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