quantum probabilistic graphical models for cognition and...

Quantum Probabilistic Graphical Models

for Cognition and Decision

Catarina Alexandra Pinto Moreira

Supervisor: Doctor Andreas Miroslaus Wichert

Thesis specifically prepared to obtain the PhD Degree in

Information Systems and Computer Engineering

Draft

August, 2017

Dedicated to all who contributed for my education

’If you can dream - and not make dreams your master;

If you can think - and not make thoughts your aim;

If you can meet with Triumph and Disaster,

And treat those two impostors just the same;

(...)

If you can force your heart and nerve and sinew

To serve your turn long after they are gone,

And so hold on when there is nothing in you

Except the Will which says to them: ”Hold on!”

(...)

Then, yours is the Earth and everything that’s in it,

And - which is more - you’ll be a Man, my son!’

’ If - ’ by Rudyard Kipling

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Title: Quantum-Like Probabilistic Graphical Models for Cognition and Decision

Name Catarina Alexandra Pinto Moreira

PhD in Information Systems and Computer Engineering

Supervisor Doctor Andreas Miroslaus Wichert

Abstract

Cognitive scientists are mainly focused in developing models and cognitive structures that are able to

represent processes of the human mind. One of these processes is concerned with human decision

making. In the last decades, literature has been reporting several situations of human decisions that

could not be easily modelled by classical models, because humans constantly violate the laws of prob-

ability theory in situations with high levels of uncertainty. In this sense, quantum-like models started to

emerge as an alternative framework, which is based on the mathematical principles of quantum mechan-

ics, in order to model and explain paradoxical findings that cognitive scientists were unable to explain

using the laws of classical probability theory.

Although quantum-like models succeeded to explain many paradoxical decision making scenarios,

they still suffer from three main problems. First, they cannot scale to more complex decision scenarios,

because the number of quantum parameters grows exponentially large. Second, they cannot be consid-

ered predictive, since they require that we know a priori the outcome of a decision problem in order to

manually set quantum parameters. And third, the way one can set these quantum parameters is still an

unexplored field and still an open research question in the Quantum Cognition literature.

This work focuses on quantum-like probabilistic graphical models by surveying the most important

aspects of classical probability theory, quantum-like models applied to human decision making and

probabilistic graphical models. We also propose a Quantum-Like Bayesian Network that can easily

scale up to more complex decision making scenarios due to its network structure. In order to address

the problem of exponential quantum parameters, we also propose heuristic functions that can set an

exponential number of quantum parameters without a priori knowledge of experimental outcomes. This

makes the proposed model general and predictive in contrast with the current state of the art models,

which cannot be generalised for more complex decision making scenarios and that can only provide an

explanatory nature for the observed paradoxes.

Keywords: Quantum Cognition, Quantum-Like Bayesian Networks, Quantum Probability, Quan-tum Interference Effects, Quantum-Like Models

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Tı́tulo Modelos Gráficos Probabilı́sticos Quânticos para Cognição e Decisão

Nome Catarina Alexandra Pinto Moreira

Doutoramento em Engenharia Informática e de Computadores

Orientador Doutor Andreas Miroslaus Wichert

Resumo

Os cientistas cognitivos concentram-se principalmente no desenvolvimento de modelos e estruturas

cognitivas capazes de representar processos da mente humana. Um desses processos está rela-

cionado com o facto de como os humanos tomam decisões. Nas últimas décadas, a literatura tem

relatado várias situações de decisões humanas que não podem ser facilmente modeladas por mode-

los clássicos, porque os humanos violam constantemente as leis da teoria da probabilidade clássica

em situações com altos nı́veis de incerteza. Nesse sentido, os modelos quânticos começaram a surgir

como uma abordagem alternativa que se baseia nos princı́pios matemáticos da mecânica quântica para

modelar e explicar situações paradoxais que os cientistas cognitivos não conseguem explicar usando

as leis da teoria da probabilidade clássica.

Embora os modelos quânticos tenham conseguido explicar muitos cenários paradoxais de decisão

humana, eles ainda sofrem de três problemas principais. Primeiro, eles não podem escalar para

cenários de decisão mais complexos, porque o número de parâmetros quânticos cresce de uma forma

exponencial relativamente à complexidade do problema de decisão. Em segundo lugar, eles não podem

ser considerados preditivos, uma vez que exigem que conheçamos a priori o resultado de um problema

de decisão para definir manualmente os parâmetros quânticos que servem para explicar os resultados

paradoxais. E em terceiro lugar, a forma como se pode definir esses parâmetros quânticos é um campo

inexplorado e ainda é uma questão de investigação aberta na literatura modelos cognitivos quânticos.

Este trabalho centra-se em modelos probabilı́sticos gráficos quânticos, consistindo num levanta-

mento dos aspectos mais importantes da teoria da probabilidade clássica, modelos quânticos aplica-

dos à tomada de decisão humana e em modelos probabilı́sticos gráficos clássicos. Também propomos

uma rede Bayesiana quântica que pode escalar facilmente para cenários de decisão mais complexos

devido à sua estrutura de rede. De forma a abordar o problema de atribuição de valores a um número

exponencial de parâmetros quânticos, também propomos funções heurı́sticas que podem definir um

conjunto exponencial de parâmetros quânticos sem conhecimento a priori de resultados experimentais.

Isso torna o modelo proposto geral e preditivo em contraste com os modelos actuais do estado da

arte, que não podem ser generalizados para cenários de tomada de decisão mais complexos e que só

podem fornecer uma natureza explicativa para os paradoxos observados.

Palavras-chave: Cognição Quântica, Redes Bayesianas Quânticas, Probabilidade Quântica,Efeitos de Interferência Quântica, Modelos Quânticos

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Tı́tulo Modelos Gráficos Probabilı́sticos Quânticos para Cognição e Decisão

Nome Catarina Alexandra Pinto Moreira

Doutoramento em Engenharia Informática e de Computadores

Orientador Doutor Andreas Miroslaus Wichert

Resumo Extendido

A cognição quântica é uma área de investigação que visa usar os princı́pios matemáticos da mecânica

quântica para modelar sistemas cognitivos para a tomada de decisões humanas. Dado que a teoria da

probabilidade clássica é muito rı́gida no sentido de que ela apresenta muitas restrições e pressupostos

(princı́pio da trajetória única, obedece a teoria dos conjuntos, etc.), torna-se muito limitado (ou mesmo

impossı́vel) desenvolver modelos simples que possam capturar julgamentos humanos e decisões, uma

vez que as pessoas podem violar as leis da lógica e da teoria da probabilidade [33, 37, 6].

A teoria da probabilidade quântica beneficia de muitas vantagens relativamente à teoria clássica.

Pode representar eventos em espaços vectoriais. Consequentemente, pode levar em consideração o

problema da ordem dos efeitos [202, 188] e representar as amplitudes dos resultados experimentais ao

mesmo tempo através de numa superposição. Psicologicamente, o efeito de superposição pode estar

relacionado ao sentimento de confusão, incerteza ou ambiguidade. Ou seja, pode representar a noção

de crença como um estado indefinido [34]. Além disso, esta representação do espaço vectorial não

obedece ao axioma distributivo da lógica booleana e nem à lei da probabilidade total. Isso permite a

construção de modelos mais gerais que podem explicar matematicamente fenómenos cognitivos, como

erros de conjunção/disjunção [40, 73] ou violações do Princı́pio da Certeza [164, 110], que é o foco

principal deste trabalho.

Um problema dos actuais sistemas probabilı́sticos é o facto de não podem fazer previsões pre-

cisas em situações em que as leis da probabilidade clássica são violadas. Estas situações ocorrem

frequentemente em sistemas que tentam modelar decisões humanas em cenários onde o princı́pio da

Certeza [170] é violado. Este princı́pio é fundamental na teoria da probabilidade clássica e afirma que

se alguém preferir a acção A relativamente à acção B no estado do mundo X, e se alguém também

preferir a acção A relativamente a B sob o estado complementar do Mundo ¬X, então subentende-

se que se deve preferir sempre a acção A relativamente a B mesmo quando o estado do mundo

não é conhecido. Violações ao Princı́pio da Certeza implicam violações à lei da probabilidade total

clássica [193, 196, 198, 9, 26].

Desta forma, neste trabalho, é proposta uma Rede Bayesiana quântica, inspirada nos formalismos

de Integrais de caminho de Feynman [72].

Uma rede Bayesiana pode ser entendida como um gráfico acı́clico direcionado, no qual cada nó

representa uma variável aleatória e cada uma das arestas representa uma influência direta do nó de

origem para o nó alvo (dependência condicional). Por sua vez, os integrais do caminho de Feynman

representam todos os caminhos possı́veis que uma partı́cula pode percorrer para alcançar um ponto de

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destino, levando em consideração que todos esses caminhos podem produzir efeitos de interferência

quântica entre eles.

A criação deste tipo de redes Bayesianas quânticas, juntamente com a aplicação dos integrais

de caminho de Feynman, geram algumas dificuldades, nomeadamente a quantidade exponencial de

parâmetros livres que resultam dos efeitos de interferência quântica. A estes parâmetros é preciso

atribuir valores que permitam acomodar os cenários de decisão onde o princı́pio da certeza é violado.

Para colmatar este problema, propomos também um conjunto de heurı́sticas de similaridade para

calcular esse número exponencial de parâmetros de interferência quânticos. Note-se que uma heurı́stica

é simplesmente um atalho que geralmente fornece bons resultados em muitas situações, mas com o

custo de ocasionalmente não nos dar resultados muito precisos [173].

Note-se que os modelos atuais da literatura exigem uma busca manual de parâmetros que podem

levar aos resultados desejados. Ou seja, é necessário sabermos o resultado do cenário de decisão à

priori para manualmente se atribuı́rem valores a esses parâmetros [93, 96, 101, 164, 44, 41]. Com a

rede proposta, pretende-se um modelo escalável e preditivo ao contraste dos modelos actuais que têm

uma natureza explicativa.

As heurı́sticas que propomos neste trabalho são de três tipos: (1) baseadas na distribuição prob-

abiı́sticas dos dados, (2) baseadas nos conteúdos dos dados e (3) baseadas em relações semânticas.

Heurı́stica de Similaridade Baseada em Distribuições Probabilı́sticas

O objetivo da heurı́stica de similaridade é determinar um ângulo entre os vectores probabilı́sticos as-

sociados à marginalização das atribuições positivas e negativas da variável de consulta. Em outras

palavras, ao realizar uma inferência probabilı́sticas a partir de uma tabela de distribuição de proba-

bilidade conjunta, selecionamos nesta tabela todas as probabilidades que combinam as atribuições

da variável de consulta e, se for dado, as variáveis observadas. Se somarmos essas probabilidades,

acabamos com uma inferência de probabilidade clássica final. Se acrescentarmos um termo de in-

terferência a essa inferência clássica, acabaremos com uma inferência probabilı́stica quântica. Neste

caso, podemos usar esses vetores de probabilidade para obter informações adicionais para calcular os

parâmetros de interferência quântica. A ideia geral da heurı́stica de similaridade é usar as distribuições

de probabilidade marginal como vetores de probabilidade e medir sua similaridade através da lei dos

Cossenos, que é uma medida de similaridade bem conhecida no domı́nio da Ciência da Computação e

é amplamente utilizada na Recuperação de Informação [23]. De acordo com esse grau de similaridade,

aplicaremos uma função de mapeamento com uma natureza heurı́stica, que produzirá o valor para o

parâmetro de interferência quântico, tendo em conta um estudo prévio relativamente à distribuição prob-

abilı́stica dos dados de várias experiências relatadas por toda a literatura. Os resultados mostraram um

erro médio entre 6.4% a 7.9% na previsão das decisões humanas em várias experiências da literatura

onde foram reportadas violações ao princı́pio da certeza.

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Heurı́stica de Similaridade Baseada em Distribuições Probabilı́sticas

Esta heurı́stica representa objectos (ou eventos num espaço vetorial N -dimensional. Isto permite a

sua comparação através de funções de similaridade. O valor da similaridade é usado para calcular

os parâmetros de interferência quântica. Tal como no trabalho de Pothos et al. [166], não estamos a

restringir o nosso modelo a um vector em num espaço psicológico multidimensional, mas a um espaço

multidimensional arbitrário.

As similaridades calculadas entre dois vectores que representam conteúdos de eventos (neste caso,

os eventos são imagens e os seus conteúdos são os pixéis que as compões) podem ser usadas para

definir parâmetros de interferência quântica, uma vez que ambos são compostos pelo cálculo do produto

interno entre duas variáveis aleatórias. Isto sugere uma equivalência matemática entre os parâmetros

θ calculados a partir da similaridade do Cosseno e os parâmetros quantitativos θ correspondentes aos

efeitos de interferência quântica. Essa suposição é baseada no livro de Busemeyer & Bruza [34], onde

se afirma que o parâmetro θ que surge em efeitos de interferência quântica corresponde à fase do

ângulo do produto interno entre os projetores de duas variáveis aleatórias. Os autores também afir-

mam que o produto interno fornece uma medida de similaridade entre dois vectores (onde cada vector

corresponde a uma superposição de eventos). Se os vectores tiverem o comprimento unitário, então, a

semelhança do Coseno colapsa para o produto interno. Dadas todas essas relações, podemos assumir

que as semelhanças computadas entre dois vetores que representam imagens (usadas na experiência

de Busemeyer et al. [41]) podem ser usadas para definir parâmetros de interferência quântica.

Os resultados das simulações aplicados ao trabalho de Busemeyer et al. [41] demonstraram que a

heurı́stica proposta foi capaz de reproduzir as observações experimentais das violações do princı́pio da

certeza com uma pequena percentagem de erro (entre 4% e 5%).

Heurı́stica de Similaridade Semântica

Esta heurı́stica procura determinar o impacto de relações de dependência semântica entre eventos.

Estas semelhanças semânticas adicionam novas dependências entre os nós das redes Bayesianas

que não incluem necessariamente relações causais directas. Usaremos essas informações semânticas

adicionais para calcular os efeitos de interferência quântica, a fim de acomodar as violações ao princı́pio

da certeza.

Sob o princı́pio da causalidade, dois eventos que não estão causalmente conectados não devem

produzir nenhum efeito. Quando alguns eventos acausais ocorrem produzindo um efeito, é chamado

de coincidência. Carl Jung, acreditava que todos os eventos tinham que estar conectados uns aos

outros, não num cenário causal, mas sim através do seu significado, sugerindo algum tipo de relação

semântica entre eventos. Esta noção é conhecida como o princı́pio da sincronicidade [87].

Definimos a heurı́stica de similaridade semântica de forma semelhante ao princı́pio da sincronici-

dade: duas variáveis são ditas sincronizadas, se compartilhem uma conexão semântica entre eles.

Essa conexão pode ser obtida através da representação de uma rede semântica das variáveis em

xi

questão. Isso permitirá o surgimento de novas dependências significativas que seriam inexistentes

ao considerar apenas relações causa/efeito. Os parâmetros quânticos são então atribuı́dos usando

esta informação adicional de forma a que o ângulo formado por essas duas variáveis, num espaço de

Hilbert, seja o menor possı́vel (alta similaridade), dessa forma forçando os eventos acausais a serem

correlacionados.

Os resultados das simulações aplicadas ao trabalho de Busemeyer et al. [41] demonstraram que a

heurı́stica proposta foi capaz de reproduzir as observações experimentais das violações do princı́pio da

certeza com uma pequena percentagem de erro (entre 3% e 6%).

Palavras-chave: Cognição Quântica, Redes Bayesianas Quânticas, Probabilidade Quântica,Efeitos de Interferência Quântica, Modelos Quânticos

xii

Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

1 Introduction 1

1.1 Violations to Normative Theories of Rational Choice . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Emergence of Quantum Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Motivation: Violations to the Sure Thing Principle . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Why Quantum Cognition? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Challenges of Current Quantum-Like Models . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Thesis Proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6.1 Why Bayesian Networks? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6.2 Quantum-Like Bayesian Networks for Disjunction Errors . . . . . . . . . . . . . . . 8

1.6.3 Comparison with Existing Quantum-Like Models . . . . . . . . . . . . . . . . . . . 9

1.7 Advantages of Quantum-Like Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.7.1 Research Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.8 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.8.1 Conference Papers, Extended Abstracts and Posters . . . . . . . . . . . . . . . . . 11

1.9 Organisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Quantum Cognition Fundamentals 15

2.1 Introduction to Quantum Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 Representation of Quantum States . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.2 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.1.3 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.1.4 System State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.5 State Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.6 Compatibility and Incompatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Interference Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

xiii

2.2.1 The Double Slit Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2.2 Derivation of Interference Effects from Complex Numbers . . . . . . . . . . . . . . 24

2.3 Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Path Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4.1 Single Path Trajectory Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.2 Multiple Indistinguishable Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.4.3 Multiple Distinguishable Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5 Born’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.6 Why Complex Numbers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.7 Summary and Final Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Fundamentals of Bayesian Networks 35

3.1 The Naı̈ve Bayes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2.1 Example of Inferences in Bayesian Networks . . . . . . . . . . . . . . . . . . . . . 38

3.3 Reasoning Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 Causal Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.2 Evidential Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.3 Intercausal Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Flow of Probabilistic Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


4 Paradoxes and Fallacies for Cognition and Decision-Making 45

4.1 Utility Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1.1 Expected Utility Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1.2 Subjective Expected Utility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Paradoxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2.1 Ellsberg Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2 Allais Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.3 Three Color Ellsberg Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3 Conjunction and Disjunction Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 The Linda Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.4 Disjunction Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4.1 The Two Stage Gambling Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4.2 The Prisoner’s Dilemma Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.5 Order of Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56


xiv

5 Related Work 61

5.1 Disjunction Fallacy: The Prisoner’s Dilemma Game . . . . . . . . . . . . . . . . . . . . . . 62

5.2 A Classical Markov Model of the Prisoner’s Dilemma Game . . . . . . . . . . . . . . . . . 62

5.3 The Quantum-Like Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3.1 Contextual Probabilities: The Växjö Model . . . . . . . . . . . . . . . . . . . . . . . 64

5.3.2 The Hyperbolic Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3.3 Quantum-Like Probabilities as an Extension of the Växjö Model . . . . . . . . . . 67

5.3.4 Modelling the Prisoner’s Dilemma using the Quantum-Like Approach . . . . . . . . 68

5.4 The Quantum Dynamical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.5 The Quantum Prospect Decision Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.5.1 Choosing the Uncertainty Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5.2 The Quantum Prospect Decision Theory Applied to the Prisoner’s Dilemma Game 74

5.6 Probabilistic Graphical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.6.1 Classical Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.6.2 Classical Bayesian Networks for the Prisoner’s Dilemma Game . . . . . . . . . . . 75

5.6.3 Quantum-Like Bayesian Networks in the Literature . . . . . . . . . . . . . . . . . . 77

5.7 Discussion of the Presented Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.7.1 Discussion in Terms of Interference, Parameter Tuning and Scalability . . . . . . . 78

5.7.2 Discussion in Terms of Parameter Growth . . . . . . . . . . . . . . . . . . . . . . . 81

5.8 The Quantum-Like Approach Over the Literature . . . . . . . . . . . . . . . . . . . . . . . 82

5.9 The Quantum Dynamical Model Over the Literature . . . . . . . . . . . . . . . . . . . . . . 83

5.10 A Model of Neural Oscillators for Quantum Cognition and Negative Probabilities . . . . . 84

5.11 A Quantum-Like Agent-Based Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85


6 Quantum-Like Bayesian Networks for Cognition and Decision 89

6.1 Classical Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1.1 Classical Conditional Independece . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.1.2 Classical Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.1.3 Example of Application in the Two-Stage Gambling Game . . . . . . . . . . . . . . 90

6.1.4 Classical Full Joint Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.1.5 Classical Marginalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2 Quantum-Like Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2.1 Quantum Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2.2 Quantum State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

6.2.3 Quantum-Like Full Joint Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.2.4 Quantum-Like Marginalisation: Exact Inference . . . . . . . . . . . . . . . . . . . . 96

6.3 The Impact of the Phase θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.4 A Cognitive Interpretation of Quantum-Like Bayesian Networks . . . . . . . . . . . . . . . 100

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6.5 Summary of the Quantum-Like Bayesian Network Model . . . . . . . . . . . . . . . . . . . 100

6.6 Inference in More Complex Networks: The Burglar/Alarm Network . . . . . . . . . . . . . 103

6.7 Discussion of Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105


7 Heuristical Approaches Based on Data Distribution 109

7.1 The Vector Similarity Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.1.1 Acquisition of Additional Information . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.1.2 Definition of the Heuristical Function . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.1.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

7.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.2 Example of Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

7.3 Similarity Heuristic Applied to the Prisoner’s Dilemma Game . . . . . . . . . . . . . . . . 117

7.3.1 The Special Case of Crosson’s (2009) Experiments . . . . . . . . . . . . . . . . . 119

7.3.2 Analysing Li’s et al. (2002) Experiments . . . . . . . . . . . . . . . . . . . . . . . . 120

7.4 Similarity Heuristic Applied to the Two Stage Gambling Game . . . . . . . . . . . . . . . . 122

7.5 Comparing the Similarity Heuristic with other Works of the Literature . . . . . . . . . . . . 123


8 Heuristical Approaches Based on Contents of the Data 127

8.1 A Vector Similarity Model to Extract Quantum Parameters . . . . . . . . . . . . . . . . . . 128

8.1.1 Using Cosine Similarity to Determine Quantum Parameters . . . . . . . . . . . . . 129

8.2 Application to the categorisation-Decision Experiment . . . . . . . . . . . . . . . . . . . . 130

8.2.1 Categorisation - Decision Making Experiment . . . . . . . . . . . . . . . . . . . . . 130

8.2.2 Modelling the Problem using Quantum-Like Bayesian Networks . . . . . . . . . . . 132

8.2.3 Computation of the Probability of Narrow Faces . . . . . . . . . . . . . . . . . . . . 132

8.2.4 Computing Quantum Interference Terms . . . . . . . . . . . . . . . . . . . . . . . . 133

8.2.5 The Impact of the Conversion Threshold . . . . . . . . . . . . . . . . . . . . . . . . 134

8.2.6 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

8.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138


9 Heuristical Approaches Based on Semantic Similarities 141

9.1 Synchronicity: an Acausal Connectionist Principle . . . . . . . . . . . . . . . . . . . . . . 142

9.2 Combining Causal and Acausal Principles for Quantum Cognition . . . . . . . . . . . . . 142

9.2.1 Semantic Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.2.2 The Semantic Similarity Heuristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

9.3 The Semantic Similarity Heuristic in the Categorisation/Decision Experiment . . . . . . . 144

9.3.1 Application of the Synchronity Heuristic: Narrow Faces . . . . . . . . . . . . . . . 145

9.3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

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9.4 Application to More Complex Bayesian Networks: The Lung Cancer Network . . . . . . . 146

9.4.1 Deriving a Semantical Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

9.4.2 Inference in Quantum Bayesian Networks . . . . . . . . . . . . . . . . . . . . . . . 147

9.4.3 Results with No Evidences Observed: Maximum Uncertainty . . . . . . . . . . . . 147

9.4.4 Results with One Piece of Evidence Observed . . . . . . . . . . . . . . . . . . . . 148

9.5 Application to More Complex Bayesian Networks: The Burglar / Alarm Network . . . . . . 149

9.5.1 Semantic Networks: Incorporating Acausal Connections . . . . . . . . . . . . . . . 150

9.6 Summary and FInal Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

10 Classical and Quantum Models for Order Effects 153

10.1 The Gallup Poll Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

10.2 A Quantum Approach for Order Effectsl . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

10.2.1 The Quantum Projection Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

10.2.2 Discussion of the Quantum Projection Model . . . . . . . . . . . . . . . . . . . . . 159

10.3 The Relativist Interpretation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 160

10.4 Do We Need Quantum Theory for Order Effects? . . . . . . . . . . . . . . . . . . . . . . . 162

10.4.1 A Classical Approach for Order Effects . . . . . . . . . . . . . . . . . . . . . . . . . 162

10.4.2 Analysis of the Classical Projection Model . . . . . . . . . . . . . . . . . . . . . . . 164

10.4.3 Explaining Serveral Order Effects using the Classical and Quantum Projection

Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

10.4.4 Occam’s Razor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166


11 Classical Models with Hidden Variables 169

11.1 Latent Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

11.2 Classical Bayesian Network with Latent Variables . . . . . . . . . . . . . . . . . . . . . . . 172

11.2.1 Estimating the Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

11.2.2 Increasing the Dimensionality of a Classical Bayesian Network . . . . . . . . . . . 179

11.3 Quantum-Like Bayesian Networks as an Alternative Model . . . . . . . . . . . . . . . . . 180

11.4 Summary and Final Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

12 Conclusions 187

13 Future Work 191

13.1 A Quantum-Like Analysis of a Real Life Financial Scenario: The Dutch’s Bank Loan Ap-

plication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

13.2 Quantum-Like Influence Diagrams: Incorporating Expected Utility in Quantum-Like Bayesian

Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

13.3 Neuroeconomics: quantum probabilities towards a unified theory of decision making . . . 193

xvii

Bibliography 194

xviii

List of Tables

3.1 Summary of all possible active trails in a Bayesian Network . . . . . . . . . . . . . . . . . 43

4.1 Allais Paradox Experiement 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Allais Paradox Experiement 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.3 Three color ellesberg paradox experiment 1. . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.4 Three color ellesberg paradox experiment 2. . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.5 Results of the two-stage gambling game reported by different works from the literature. . 54

4.6 Works of the literature reporting the probability of a player choosing to defect under sev-

eral conditions. a corresponds to the average of the results reported in the first two payoff

matrices of the work of Crosson [55]. b corresponds to the average of all seven experi-

ments reported in the work of Li & Taplin [125]. . . . . . . . . . . . . . . . . . . . . . . . . 55

4.7 Summary of the results obtained in the work of Moore [134]. . . . . . . . . . . . . . . . . 57

4.8 Results obtained from the medical decision experiment in Bergus et al. [25]. . . . . . . . . 57

4.9 Results reported by Trueblood & Busemeyer [188] of the experiments performed by McKen-

zie et al. [132]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.1 Average results of several different experiments of the Prisoner’s Dilemma Game reported

in Section 4.4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.2 Classical full joint probability distribution representation of the Bayesian Network in Fig-

ure 5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.3 Relation between classical and quantum probabilities used in the work of Leifer & Poulin

[124]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.4 Comparison of the different models proposed in the literature. . . . . . . . . . . . . . . . . 79

6.1 Fulll joint distribution of the Bayesian Newtwork in Figure 6.1 representing the average

results reported over the literature for the Two Stage Gambling Game (Table 4.5). The

random variable G1 corresponds to the outcome of the first gamble and the variable G2

corresponds to the decision of the player of playing/not playing the second gamble. . . . . 91





xix

6.3 Probabilities obtained when performing inference on the classical Bayesian Network of

Figure 6.6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.4 Probabilities obtained when performing inference on the quantum Bayesian Network of

Figure 6.7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.5 Optimum θ’s found for each variable from the burglar/alarm bayesian network (Figure 6.6). 105

7.1 Table representation of a quantum full joint probability distribution. . . . . . . . . . . . . . 110





7.3 Analysis of the quantum θ parameters computed for each work of the literature using

the proposed similarity function. Expected θ corresponds to the quantum parameter that

leads to the observed probability value in the experiment. Computed θ corresponds to the

quantum parameter computed with the proposed heurisitc. b corresponds to the average

of all seven experiments reported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.4 Results for the two games reported in the work of Crosson [55] for the Prisoner’s Dilemma

Game for several conditions: when the action of the second player was guessed to be

Defect (Guessed to Defect), when the action of the second player was guessed to be

Cooperate (Guessed to Collaborate), and when the action of the second player was not

known(Unknown). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

7.5 Experimental results reported in work of Li & Taplin [125] for the Prisoner’s Dilemma

game for several conditions: when the action of the second player is known to be Defect

(Known to Defect), when the action of the second player is known to be Cooperate (Known

to Collaborate), and when the action of the second player is not known(Unknown). The

column Violations of STP corresponds to determining if the collected results are violating

the Sure Thing Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.6 Experimental results reported in work of Li & Taplin [125] for the Prisoner’s Dilemma

game. The entries highlighted correspond to games that are not violating the Sure Thing

Principle. Expected θ corresponds to the quantum parameter that leads to the observed

probability value in the experiment. Computed θ corresponds to the quantum parameter

computed with the proposed heurisitc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.7 Comparison between the Quantum Prospect Decision Theory (QPDT) model and the

proposed Quantum-Like Bayesian Network (QLBN) for different works of the literature

reporting violations to the Sure Thing Principle. b corresponds to the average of all seven

experiments reported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

7.8 Comparison between the Quantum Prospect Decision Theory (QPDT) model and the

proposed Quantum-Like Bayesian Network (QLBN) for all the different experiments per-

formed in the work of Li & Taplin [125]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

xx

8.1 Empirical data collected in the experiment of Busemeyer et al. [41]. . . . . . . . . . . . . 131

8.2 Results from the application of the Quantum Like Bayesian Network (QLBN) model to the

categorisation / Decision experiment and comparison with the Quantum Dynamical Model

(QDM) proposed in the work of Busemeyer et al. [41]. . . . . . . . . . . . . . . . . . . . . 137

9.1 Full joint probability distribution. Pr(C,D) corresponds to the classical probability and

ψ(C,D) corresponds to the respective quantum amplitude. . . . . . . . . . . . . . . . . . 145

9.2 Comparison between a Quantum Markov Model and the proposed Bayesian Network. . . 146

9.3 Probabilities obtained when performing inference on the Bayesian Network of Figure 9.4. 149

9.4 Probabilities obtained when performing inference on the Bayesian Network of Figure 9.6. 151

10.1 Summary of the results obtained in the work of [134] for the Clinton-Gore Poll, showing

an Assimilation Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

10.2 Summary of the results obtained in the work of [134] for the Gingrich-Dole Poll, showing

a Contrast Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.3 Summary of the results obtained in the work of [134]. The table reports the probability of

answering All or Many to the questions. The results show the occurrence of an Additive

Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

10.4 Summary of the results obtained in the work of [134] for the Rose-Jacjson Poll, showing

a Subtractive Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

10.5 Prediction of the geometric approach using different φ rotation parameters to explain the

different types of order effects reported in the work of [134]. The columns Pr(1st ans) vs

Pr(1st ans exp) represent the answer to the first question obtained using the projection

models and the value reported in [134], respectively. Pr(2nd ans) vs Pr(2nd ans exp)

represent the answer to the second question obtained using the projection models and

the value reported in [134]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

11.1 Full joint probability distribution for the general Bayesian Network from Figure 11.2, which

models the Prisoner’s Dilemma game. Note that rs stands for risk seeking, ra for risk averse,

d for defect and c for cooperate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

11.2 Full joint probability distribution table of the Quantum-Like Bayesian Network in Figure 11.5.182

11.3 Analysis of the quantum θx parameters computed for each work of the literature in order

to reproduce the observed and unobserved conditions of the Prisoner’s Dilema Game. b

corresponds to the average of all seven experiments reported. . . . . . . . . . . . . . . . 183

xxi

List of Figures

1.1 Representation of the knowable conditions of the Two Stage Gambling Game experiment

of Tversky & Shafir [198]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Representation of the unknowable conditions of the Two Stage Gambling Game experi-

ment of Tversky & Shafir [198]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Representation of the unknowable conditions of the Two Stage Gambling Game experi-

ment conducted by Tversky & Shafir [198]. . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Sample Space (classical probability theory) . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Hilbert Space (quantum probabilty theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Example of a representation of an event on a Hilbert Space . . . . . . . . . . . . . . . . . 19

2.4 Example of a quantum system state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5 The double slit experiment. Electrons are fired and they can pass through one of the

slits (either s1ors2) to reach a detector screen in points d1 or d2. If we measure from

which slit the electron went through, then the pattern in the detectetor will have the shape

and size of the two slits, suggesting a particle baheviour of the electron. If we do not

measure from which slit the electron is going through, then the electron behaves as a

wave and produces an interference pattern in the detector screen, with one point detecting

constructive interference and another point detecting destructive interference. . . . . . . . 24

2.6 Classical Principle of Least Action. The path that a particle chooses between a starting

and ending position is always the one that requires the least energy (left). Quantum

version of the Principle of Least Action. A particle can be on different paths at the same

time and use them to find the optimal path (the one that requires less energy) between a

starting and final position (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.7 Single Path Trajectory (left). Multiple distinguishable paths (center). Multiple undistin-

guishable paths (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.8 Representation of the projections, Pi, of a qubit ψ, to either the |0〉 state subspace S0 or

the |1〉 state subspace S1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.9 Example of a distance between two points in L1-norm, also known as the Manhatten

distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.10 Example of a distance between two points in L2-norm, also known as the Euclidean dis-

tance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

xxiii

3.1 Naı̈ve Bayes Model, where node C represents the class variable and the set of random

variables {X1, X2, ..., Xn} represent the features. . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 The Burglar Bayesian Network proposed in the book of [168] . . . . . . . . . . . . . . . . 38

3.3 Difference between causal reasoning and evidential reasoning . . . . . . . . . . . . . . . 40

3.4 Indirect Causal Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.5 Indirect Evidential Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Common Cause Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.7 V-Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.1 Linda is feminist and bank teller. Notice that Pr(F ∩ B) has always to be smaller than

Pr(B). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2 Linda is feminist and bank teller. Notice that Pr(F ∪B) has always to be bigger than Pr(F ). 52

4.3 The two-stage gambling experiment proposed by Tversky & Shafir [198] . . . . . . . . . . 53

4.4 Example of a payoff matrix for the Prisoner’s Dilemma Game. . . . . . . . . . . . . . . . . 54

4.5 The Prisoner’s Dilemma game experiment proposed by Tversky & Shafir [198] . . . . . . 55

5.1 Illustration of the probabilities that can be obtained by varying the parameters γ and µd. . 71

5.2 Illustration of the probabilities that can be obtained by varying the parameters γ and µc. . 71

5.3 Illustration of the probabilities that can be obtained by varying the parameters γ and µB . . 71

5.4 Bayesian Network representation of the Average of the results reported in the literature

(last row of Table 8.2). The random variables that were considered are P1 and P2, corre-

sponding to the actions chosen by the first participant and second participant, respectively. 76

6.1 Classical Bayesian Network representation of the average results reported over the liter-

ature for the Two Stage Gambling Game (Section 4.4.1, Table 4.5). . . . . . . . . . . . . . 90

6.2 Quantum-Like Bayesian Network representation of the average results reported over the

literature for the Two Stage Gambling Game (Section 4.4.1, Table 4.5). The ψ(x) repre-

sents a complex probability amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.3 Example of constructive interference: two waves collide forming a bigger wave. . . . . . . 98

6.4 Example of destructive interference: two waves collide cancelling each other. . . . . . . . 98

6.5 The various quantum probability values that can be achieved by variying the angle θ in

Equation 6.14. Note that quantum probability can achieve much higher/lower values than

the classical probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.6 Burglar/Alarm classical Bayesian Network proposed in the book of Russel & Norvig [168] 104

6.7 Quantum-like counterpart of the Burglar/Alarm Bayesian Network proposed in the book

of Russel & Norvig [168] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

6.8 Possible probabilities when querying ”MaryCalls = t” with no evidence. Parameters used

were: {θ1, θ2, θ3, θ5, θ7, θ8} → {0, 0, 0, 0, 3.1, 0}. Maximum probability for {θ1, θ2} → {0, 3.1}. 106

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6.9 Possible probabilities when querying ”Burglar = t” with no evidence. Parameters used

were: {θ1, θ2, θ3, θ5, θ6, θ7} → {0, 0, 0, 6.2, 0.1, 3.1}. Maximum probability for {θ4, θ8} →

{0, 3.2}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.10 Possible probabilities when querying ”JohnCalls = t” with no evidence. Parameters used

were: {θ1, θ3, θ4, θ5, θ7, θ8} → {1.9, 0, 2.3, 0.5, 4.5, 2.4}. Maximum probability for {θ2, θ6} →

{2.3, 5.5}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

6.11 Possible probabilities when querying ”Alarm = t” with no evidence. Parameters used were:

{θ1, θ3, θ4, θ5, θ7, θ8} → {0, 0, 0.8, 6.2, 3.1, 4.3}. Maximum probability for {θ2, θ6} → {0.2, 0.5}.106

7.1 Vector representation of two vectors representing a certain state. . . . . . . . . . . . . . . 111

7.2 Illustration of the different 2-dimensional vectors that will be generated for each step of

iteration during the computation of the quantum interference term. . . . . . . . . . . . . . 111

7.3 Vector representation of vectors G2play and G2nplay plus the euclidean distance vector c. 116

7.4 Comparison of the results obtained, for different works of the literature concerned with the

Prisoner’s Dilemma game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.5 Possible probabilities that can be obtained from Game 1 (left), Game 2 (center) and the

average of the Games of the work of Crosson [55], using the quantum law of total probability.119

7.6 Comparison of the results obtained, for different experiments reported in the work of Li &

Taplin [125] in the context of the Prisoner’s Dilemma game. . . . . . . . . . . . . . . . . . 120

7.7 Possible probabilities that can be obtained in Game 2 of the work of Li & Taplin [125]

(left). Possible probabilities that can be obtained in Game 6 of the work of Li & Taplin

[125] (center). Possible probabilities that can be obtained in the work of Busemeyer et al.

[39] (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

7.8 Comparison of the results obtained, for different works of the literature concerned with the

Two-Stage Gambling game. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.9 Error percentage obtained in each experiment of the Two Stage Gambling game. . . . . . 122

7.10 Possible probabilities that can be obtained in the work of Lambdin & Burdsal [122]. The

probabilities observed in their experiment and the one computed with the proposed quan-

tum like Bayesian Network are also represented. . . . . . . . . . . . . . . . . . . . . . . . 122

8.1 Vector normalization to obtain quantum destructive interferences. . . . . . . . . . . . . . . 129

8.2 Example of Wide faces used in the experiment of Busemeyer et al. [41]. . . . . . . . . . . 130

8.3 Example of Narrow faces used in the experiment of Busemeyer et al. [41]. . . . . . . . . . 130

8.4 Summary of the probability distribution of the Good / Bad faces in the experiment of Buse-

meyer et al. [41]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

8.5 Representation of the Narrow faces experiment (left) and Wide faces experiment (right)

in a Bayesian Network with classical probabilities and quantum amplitudes. The classical

probabilities are given by Pr(X) and the quantum amplitudes by ψx. . . . . . . . . . . . . 132

8.6 Conversion of a dataset image into a binary image. Conversion with a small threhold (left).

Conversion with a high threhold (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

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8.7 Impact of the threshold when converting an image into a binary image. Threhold ranges

from 0.2 (left) to 0.8 (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

8.8 Distribution of Pr(Attack) using threshold 0.2. . . . . . . . . . . . . . . . . . . . . . . . . 135







8.15 Probability distribution of the 100 simulations performed when converting a grayscale im-

age into a binary one with a threshold of 0.4. . . . . . . . . . . . . . . . . . . . . . . . . . 136

9.1 Encoding of the synchronised variables with their respective angles (left). Two synchro-

nised events forming an angle of π/4 between them (right). . . . . . . . . . . . . . . . . . 143

9.2 Representation of the Synchronicity heuristic in the Hilbert Space. Vector i corresponds

to the event C = Good, D = Attack. Vector j corresponds to the event C = Bad,

D = Attack. The computed angle for the Attack (left) and Wthdraw (right) actions is

θ = 3π/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

9.3 Semantic Network for the Lung Cancer Bayesian Network. . . . . . . . . . . . . . . . . . 147

9.4 Lung Cancer Bayesian Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.5 Probabilities obtained using classical and quantum inferences for different queries for the

Lung Cancer Bayesian Network (Figure 9.4). . . . . . . . . . . . . . . . . . . . . . . . . . 148

9.6 Example of a Quantum-Like Bayesian Network [168]. ψ represents quantum amplitudes.

Pr corresponds to the real classical probabilities. . . . . . . . . . . . . . . . . . . . . . . . 150

9.7 Semantic Network representation of the network in Figure 9.6. . . . . . . . . . . . . . . . 150

9.8 Results for various queries comparing probabilistic inferences using classical and quan-

tum probability when no evidences are observed: maximum uncertainty. . . . . . . . . . . 151

10.1 Example of the application of the quantum projection approach for a sequece of two bi-

nary questions A and B. We start in a superposition state and project this state into the

yes basis of question A (left). Then, starting in this basis, we project into the basis corre-

sponding to the answer yes of question B (center). We can then have a different result if

we reverse the order the projections (right). . . . . . . . . . . . . . . . . . . . . . . . . . . 157

10.2 Relation between the rotation parameter φ and the quantum probability amplitude s0 of

Equation 10.15. The amplitude s1 was set to s1 = 1− s0. We can simulate several order

effects by varying the parameter φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

10.3 Relation between the rotation parameter φ and the quantum probability amplitude s0 of

Equation 10.12. The amplitude s1 was set to s1 = 1− s0. We can simulate several order

effects by varying the parameter φ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

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10.4 Example of the Relativistic Interpretation of Quantum Parameters. Each person reasons

according to a N-dimensional personal basis state without being aware of it. The repre-

sentation of the beliefs between different people consists in rotating the personal belief

state by φ radians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

11.1 Example of a Bayesian Network with a latent variable H and a random variable X. . . . . 171

11.2 A classical Bayesian Network with a latent variables to model the Prisoner’s Dilemma

game. P1 and P2 are both random variables. P1 represents the decision of the first player

and P2 represents the decision of the second player (either to cooperate or to defect). H

is the hidden state or latent variable and represents some unmeasurable factor that can

influence the participant’s decisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

11.3 Classical Bayesian Network to model the observed conditions for the Prisoner’s Dilemma

Game. OutP1 and P2 are both random variables that represent the outcome (or decision)

of the first player and the decision of the second player. The decisions can either be

defect, which is represented by d or cooperate, represented by c. H2 represents a latent

(hidden) unmeasurable variable that corresponds to the personality of the second player:

either risk averse (ra) or risk seeking (rs). . . . . . . . . . . . . . . . . . . . . . . . . . . 177

11.4 A general classical Bayesian Network with two latent variables, H1 and H2, to express

both unobserved and observed conditions for the Prisoner’s Dilemma game. Random

variables P1U and P1 represent the first player’s decision according to the unobserved

and observed conditions, respectively. Random variables P2U and P2 represent the sec-

ond player’s decision according to the unobserved and observed conditions, respectively.

The assignments ra stand for risk averse, rs risk seeking, d defect and c cooperate. . . . 179

11.5 Example of a Quantum-Like Bayesian Network. The terms ψ correspond to quantum

probability amplitudes. The variables P1 and P2 correspond to random variables repre-

senting the first and the second player, respectively. . . . . . . . . . . . . . . . . . . . . . 181

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xxviii

Chapter 1

Introduction

It is the purpose of this thesis to explore the applications of the formalisms of quantum mechanics in

areas outside of physics. More specifically, it is proposed a quantum-like decision model based on a

network structure to accommodate and predict several paradoxical findings that were reported over the

literature [193, 89, 195, 197, 198]. Note that, the term quantum-like is simply the designation that is

employed to refer to any model, which is applied in the domains outside of physics and that makes use

of the mathematical formalisms of quantum mechanics, abstracting them from any physical meanings

or interpretations. The paradoxes reported over the literature suggest that human behaviour does not

follow normative rational choices. In other words, people usually do not choose the preferences which

lead to a maximum utility in a decision scenario and consequently are consistently violating the axioms of

expected utility functions and the laws of classical probability theory. When observations contradict one

of the most significant and predominant decision theories, like the Expected Utility Theory, then it often

suggests that there is something missing in the theory. When dealing with preferences under uncertainty,

it seems that models based on normative theories of rational choice tend to tell how individuals must

choose, instead of telling how they actually choose [129].

It is the purpose of this thesis to provide a set of contributions of quantum based models applied to

decision scenarios as an alternative mathematical approach to human decision-making and cognition in

order to better understand the structure of human behaviour.

1.1 Violations to Normative Theories of Rational Choice

The process of decision-making is a research field that has always triggered a vast amount of interest

among several fields of the scientific community. Throughout time, many frameworks for decision-making

have been developed. In the beginning of the 1930’s, economical models focused in the mathematical

structures of preferences, which take choices as primitives and investigate whether these choices can

be represented by some utility function. The biggest consequence of this approach is the separation of

economics from psychology. This means that human psychological processes started to be irrelevant as

long as human decision-making obeys to some set of axioms [77]. According to these strong normative

1

models, human behaviour is assumed to maximise his/her utility function and by doing so, the person

would be acting in a rational manner. It was in 1944, that the Expected Utility theory was axiomatised

by the mathematician John von Neumann and the economist Oskar Morgenstern, and became one of

the most significant and predominate rational theories of decision-making [201]. The Expected utility

hypothesis is characterised by a specific set of axioms that enable the computation of the person’s

preferences with regard to choices under risk [74]. By risk, we mean choices that can be measured and

quantified. Putting in other words, choices based on objective probabilities. However, in 1953, Allais

proposed an experiment that showed that human behaviour does not follow these normative rules and

violates the axioms of Expected Utility, leading to the well known Allais paradox [13]. Later, in 1954, the

mathematician Leonard Savage proposed an extension of the Expected Utility hypothesis, giving origin

to the Subjective Expected Utility [170]. Instead of dealing with decisions under risk, the Subjective Utility

theory deals with uncertainty. Uncertainty is specified by subjective probabilities and is understood as

choices that cannot be quantified and are not based on objective probabilities. But once more, in 1961,

Daniel Ellsberg proposed an experiment that showed that human behaviour also contradicts and violates

the axioms of the Subjective Expected Utility theory, leading to the Ellsberg paradox [70]. In the end,

the Ellsberg and Allais paradoxes show that human behaviour is not normative and tend to violate the

axioms of rational decision theories.

1.2 The Emergence of Quantum Cognition

Later, in the 70s, cognitive psychologists Amos Tversky and Daniel Kahneman decided to put to test the

axioms of the Expected Utility hypothesis. They performed a set of experiments in which they demon-

strated that people usually violate the Expected Utility hypothesis and the laws of logic and probability in

decision scenarios under uncertainty [193, 195, 197, 90, 89]. From these experiments, it was reported

several paradoxes, such as disjunction / conjunction fallacies, order of effects, etc.

Motivated by these findings, researchers started to look for alternative mathematical representations

in order to accommodate these violations. Although in the 40’s, Niels Bohr had defended and was con-

vinced that the general notions of quantum mechanics could be applied in fields outside of physics [150],

it was only in the 90’s, that researchers started to actually apply the formalisms of quantum mechanics

to problems concerned with social sciences. It was the pioneering work of Aerts & Aerts [7] that gave

rise to the field Quantum Cognition. In their work, Aerts & Aerts [7] designed a quantum machine that

was able to represent the evolution from a quantum structure to a classical one, depending on the de-

gree of knowledge regarding the decision scenario. The authors also made several experiments to test

the variation of probabilities when posing yes/no questions. According to their experiment, most partici-

pants formed their answer at the moment the question was posed. This behaviour goes against classical

theories, because in classical probability, it would be expected that the participants have a predefined

answer to the question (or a prior) and not form it at the moment of the question. A further discussion

about this study can be found in the works of [4, 11, 12, 76, 8].

Quantum cognition has emerged as a research field that aims to build cognitive models using the

2

mathematical principles of quantum mechanics. Given that classical probability theory is very rigid in the

sense that it poses many constraints and assumptions (single trajectory principle, obeys set theory, etc.),

it becomes too limited (or even impossible) to provide simple models that can capture human judgments

and decisions since people are constantly violating the laws of logic and probability theory [33, 37, 6].

1.3 Motivation: Violations to the Sure Thing Principle

Although there are many paradoxical situations reported all over the literature, in this work we focus on

one of the most predominant human decision-making errors that still persists nowadays: the disjunction

effect [198]. The disjunction effect occurs whenever the Sure Thing Principle is violated. This principle is

fundamental in classical probability theory and states that, if one prefers action A over B under the state

of the world X, and if one also prefers A over B under the complementary state of the world X, then

one should always prefer action A over B even when the state of the world is not known [170]. Violations

of the Sure Thing Principle imply violations of the classical law of total probability.

In order to put to test the Sure Thing Principle, Tversky & Shafir [198] conducted an experiment,

which is called the Two Stage Gambling Game. Under this experiment, participants were asked to make

a set of two consecutive gambles. At each stage, they were asked to make the decision of whether or

not to play a gamble that has an equal chance of winning $200 or losing $100. Three conditions were

verified:

1. Participants were informed if they had won the first gamble;

2. Participants were informed if they had lost the first gamble;

3. Participants did not know the outcome of the first gamble;

The results obtained showed that participants who knew they had won the first gamble, decided to

play the second gamble. Participants who knew they had lost the first gamble also decided to play the

second gamble. We will address to these two conditions as the knowable conditions. Through Savage’s

Sure Thing Principle, it would be expected that the participants would choose to play the second gamble

even when they did not know the outcome of the first gamble. However, the results obtained showed that

the majority of the participants became risk averse and chose not to play the second gamble, leading to

a violation of the Sure Thing Principle. We will refer to this experimental condition as the unknowable

condition. Figures 1.1 and 1.2 represent the knowable and unknowable conditions, respectively.

Tversky & Shafir [198] explained these findings in the following way: when the participants knew that

they had won, then they had extra house money to play with and decided to play the second gamble.

When the participants knew that they lost, then they decided to play again with the hope of recovering

the lost money. But when the participants did not know if they had won or lost the gamble, then these

thoughts did not arise in their minds and consequently they ended up not to playing the second gamble.

Under a mathematical point of view, a person acts in a rational and consistent way, if under the

unknowable condition, he/she chooses to play the second gamble. Let Pr (G2 = play|G1 = win) and

3

Figure 1.1: Representation of the knowableconditions of the Two Stage Gambling Gameexperiment of Tversky & Shafir [198].

Figure 1.2: Representation of the unknowableconditions of the Two Stage Gambling Gameexperiment of Tversky & Shafir [198].

Pr (G2 = play|G1 = win) be the probability of a player choosing to play the second gamble given that

it is known that he won / lost the first gamble, respectively. And let Pr (G2 = play) be the probability of

the second player choosing to play without knowing the outcome of the first gamble. Assuming a neutral

prior and that the gamble is fair and not biased (50% chance of either winning or losing the first gamble),

it would be expected that:

Pr (G2 = play|G1 = win) ≥ Pr (G2 = play) ≥ Pr (G2 = play|G1 = lose)

However, this is not consistent with the experimental results reported in the work of Tversky & Shafir

[198]. What it was perceived in their experiments was that the probability of the unknowable condition

got extremely low compared to the known conditions.

Pr (G2 = play|G1 = win) ≥ Pr (G2 = play|G1 = lose) ≥ Pr (G2 = play)

This led to a violation of the laws of classical probability theory. Classical mechanics was also not able

to accommodate many paradoxical findings that were being observed in several experimental settings.

This gave rise to the axiomatisation of the theory of quantum mechanics. In this thesis, we explore

these paradoxical scenarios in the same way by using quantum probability theory as an alternative

mathematical formalism. Under a quantum cognition perspective, the third experimental condition, the

unknown condition, could be mathematically explained by quantum interference effects. In quantum

mechanics, electrons which are in an undefined state can interfere with each other. Under a quantum

cognitive point of view, if we consider that the beliefs of the participants are in an undefined state, then

they can also interfere with each other causing the final probabilities to be disturbed either increasing

them (constructive interferences) or decreasing them (destructive interferences). This last one is the

type of interference that results in violations to the Sure Thing Principle. Figure 6.2 represents the

third experimental condition from Tversky & Shafir [198] under a quantum cognitive point of view with

interference effects being generated when the outcome of the first gamble is not known.

4

Figure 1.3: Representation of the unknowable conditions of the Two Stage Gambling Game experimentconducted by Tversky & Shafir [198].

1.4 Why Quantum Cognition?

It is not the purpose of this thesis to argue whether quantum-like models should be preferred over

classical models. Just like it will be addressed in future chapters of this work, the advantages of the

applications of quantum-like models depend on the type of the decision problem (Chapters 10 and 11).

Following the lines of though of Sloman [179], people have to deal with missing / unknown information.

This lack of information can be translated into the feelings of ambiguity, uncertainty, vagueness, risk,

ignorance, etc [216], and each of them may require different mathematical approaches to build adequate

cognitive / decision problems. Quantum probability theory can be seen as an alternative mathematical

approach to model such cognitive phenomena.

Some researchers argue that quantum-like models do not offer many underlying aspects of human

cognition (like perception, reasoning, etc). They are merely mathematical models used to fit data and

for this reason they are able to accommodate many paradoxical findings [123]. Indeed quantum-like

models provide a more general probability theory that use quantum interference effects to model de-

cision scenarios, however they are also consistent with other psychological phenomena (for instance,

order effects) [179]. In the book of Busemeyer & Bruza [34], for instance, the feeling of uncertainty or

ambiguity can be associated to quantum superpositions, in which assumes that all beliefs of a person

occur simultaneously, instead of the classical approach which considers that each belief occurs in each

time frame. The book of Busemeyer & Bruza [34] provides a set of quantum phenomena that can be

associated to psychological processes that support the application of quantum-like models to cognitive

models.

• Violation of Classical Laws: The biggest motivation for the application of quantum formalisms in

areas outside of physics is the need to explain paradoxical findings that are hard to explain through

classical theory: violations to the Sure Thing Principle, disjunction/conjunction errors, Ellsberg /

Allais paradoxes. Quantum theory is a more general framework that allows the accommodation

and explanation of scenarios violating the laws of classical probability and logic [33, 31, 38].

• Superposition: Under a classical point of view, cognitive models assume that, at each moment of

5

time, a person is in a definite state. For instance, while making a judgement whether of not buying a

car, a person is either in a state corresponding to the judgment buy car or in the state not buy car for

each instance of time. In quantum cognition, it is assumed that the human thought process works

like a wave until a decision is made. A person can be in an indefinite state, that is, due to the wave-

like structure, a person can be in a superposition of thoughts. For each instance of time, a person

can be in the state buy car and not buy car. This wave-like paradigm enables the representation

of conflicting, ambiguous and uncertain thoughts more clearly as well as vagueness [28].

• The Principle of Unicity: In classical theory, when the path of a particle is unknown, it is assumed

that the particle either goes from one path or another with probability 1/2. In quantum theory,

when the path is unknown, the particle enters in a superposition state, taking all paths at the same

instance of time, generating interference effects that alter the final probabilities of the particle.

• Sensitivity to Measurement: In quantum mechanics, the act of measuring disturbs a quantum

superposition state making it collapse into one definite state. A measurement on a system rather

creates than records a property of the system [162]. In the scope of quantum cognition, the

measurement process can be used to explain decisions if we assume that human thoughts are

represented by a wave in superposition. For instance, if we ask a person if he/she will buy a

car, immediately before the question is posed, the person is in a superposition state. When the

question is posed, the superposition state will collapse into either one of two states: one in which

the answer is yes, the other in which the answer is no. The answer is created from the interaction

of the superposition state and the question. In classical mechanics, this act of creation does not

exist. Since a state is always considered definite, then the properties of a system are recorded

rather than created.

• Measurement Incompatibility: In classical theory, the act of asking a sequence of two questions

should yield the same answers as in the situation where the questions are posed in reversed

order. Empirical experiments have shown that this is not the case, and the act of answering the

first question changes the context of the second question, yielding people to give different answers

according to the order in which the questions are posed. Quantum theory allows the explanation

of order effects intuitively, since operations in quantum theory are non-commutative.

1.5 Challenges of Current Quantum-Like Models

Although recent research shows the successful application of quantum-like models in many different

decision scenarios of the literature [20, 16, 54, 50, 187], there are still many concerns that challenge

and put some resistance in the acceptance and usage of these models. Some of the current challenges

that quantum-like models face can be summarised in the following points.

• Prediction. Although many cognitive models have been successfully applied to accommodate

several paradoxical findings, they cannot be considered predictive. Most of the quantum-like mod-

els proposed need to have a priori knowledge of the outcome of the probability distributions of

6

the experiment in order to fit parameters to explain the paradoxical results. For this reason, it is

considered that these models have an explanatory nature rather than a predictive one.

• Scalability. Although there are many experiments that report paradoxical findings [164, 41, 120,

122], these experiments consist of very small scenarios that are modelled by, at most, two random

variables. Therefore, many of the proposed models in the literature are only effective under such

small scenarios and become intractable (or even intractable) for more complex situations. The

number of quantum interference parameters grows exponentially large [93, 96, 101] or there are

computational constraints in the computation of very large unitary operators [164, 44, 41].

• What can be considered Quantum-Like? Since the emergence of the Quantum Cognition field,

many researchers have been attempting to apply the mathematical formalisms of quantum me-

chanics in many different research areas, ranging from Biology [20, 16], Economics [102, 82],

Finance [81] Perception [54, 50], Jury duty [187] to domains such as Information Retrieval [133].

Regarding this last field, it has been proposed quantum-like versions of geometric-based pro-

jection models, which measure the similarity between entities (either documents, concepts, etc).

However, it is still not clear if applying a quantum-like projective approach has any advantages

towards the classical models, since the way that these models accommodate the paradoxical find-

ings is through a rotation of the vector space instead of the usage of quantum interferences [144].

• Classical vs. Quantum-Like. Recent research shows that quantum-based probabilistic models

are able to explain and accommodate decision scenarios that cannot be explained by pure clas-

sical models [38, 31]. However, there is still a big resistance in the scientific literature to accept

these quantum-based models [123, 184]. Many researchers believe that one can model scenarios

that violate the laws of probability and logic through traditional classical decision models [151].

Classical models can indeed simulate many of the paradoxical findings reported all over the litera-

ture [144]. This rises the question of the advantages or even the applicability of quantum models

over classical ones.

• Non-Kolmogorovian Models. Quantum-like models make use of quantum interference effects

in order to accommodate paradoxical decision-scenarios [165]. Since pure classical probabilistic

models are constrained to the limitations of set theory, it is difficult (or even impossible) to represent

these paradoxes. But if the limitations are in the constraints of set theory, then non-Kolmogorovian

theories such as Dempster-Shafer (D-S) theory [171] should also be able to accommodate the

same decision problems that quantum-like models are able to. The Dempster-Shafer theory of

evidence differs from classical set theory in the sense that it is possible to associate measures

of uncertainty to sets of hypotheses, this way enabling the theory to distinguish between uncer-

tainty from ignorance [128]. This distinction has been proved all over the literature and has shown

accurate predictions in sensor fusion models [136]. The uncertainty in the D-S theory is speci-

fied by allowing the representation of probabilities to sets of events, instead of being constraint

to specify the probabilities to atomic events (like in classical probability theory). It is still an open

research question in the literature if there are any relations between quantum-like models and

7

other non-Kolmogorovian probability theories. If this holds to be true, then the accommodation of

the paradoxical decision scenarios does not come from the unique characteristics of quantum-like

models and quantum interference effects, but because of the limitations of set theory.

1.6 Thesis Proposal

In order to overcome the above challenges, in this thesis it is proposed a quantum-like Bayesian Network

formalism, which consists in replacing classical probabilities by quantum complex probability amplitudes.

However, since this approach also suffers from the problem of exponential growth of quantum param-

eters that need to be fit, it is also proposed a similarity heuristic [173] that automatically computes this

exponential number of quantum parameters through vector similarities. A Bayesian Network can be un-

derstood as an acyclic directed graph, in which each node represents a random variable and each edge

represents a direct causal influence from the source node to the target node (conditional dependence).

Under the proposed network, if a node (event) is unobserved, then it can enter in a superposition state

and produce interference effects. These effects provide some explanation in terms of cognition, since

they can be seen as the feeling of confusion or ambiguity [34].

1.6.1 Why Bayesian Networks?

Bayesian Networks are one of the most powerful structures known by the Computer Science community

for deriving probabilistic inferences (for instance, in medical diagnosis, spam filtering, image segmenta-

tion, etc) [116]. The reason why Bayesian Networks were chosen is because it provides a link between

probability theory and graph theory. And a fundamental property of graph theory is its modularity: one

can build a complex system by combining smaller and simpler parts. It is easier for a person to combine

pieces of evidence and to reason about them, instead of calculating all possible events and their respec-

tive beliefs [79]. In the same way, Bayesian Networks represent the decision problem in small modules

that can be combined to perform inferences. Only the probabilities which are actually needed to perform

the inferences are computed.

This process can resemble human cognition [79]. While reasoning, humans cannot process all

possible information, because of their limited capacity [90]. Consequently, they combine several smaller

pieces of evidence in order to reach a final decision. A Bayesian Network works exactly in the same

way. It provides a relation mechanism between human cognition and inductive inference [161].

1.6.2 Quantum-Like Bayesian Networks for Disjunction Errors

In this thesis, it is addressed the problem of violations to the Sure Thing Principle (which are a conse-

quence of disjunction errors) by examining two major problems in which these violations were verified:

the Prisoner’s Dilemma game and the Two Stage Gambling game. These violations were initially re-

quantum probabilistic graphical models for cognition and...

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