Quantum cognition quo vadis? A classicprobability account of mirrored order
effects in human judgments
Henrik Singmann1, David Kellen2, William H. Batchelder3
1University
of Zurich, Switzerland
2Syracuse University, USA
3University of California, Irvine, USA
Quantum Models of Cognition
© some crackpot website
Quantum Models of Cognition
• Formal approach for modeling psychological phenomena that
do not follow classic probability theory (i.e., Kolmogorov axioms):
• Linda problem: conjunction or disjuntion fallacy.
• Question order effects and QQ-equality.
• Formalization to describe context effects:
• Humans can be in indefinite state and act of judgment can create a
specific state.
• Judgments can be non-commutative and influence each other.
• Complemetarity of measurements (Wang & Busemeyer, 2015):
Only one question at the same time.
• Quantum probability:
• Probabilities are amplitudes in a vector space.
• Arise from projections in a vector (Hilbert) space.
Order Effects in Opinion Polls
• A: Is Bill Clinton trustworthy?
• B: Is Al Gore trustworthy?
• Direction P: Clinton → Al Gore
P: A → B
P*.1 (Gore "yes")
P*.2 (Gore "no")
P1.* (Clinton "yes")
P2.* (Clinton "no")
• Direction Q: Al Gore → Clinton
Q: B → A
Q*.1 (Gore "yes")
Q*.2 (Gore "no")
Q1.* (Clinton "yes")
Q2.* (Clinton "no")
Wang, Solloway, Shiffrin, & Busemeyer, 2014, PNAS
Order Effects in Opinion Polls
• A: Is Bill Clinton trustworthy?
1. Question difference: 22%
• B: Is Al Gore trustworthy?
• Direction P: Clinton → Al Gore
P: A → B
P*.1 (Gore "yes")
P*.2 (Gore "no")
54%
P1.* (Clinton "yes")
P2.* (Clinton "no")
• Direction Q: Al Gore → Clinton
Q: B → A
Q*.1 (Gore "yes")
Q*.2 (Gore "no")
Q1.* (Clinton "yes")
Q2.* (Clinton "no")
76%
Wang, Solloway, Shiffrin, & Busemeyer, 2014, PNAS
Order Effects in Opinion Polls
• A: Is Bill Clinton trustworthy?
1. Question difference: 22%
2. Question difference: 8%
• B: Is Al Gore trustworthy?
• Direction P: Clinton → Al Gore
P: A → B
P*.1 (Gore "yes")
P*.2 (Gore "no")
54%
P1.* (Clinton "yes")
P2.* (Clinton "no")
67%
• Direction Q: Al Gore → Clinton
Q: B → A
Q*.1 (Gore "yes")
Q*.2 (Gore "no")
59%
Q1.* (Clinton "yes")
Q2.* (Clinton "no")
76%
Wang, Solloway, Shiffrin, & Busemeyer, 2014, PNAS
Order Effects in Opinion Polls
• Direction P: Clinton → Al Gore
P: A → B
P*.1 (Gore "yes")
P*.2 (Gore "no")
P1.* (Clinton "yes")
49%
5%
P2.* (Clinton "no")
18%
28%
• Direction Q: Al Gore → Clinton
Q: B → A
Q*.1 (Gore "yes")
Q*.2 (Gore "no")
Q1.* (Clinton "yes")
56%
3%
Q2.* (Clinton "no")
20%
21%
• Difference:
P-Q
P*.1 - Q*.1
P*.2 - Q*.2
P1.* - Q1.*
-7%
2%
P2.* - Q2.*
-2%
7%
QQ-values:
-7 + 7 = 0
-2 + 2 = 0
Wang, Solloway, Shiffrin, & Busemeyer, 2014, PNAS
QQ (Quantum Question) Equality
• 72 data sets
• for most N > 1000
• QQ prediction of Quantum Model (parameter free):
• Probability of same response is invariant of question order.
• Probability of different response is invariant of question order.
Wang, Solloway, Shiffrin, & Busemeyer, 2014, PNAS
QQ-test model
Σ𝐺 2 = 75.9
Wang, Solloway, Shiffrin, & Busemeyer, 2014, PNAS
Classic Probability Account of QQ-Equality:
Repeat-Choice Model ℳ𝟎
• Si,j: Preference states, 3 parameters
• aO,i,j: Cond. probability that subsequent response is function of first, 8 p.
• rO,i,j: Cond. probability that second response is identical to first
(assimilation) or opposite (contrast), 8 p.
= 19 parameters
Hierarchy of Repeat-Choice Models
Classic Probability Account of QQ-Equality
Repeat-Choice Model ℳ𝟏 : Order Independent
=
=
=
=
• Si,j: Preference states, 3 parameters
• ai,j: Cond. probability that subsequent response is function of first, 4 p.
• ri,j: Cond. probability that second response is identical to first
(assimilation) or opposite (contrast), 4 p.
= 11 parameters
Classic Probability Account of QQ-Equality
Repeat-Choice Model ℳ𝟐 : Order Dependent
• Si,j: Preference states, 3 parameters
• ai,j: Cond. probability that subsequent response is function of first, 4 p.
• ri,j: Cond. probability that second response is identical to first
(assimilation) or opposite (contrast), 4 p.
= 11 parameters
Classic Probability Account of QQ-Equality
Comparing ℳ𝟏 and ℳ𝟐
Σ𝐺 2 = 75.9
Δ𝐺 2 = 3.1
Hierarchy of Repeat-Choice Models
Δ𝐺 2 = −75.9
Δ𝐺 2 = 0
Δ: 0
Δ: 3.1
Δ: 0.3
Δ: 9.8
Δ: 3.1
Δ: 9.8
Δ > 260
Δ: 57.8
Δ > 1360
Δ > 1102
Δ: 12.4
Mixtures: ℳ𝟏𝒂𝒓,𝑺 /ℳ𝟐𝒂𝒓,𝑺 (1000 simulations)
N per simulation: 400
𝜋: mixture weight
Fitting the generating
model to aggregated
data:
Summary
• QQ equality is intriguing (parameter free) prediction of Quantum Model
• QQ equality holds as parameter free prediction for simple repeat-choice
model if repeat-choice probabilitites are independent of question order.
• Repeat-choice model with more realistic assumption that repeat-choice
probabilities are determined by question order provides at least as good
account.
• Problems with aggregated data used here:
• Ergodicity: Conflating inter-individual variability with intra-individual
variability (Molenaar, 2004).
• Aggregation does not affect QQ equality, but parameters and fit of
simple repeat-choice model if data comes from mixture.
• In A-B-A question sequence, quantum models can only predict no
change to A when predicting no order effect (Khrennikov, Basieva,
Dzhafarov, & Busemeyer, 2014)
• Manuscript (not latest version): http://singmann.org/publications/