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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 ...

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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- 1. On the Relevance of Quantum Probability in Decision Theory: Sequential Decisions, Contexts, and Uncertainty Seminar at the Nanyang Technological University Peter Wittek ˚ University of Boras November 8, 2013
- 2. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions Why Quantum Probability Theory It relaxes assumptions while also being conceptually simpler. Theory generator – not just a theory. Contextuality is key in many decision problems: Pothos, E. M. & Busemeyer, J. R. Can quantum probability provide a new direction for cognitive modeling? Behavioral and Brain Sciences, 2013, 36, pp. 255–274. Busemeyer, J. R.; Pothos, E. M.; Franco, R. & Trueblood, J. A quantum theoretical explanation for probability judgment errors. Psychological Review, 2011, 118, pp. 193–218. Peter Wittek Quantum Probability and Decision Theory
- 3. 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
- 4. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook 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 Conclusions
- 5. 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: 1 1 p(A ∩ B) = p(A)p(B) = 1 2 = 1 6 = p(B)p(A) = p(B ∩ A). 6 2 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
- 6. 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. Y Y R-20 Rotated Y R-20 Rotated R30 Original Original Original Final Final R30 R-20 Rotated X X Peter Wittek Quantum Probability and Decision Theory X
- 7. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Noncommutative Algebras Not all operations commute. Add projectors: PX : orthogonal projection to the X axis. Y Original PX Projected X Peter Wittek Quantum Probability and Decision Theory Conclusions
- 8. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Noncommutative Algebras and Subspaces The ﬁnal result is different. Y Y R-20 Rotated Original Original Final PX PX R-20 Projected Final X X Projections to subspaces. Peter Wittek Quantum Probability and Decision Theory Conclusions
- 9. 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) = 1 1 p(A = 2) = 6 , p(A = 3) = 6 , 1 1 p(A = 4) = 6 , p(A = 5) = 6 , p(A = 6) = 1 . 6 1 6, 0.30 0.25 0.20 0.15 0.10 0.05 1 2 3 Quantum notation: ket √ 1/ 6 1/√6 √ 1/ 6 |ψ = √ = 1/√6 1/ 6 √ 1/ 6 1 1 1 1 √ . 6 1 1 1 Peter Wittek Quantum Probability and Decision Theory 4 5 6
- 10. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions High-Dimensional Vectors 1 1 1 1 |ψ = √6 = 1 1 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 1 1 1 1 1 √ + √ + √ + √ + √ + √ . 6 0 6 0 6 0 6 1 6 0 6 0 0 0 0 0 1 0 0 0 0 0 0 1 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
- 11. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Probabilities and Projections Calculate the probability of throwing ‘3’. 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 The projector is P3 = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 Apply it to the state: P3 |ψ = √6 . 0 0 0 0 0 . 0 0 0 0 Take the norm of this vector to get the probability: ||P3 |ψ || = 1/6. Peter Wittek Quantum Probability and Decision Theory Conclusions
- 12. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Superposition ¨ Forget Schrodinger’s cat |ψ = i αi |xi . The |xi components are physical possibilities. Energy levels, for instance. Peter Wittek Quantum Probability and Decision Theory Conclusions
- 13. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Alternative View on the Same State What are we measuring? Change to measure the momentum: |ψ = i βi |pi . Incompatible measurement – cannot measure simultaneously both. Reference frame. Peter Wittek Quantum Probability and Decision Theory Conclusions
- 14. 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. σx σp ≥ 2 . In the strictest physical sense, it holds to classical systems as well. It is also a mathematical result: It is a consequence of noncommuting probabilities. Peter Wittek Quantum Probability and Decision Theory
- 15. 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 recent observations. 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 task Initial Second Final H&P-ﬁrst P(UTI) = 0.674 P(UTI|H&P) = 0.778 P(UTI|H&P, Lab) = 0.509 H&P-last P(UTI) = 0.678 P(UTI|Lab) = 0.440 P(UTI|Lab, H&P) = 0.591 1 Trueblood, J. & Busemeyer, J. A quantum probability account of order effects in inference. Cognitive Science, 2011, 35, pp. 1518–1552. Peter Wittek Quantum Probability and Decision Theory
- 16. 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(H|B) p(A|H∩B) = p(H|B ∩ A). p(B|A) p(A|B) p(H|A ∩ B) = p(H|B ∩ A) – Bayesian inference is insensitive to the order of evidence. Peter Wittek Quantum Probability and Decision Theory
- 17. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Luders’ Rule ¨ Noncommuting algebra. Projection to a subspace: |ψA = ||P 1 |ψ | PA=1 ||ψ . A=1 A context is implied – a subspace is a context. Subsequent measurement: 1 PB=1 |ψA=1 = ||PA=1 |ψ || PB=1 PA=1 |ψ . In general, PB=1 PA=1 = PA=1 PB=1 . Peter Wittek Quantum Probability and Decision Theory Conclusions
- 18. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions What Is Foraging? Optimal Foraging Theory: A successful approach in understanding animal decision making. Assumption: organisms aim to maximise their net energy intake per unit time. Food sources are available in patches, which vary in quality. Switching between patches comes with a cost. Peter Wittek A bumblebee worker ﬁnds a rewarding “ﬂower.” Photograph by Jay Biernaskie. From Biernaskie, J.; Walker, S. & Gegear, R. Bumblebees learn to forage like Bayesians. The American Naturalist, 2009, 174, pp. 413–423. Quantum Probability and Decision Theory
- 19. 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 or moving 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
- 20. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Contextuality “[C]ontext-dependent utility results from the fact that perceived utility depends on background opportunities.” The sequence of optimal decisions depends on the attributes of the present opportunity and its background options. Examples: Honey bees, rufous humming birds, gray jays, European starlings, etc. Humans alternate between sequential and simultaneous decision making. Peter Wittek Quantum Probability and Decision Theory Conclusions
- 21. 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 across all other patches with two possible outcomes. A corresponds to ambiguity. B corresponds to risk. Peter Wittek Quantum Probability and Decision Theory
- 22. 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 = αij |Aij (1) i,j The square norm of the corresponding projected vector will be the quantum probability of h1 ∧ a1 : ||P11 |ψ || = |α11 |2 . Observation B: the state of belief is a superposition of four different basis vectors: |ψ = i,j βij |Bij . Under observation A, the forager bases its decision on local information. Under B, it looks at a global perspective. Peter Wittek Quantum Probability and Decision Theory
- 23. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Simulation Results horizon = 1 horizon = 3 q 0.4 0.6 net food intake net food intake q q q q 0.3 q 0.4 q q q 0.2 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 risk aversion 0.4 0.5 0.6 0.7 0.8 0.9 1 risk aversion (a) Horizon=1 (b) Horizon=3 horizon = 5 horizon = 7 q q q q q q q 0.6 0.6 q net food intake net food intake q q q q 0.4 0.4 0.2 0.2 q q q q 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 risk aversion (c) Horizon=5 Peter Wittek 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 risk aversion (d) Horizon=7 Quantum Probability and Decision Theory Conclusions
- 24. 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 two observables in general. The forager needs to leave the current patch to assess the quality of other patches: Inherent uncertainty in the decision irrespective of the quantity of information gained about either A or B. With regard to risk and ambiguity, the uncertainty principle holds: σA σB ≥ c, (2) Where c > 0 is a constant. Peter Wittek Quantum Probability and Decision Theory
- 25. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Neural Mechanisms in Humans Humans alternate between two models of choice: Comparative decision making. Foraging-type decisions. Different neural mechanisms support the two models Kolling, N.; Behrens, T.; Mars, R. & Rushworth, M. Neural Mechanisms of Foraging. Science, 2012, 336, pp. 95–98. Peter Wittek Quantum Probability and Decision Theory Conclusions
- 26. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Foraging Is Universal Short-term exploitative competition of stock traders. Social foraging. Consumer decisions. Searching in semantic memory. exploitation index buy trades d= –1 d= 1 sell stock 08.00 AAPL 10.00 GOOG YHOO 12.00 14.00 AAPL 16.00 time of the day Peter Wittek Quantum Probability and Decision Theory Conclusions
- 27. Introduction Mathematical Background Intuition from Physics Decision Theory Foraging Theory Outlook Conclusions Simultaneous Decisions Foraging theory is extremely successful in describing animal behaviour. Yet, impact on understanding human behaviour is far more limited. 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 comparative decisions? Evolutionary reasons to comparative decisions. Peter Wittek Quantum Probability and Decision Theory
- 28. 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 to an uncertainty principle. Comparative decisions do not have such constraints. Wide range of applications – a theory generator. Peter Wittek Quantum Probability and Decision Theory

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