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The order of information in the presence of uncertainty plays a fundamental role in decision making. Yet, modelling such processes by classical Bayesian inference is difficult. Using judgement errors and optimal foraging as examples, this talk describes quantum probability theory to model decision problems. Subsequent observations change the decision maker's context, imposing a restricted space for decisions. If consecutive observations are incompatible -- they relate to different aspects of a system -- then the order of the observations will matter. Departing from Heisenberg's uncertainty principle, risk and ambiguity cannot be simultaneously minimised in this framework, hence putting a formal limit on rationality in sequential decision making. This pattern is universal and helps explaining similar phenomena in a wide range of decision problems, and it also aids our understanding why simultaneous decision making evolved.
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On the Relevance of Quantum Probability inDecision Theory: Sequential Decisions,
Contexts, and Uncertainty
Seminar at the Nanyang Technological University
Peter Wittek
University of Boras
November 8, 2013
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Why Quantum Probability Theory
It relaxes assumptions while also being conceptuallysimpler.Theory generator – not just a theory.Contextuality is key in many decision problems:
Pothos, E. M. & Busemeyer, J. R. Can quantum probabilityprovide a new direction for cognitive modeling? Behavioraland Brain Sciences, 2013, 36, pp. 255–274.Busemeyer, J. R.; Pothos, E. M.; Franco, R. & Trueblood, J.A quantum theoretical explanation for probability judgmenterrors. Psychological Review, 2011, 118, pp. 193–218.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Motivating Example
Conjunction fallacy:Linda is a bank teller.Linda is a bank teller and a feminist.
Prob(bank teller)<Prob(bank teller and feminist).Classical probability fails to account for the phenomenon.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Outline
Quantum (or contextual) probability theory:Mathematical background.Intuition from physics.
Decision theory: judgment errors.Optimal foraging theory and uncertainty.Open question: what’s next?
A theory is as good as its explanatory power.Simplification.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Commutative Algebras
5 + 4 = 4 + 5 – addition of numbers is commutative.2 ∗ 3 = 3 ∗ 2 – multiplication of numbers is commutative.Take a dice and a coin:
A: getting “3” on the dice; B: getting “heads” on the coin.Independent events:p(A ∩ B) = p(A)p(B) = 1
612 = 1
216 = p(B)p(A) = p(B ∩ A).
True for non-independent events as well:p(A ∩ B) = p(B ∩ A).Conjunction in classical probability theory is commutative.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Commutative Algebras and Geometry
Rotations:R−20: rotation by -20 degrees; R30: rotation by 30 degrees.
Original
Rotated R-20
X
Y
Original
Rotated R-20
X
Y
R30
Final
Original
Rotated
R-20
X
Y
R30Final
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Noncommutative Algebras
Not all operations commute.Add projectors:
PX : orthogonal projection to the X axis.
Original
Projected
X
Y
PX
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Noncommutative Algebras and Subspaces
The final result is different.
Original
Rotated R-20
X
Y
Final
PX
Original
Projected
X
Y
PX
R-20
Final
Projections to subspaces.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Probability Vectors and Ket Notation
Uniform distribution: for example,throwing a dice.
Classical notation: p(A = 1) = 16 ,
p(A = 2) = 16 , p(A = 3) = 1
6 ,p(A = 4) = 1
6 , p(A = 5) = 16 ,
p(A = 6) = 16 .
1 2 3 4 5 6
0.05
0.10
0.15
0.20
0.25
0.30
Quantum notation: ket
|ψ〉 =
1/√
61/√
61/√
61/√
61/√
61/√
6
= 1√
6
111111
.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
High-Dimensional Vectors
|ψ〉 = 1√6
111111
=
1√6
100000
+ 1√6
010000
+ 1√6
001000
+ 1√6
000100
+ 1√6
000010
+ 1√6
000001
.
Six-dimensional vector.Also called a state.Square sum of coefficients (the vector norm) adds up to 1.What happens after you throw the dice?
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Probabilities and Projections
Calculate the probability of throwing ‘3’.
The projector is P3 =
0 0 0 0 0 00 0 0 0 0 00 0 1 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0
.
Apply it to the state: P3|ψ〉 = 1√6
001000
.
Take the norm of this vector to get the probability:||P3|ψ〉|| = 1/6.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Superposition
Forget Schrodinger’s cat|ψ〉 =
∑i αi |xi〉.
The |xi〉 components arephysical possibilities.Energy levels, for instance.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Alternative View on the Same State
What are we measuring?Change to measure the momentum:
|ψ〉 =∑
i βi |pi〉.Incompatible measurement – cannot measuresimultaneously both.Reference frame.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Heisenberg’s Uncertainty Principle
An absolute limit of how precise a measurement can get.σxσp ≥ ~
2 .
In the strictest physical sense, it holds to classical systemsas well.It is also a mathematical result:
It is a consequence of noncommuting probabilities.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
The Problem
Primacy effect and recency effect.Disproportionate importance of initial and most recentobservations.
Clinical data and diagnostic task1. Urinary tract infection:History and physical examination first, then laboratory data(H&P-first).The other way around (H&P-last).
Mean probability estimates from diagnostic taskH&P-first H&P-last
Initial P(UTI) = 0.674 P(UTI) = 0.678Second P(UTI|H&P) = 0.778 P(UTI|Lab) = 0.440Final P(UTI|H&P,Lab) = 0.509 P(UTI|Lab,H&P) = 0.591
1Trueblood, J. & Busemeyer, J. A quantum probability account of ordereffects in inference. Cognitive Science, 2011, 35, pp. 1518–1552.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Bayes’ Rule
Commuting algebra.
The rule: p(H|A ∩ B) = p(H|A)p(B|H∩A)p(B|A) .
What does this mean?Why is it a useful definition?
The problem:
p(H|A ∩ B) = p(H|A)p(B|H∩A)p(B|A) = p(H|B)p(A|H∩B)
p(A|B) = p(H|B ∩ A).
p(H|A ∩ B) = p(H|B ∩ A) – Bayesian inference isinsensitive to the order of evidence.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Luders’ Rule
Noncommuting algebra.Projection to a subspace:|ψA〉 = 1
||PA=1|ψ〉|PA=1||ψ〉.
A context is implied – a subspace is acontext.Subsequent measurement:
PB=1|ψA=1〉 = 1||PA=1|ψ〉||PB=1PA=1|ψ〉.
In general, PB=1PA=1 6= PA=1PB=1.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
What Is Foraging?
Optimal Foraging Theory:A successful approachin understanding animaldecision making.Assumption: organismsaim to maximise theirnet energy intake perunit time.Food sources areavailable in patches,which vary in quality.Switching betweenpatches comes with acost.
A bumblebee worker finds a rewarding “flower.”Photograph by Jay Biernaskie. From Biernaskie, J.;
Walker, S. & Gegear, R. Bumblebees learn to forage likeBayesians. The American Naturalist, 2009, 174,
pp. 413–423.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Types of Uncertainty
Ideas come from economics.Uncertainty: in decisions about staying at a patch ormoving on to the next one.Two fundamental types of uncertainty: ambiguity and risk.
Ambiguity: the estimation of the quality of a patch.Risk: the potential quality of other patches.
Decisions are quintessentially sequential.
Forager
Patch of resource
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Contextuality
“[C]ontext-dependent utility results from the fact thatperceived utility depends on background opportunities.”The sequence of optimal decisions depends on theattributes of the present opportunity and its backgroundoptions.Examples: Honey bees, rufous humming birds, gray jays,European starlings, etc.Humans alternate between sequential and simultaneousdecision making.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Probability Space
Two hypotheses describe the decision space of a forager:h1: Stay at the current patch.h2: Leave the patch.
Consider the following events:A: Current patch quality with two possible outcomes: a1 –the patch quality is good; a2 – the patch quality is bad.B: Quality of other patches. A collective observation acrossall other patches with two possible outcomes.
A corresponds to ambiguity.B corresponds to risk.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Belief State in Superposition
A and B are incompatible observations on a system.Forager’s state of belief is described by a state vector.Under observation A, this superposition is written as
|ψ〉 =∑i,j
αij |Aij〉 (1)
The square norm of the corresponding projected vector willbe the quantum probability of h1 ∧ a1: ||P11|ψ〉|| = |α11|2.Observation B: the state of belief is a superposition of fourdifferent basis vectors: |ψ〉 =
∑i,j βij |Bij〉.
Under observation A, the forager bases its decision onlocal information.Under B, it looks at a global perspective.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Simulation Results
●
●
●
●
●
●
●
●
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
risk aversion
net f
ood
inta
ke
horizon = 1
(a) Horizon=1
●
0.2
0.4
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
risk aversion
net f
ood
inta
ke
horizon = 3
(b) Horizon=3
●
●
●
●
●
0.2
0.4
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
risk aversion
net f
ood
inta
ke
horizon = 5
(c) Horizon=5
●●
●
●
●
●
●
●
●
●●
●
0.2
0.4
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
risk aversion
net f
ood
inta
ke
horizon = 7
(d) Horizon=7
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Uncertainty in Sequential Decisions
A state cannot be a simultaneous eigenvector of the twoobservables in general.The forager needs to leave the current patch to assess thequality of other patches:
Inherent uncertainty in the decision irrespective of thequantity of information gained about either A or B.
With regard to risk and ambiguity, the uncertainty principleholds:
σAσB ≥ c, (2)
Where c > 0 is a constant.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Neural Mechanisms in Humans
Humans alternate between two models ofchoice:
Comparative decision making.Foraging-type decisions.
Different neural mechanisms support the twomodels
Kolling, N.; Behrens, T.; Mars, R. &Rushworth, M. Neural Mechanisms ofForaging. Science, 2012, 336, pp. 95–98.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Foraging Is Universal
Short-term exploitative competition of stock traders.Social foraging.Consumer decisions.Searching in semantic memory.
d= –1
d= 1
exploitation index
stock
08.00 10.00 12.00
time of the day
14.00 16.00
AAPL GOOG YHOO AAPL
buy
sell
trad
es
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Simultaneous Decisions
Foraging theory is extremely successful in describinganimal behaviour.Yet, impact on understanding human behaviour is far morelimited.When can we put up with uncertainty?
Bounds to rationality: with foraging-type decisions,uncertainty can never be eliminated.
Is there a higher cognitive cost of making comparativedecisions?Evolutionary reasons to comparative decisions.
Peter Wittek Quantum Probability and Decision Theory
Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions
Summary
If a decision making scenario has a sequential component,quantum probability is relevant.Order effects are easy to model.Risk and ambiguity are incompatible concepts, leading toan uncertainty principle.Comparative decisions do not have such constraints.Wide range of applications – a theory generator.
Peter Wittek Quantum Probability and Decision Theory