probability theory assumes a single sample space - a complete and exhaustive description of all events

• Quantum probability allows for multiple sample spaces that are pasted together in a coherent way

Future events

Thursday, September 5, 13

Page 10: Ldb Convergenze Parallele_trueblood_03

Reason 6: Cognitive phenomena may not be decomposable

• Two seemingly distinct and separated systems behave as one: quantum correlated

• Example, words in a memory experiment

• Two theories:

1.Spreading activation model: words are discrete nodes that can be highly connected

2.Study word and its associates behave as one: a word’s associative network arises in synchrony with the word being studied

Thursday, September 5, 13

Page 11: Ldb Convergenze Parallele_trueblood_03

Examples from Cognition and Decision

• Examples of paradoxical findings from cognition and decision that can be modeled with quantum probability

1. Interference effects in perception

2. Interference of categorization on decision-making

3. The disjunction effect

4. Violations of dynamic consistency

5. Survey question order effects

6. Conceptual combinations

Thursday, September 5, 13

Page 12: Ldb Convergenze Parallele_trueblood_03

Interference Effects in Perception

• Consider two different perceptual judgment tasks:

• Task A: binary forced choice response

• Task B: confidence rating on a 7 point scale

• Participants are randomly assigned to two groups:

• Group 1: Task B

• Group 2: Task A followed by Task B

Thursday, September 5, 13

Page 13: Ldb Convergenze Parallele_trueblood_03

Response Probabilities

Thursday, September 5, 13

Page 14: Ldb Convergenze Parallele_trueblood_03

Response Probabilities

Task B

.

.

.

1

2

7

S

pB(R

B=1)

pB(R

B=7)

Task A Task B

.

.

.

11

2

1

1

2

7

S

pAB

(RB=1|R

A=1)

pAB

(RB=7|R

A=2)

pAB

(RA=2)

pAB

(RA=1)

Thursday, September 5, 13

Page 15: Ldb Convergenze Parallele_trueblood_03

Total Probability and Interference Term

• The law of total probability gives:

TP (RB = k) =2X

j=1

pAB(RA = j) · pAB(RB = k | RA = j)

• The interference effect for level k of the response to task B:

IntB(k) = pB(RB = k)� TP (RB = k)

Thursday, September 5, 13

Page 16: Ldb Convergenze Parallele_trueblood_03

Conte et al. (2009)

• Experiments on perceptual judgment tasks with pairs of ambiguous figures

Thursday, September 5, 13

Page 17: Ldb Convergenze Parallele_trueblood_03

Conte et al. (2009) Experiment 1• Task was to decide whether the objects (circles or horizontal lines) were

equal or not

Figure A Figure B

Thursday, September 5, 13

Page 18: Ldb Convergenze Parallele_trueblood_03

Experiment 1 Results

• The inference effect for level B+ (horizontal lines are equal):

IntB(+) = pB(RB = +)� TP (RB = +)

= 0.6667� 0.5000 = 0.1667

Total Probability for A followed by B

Thursday, September 5, 13

Page 19: Ldb Convergenze Parallele_trueblood_03

Interference of Categorization on Decision-making

• Townsend et al. (2000) Task:

• Categorize faces as ‘Good guy’ or ‘Bad guy’

• Decide to act Friendly (withdraw) or Aggressive (attack)

• Narrow faces had a 0.6 probability of being ‘Bad’ and wide faces had a 0.6 probability of being ‘Good’

Thursday, September 5, 13

Page 20: Ldb Convergenze Parallele_trueblood_03

Townsend et al. (2000)

• Two conditions

Thursday, September 5, 13

Page 21: Ldb Convergenze Parallele_trueblood_03

Busemeyer et al. (2009)

• Replication of Townsend (2000)

• Results:

• Total probability of attacking after categorization = 0.59

• Probability of attacking without categorization = 0.69

Interference Effect

Thursday, September 5, 13

Page 22: Ldb Convergenze Parallele_trueblood_03

Savage’s Sure Thing Principle

• Suppose

• when is the state of the world, you prefer action A over B

• when is the state of the world, you also prefer action A over B

S

• Therefore you should prefer A over B even when S is unknown

Thursday, September 5, 13

Page 23: Ldb Convergenze Parallele_trueblood_03

The Disjunction Effect• Violations of the Sure Thing Principle (Tversky & Shafir, 1992) in

the Prisoner’s Dilemma game

You DefectYou

Cooperate

Other Defects

other: 10you: 10

other: 25you: 5

Other Cooperates

other: 5you: 25

other: 20you: 20

StudyKnown to

defectKnown to cooperate

Unknown

Shafir & Tversky (1992)

97 84 63

Croson (1999)

67 32 30

Li & Taplan (2002)

83 66 60

Busemeyer et al. (2006)

91 84 66

Observed proportion of defections

Thursday, September 5, 13

Page 24: Ldb Convergenze Parallele_trueblood_03

Dynamic Consistency• Dynamic consistency: Final decisions agree with planned decisions (Barkan

and Busemeyer, 2003)

• Two stage gamble

1. Forced to play stage one, but outcome remained unknown2. Made a plan and final choice about stage two

• Plan:

• If you win, do you plan to gamble on stage two?• If you lose, do you plan to gamble on stage two?

• Final decision

• After an actual win, do you gamble on stage two?• After an actual loss, do you now choose to gamble on stage two?

Thursday, September 5, 13

Page 25: Ldb Convergenze Parallele_trueblood_03

Barkan and Busemeyer (2003) Results

Risk averse after a win

Risk seeking after a loss

Thursday, September 5, 13

Page 26: Ldb Convergenze Parallele_trueblood_03

Question Order Effects• A Gallup Poll question in 1997 (N = 1002, split sample)

• Do you generally think Bill Clinton is honest and trustworthy?

• How about Al Gore?

• Do you generally think Al Gore is honest and trustworthy?

• How about Bill Clinton?

Thursday, September 5, 13

Page 27: Ldb Convergenze Parallele_trueblood_03

Question Order Effects: Assimilation

• Proportion of “Yes” responses

• Do you generally think Bill Clinton is honest and trustworthy? (50%)

• How about Al Gore? (60%)

• Do you generally think Al Gore is honest and trustworthy? (68%)

• How about Bill Clinton? (57%)

18%

3%

Thursday, September 5, 13

Page 28: Ldb Convergenze Parallele_trueblood_03

Question Order Effects: Contrast

• Proportion of “Yes” responses

• Do you generally think Newt Gingrich is honest and trustworthy? (41%)

• How about Bob Dole? (64%)

• Do you generally think Bob Dole is honest and trustworthy? (60%)

• How about Newt Gingrich? (33%)

19%

31%

Thursday, September 5, 13

Page 29: Ldb Convergenze Parallele_trueblood_03

Conceptual Combinations

• Hampton (1988) asked subjects to rate typicality

1. rate whether an item belonged to category A

2. rate whether it belonged to category B

3. rate whether it belonged to A or B

4. rate whether it belonged to A and B

Thursday, September 5, 13

Page 30: Ldb Convergenze Parallele_trueblood_03

Results• Is drinking beer a member of games (.78)

• Is drinking beer a member of hobbies (.20)

• Is drinking beer a member of games or hobbies (.58)

• Is a spider a pet (.40)

• Is a spider a farmyard animal (.33)

• Is a spider both a pet and a farmyard animal (.65)

• Is guppy a pet (.09)

• Is guppy a fish (.08)

• Is guppy a pet fish (.39)Thursday, September 5, 13

Page 31: Ldb Convergenze Parallele_trueblood_03

Two Dimensional Example

Thursday, September 5, 13

Page 32: Ldb Convergenze Parallele_trueblood_03

Uncertainty Principle• Given True is observed

• State changes to True (S T)

• Prob True now equals 1.0

• Prob of Good equals .50

• Certain about True implies uncertain about Good and visa versa

Thursday, September 5, 13

Page 33: Ldb Convergenze Parallele_trueblood_03

Interference Principle

• If initially asked True vs False, and True is observed, state changes to True (becoming uncertain about Good)

• If next asked about Good vs Bad and Good is observed, state changes to Good (becoming uncertain about True)

Thursday, September 5, 13

Page 34: Ldb Convergenze Parallele_trueblood_03

Distributive Rule• Classic Theory

G = G⋀(T⋁F)

= (G⋀T)⋁(G⋀F)

• Quantum Theory

G = G⋀(T⋁F)

≠ (G⋀T)⋁(G⋀F)

(G⋀T) doesn’t exist

(G⋀F) doesn’t exist

Thursday, September 5, 13

Page 35: Ldb Convergenze Parallele_trueblood_03

Unicity (closure)• Classic Theory

If G, T are events then (G⋀T) is an event

• Quantum Theory

If G, T are compatible, then (G⋀T) is an event

If G, T are incompatible, then (G⋀T) doesn’t exist

Thursday, September 5, 13

Page 36: Ldb Convergenze Parallele_trueblood_03

Compatible vs Incompatible Representations

• If T,F are compatible with G, B then

Require at least a 4 dimensional space

S = ↵TG|TGi+ ↵TB |TBi+ ↵FG|FGi+ ↵FB |FBi

S = ↵T |T i+ ↵F |F i = �G|Gi+ �B |Bi

• If T, F are incompatible with G,B then

Require at least a 2 dimensional space

Thursday, September 5, 13

Page 37: Ldb Convergenze Parallele_trueblood_03

Three Dimensional Example

• Voting Event:

1. democrat (outcome D)

2. republican (outcome R)

3. independent (outcome I)

• Ideology Event:

1. liberal (outcome L)

2. conservative (outcome C)

3. moderate (outcome M)

Thursday, September 5, 13

Page 38: Ldb Convergenze Parallele_trueblood_03

Classic Set Representation

L C M

D D ∩ L D ∩ C D ∩ M

R R ∩ L R ∩ C R ∩ M

I I ∩ L I ∩ C I ∩ M

Vote

Ideology

Thursday, September 5, 13

Page 39: Ldb Convergenze Parallele_trueblood_03

Classic Probability Function

p(L) p(C) p(M)

p(D) p(D ∩ L) p(D ∩ C) p(D ∩ M)

p(R) p(R ∩ L) p(R ∩ C) p(R ∩ M)

p(I) p(I ∩ L) p(I ∩ C) p(I ∩ M)

Probabilities sum to 1 for

nine joint outcomes

Thursday, September 5, 13

Page 40: Ldb Convergenze Parallele_trueblood_03

Compatibility

• Compatible events

• Two events can be realized simultaneously

• Order of events does not matter

• Incompatible events

• Two events cannot be realized simultaneously

• Order of events does matter

} }QuantumProbability

ClassicProbability

Thursday, September 5, 13

Page 41: Ldb Convergenze Parallele_trueblood_03

Compatible Events in Quantum Probability

• State vector within nine dimensional space

↵i\j

same as classical probability

q(i \ j) = ||Pij | i||2 = ||↵i\j ||2 = p(i \ j)

Projector (9x9 matrix with all zeros except a 1 for i∩j)

| i =

2

6666666666664

↵D\L

↵D\C

↵D\M

↵R\L

↵R\C

↵R\M

↵I\L

↵I\C

↵I\M

3

7777777777775

• Probability amplitude for voting outcome i and ideology outcome j:

• Quantum probability for voting outcome i and ideology outcome j

Thursday, September 5, 13

Page 42: Ldb Convergenze Parallele_trueblood_03

Vector Space For Incompatible Events

• Represented by two basis for the same 3 dimensional vector spaceD

R

I

C

M

L • Ideology Basis:

L = liberal

C = conservative

M = moderate

• Voting Basis:

D = democrat

R = republican

I = independent

• Ideology Basis is a unitary transformation of the Voting Basis:

Id = {U |Di, U |Ri, U |Ii}

V = {|Di, |Ri, |Ii} Id = {|Li, |Ci, |Mi}

Thursday, September 5, 13

Page 43: Ldb Convergenze Parallele_trueblood_03

Calculating Quantum Probabilities

• Belief State:D

R

I

C

M

L| i = S

| i = �L|Li+ �C |Ci+ �M |Mi

= U�L|Di+ U�C |Ri+ U�M |Ii

| i = ↵D|Di+ ↵R|Ri+ ↵I |Ii

↵ = U� � = U †↵

q(D) = ||PD| i||2 = ||↵D||2

q(C) = ||PC | i||2 = ||�C ||2

• Unitary Transformations relate the probability amplitudes:

• Calculating probabilities

Thursday, September 5, 13

Page 44: Ldb Convergenze Parallele_trueblood_03

Mixed vs Superposed States

• Suppose voter is NOT independent

• Mixed State:

• 0.5 probability state = D

• 0.5 probability state = R

• Superposition:

S =1p2(|Di+ |Ri)

Thursday, September 5, 13

Page 45: Ldb Convergenze Parallele_trueblood_03

Mixed vs Superposed States• For both mixed and superposed states

• Equal probability of voting democrat or republican

• If democrat, probability moderate equals 0.5

• If republican, probability moderate equals 0.25• In a classical mixed state

• Either democrat or republican exactly

• Total probability moderate = P(D)P(M|D) + P(R)P(M|R) = (.5)x(.5) +(.5)x(.25) = .375

S =1p2(|Di+ |Ri)• In a quantum superposed state

• Neither a democrat or republican exactly

• The probability of being a moderate is zero because S is orthogonal to the moderate event

Thursday, September 5, 13

Page 46: Ldb Convergenze Parallele_trueblood_03

Thank You• What’s coming next...

• Quantum models of human judgments

• Dynamic quantum decision models

Thursday, September 5, 13


Recommended