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Quantum Probabilistic Graphical Models for Cognition and Decision 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 ii 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 iii iv 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 v vi T´ıtulo Modelos Graficos´ Probabil´ısticos Quanticosˆ para Cognic¸ao˜ e Decisao˜ Nome Catarina Alexandra Pinto Moreira Doutoramento em Engenharia Informatica´ 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 esta´ rela- cionado com o facto de como os humanos tomam decisoes.˜ Nas ultimas´ decadas,´ a literatura tem relatado varias´ situac¸oes˜ de decisoes˜ humanas que nao˜ podem ser facilmente modeladas por mode- los classicos,´ porque os humanos violam constantemente as leis da teoria da probabilidade classica´ em situac¸oes˜ com altos n´ıveis de incerteza. Nesse sentido, os modelos quanticosˆ comec¸aram a surgir como uma abordagem alternativa que se baseia nos princ´ıpios matematicos´ da mecanicaˆ quanticaˆ para modelar e explicar situac¸oes˜ paradoxais que os cientistas cognitivos nao˜ conseguem explicar usando as leis da teoria da probabilidade classica.´ Embora os modelos quanticosˆ tenham conseguido explicar muitos cenarios´ paradoxais de decisao˜ humana, eles ainda sofrem de tresˆ problemas principais. Primeiro, eles nao˜ podem escalar para cenarios´ de decisao˜ mais complexos, porque o numero´ de parametrosˆ quanticosˆ cresce de uma forma exponencial relativamente a` complexidade do problema de decisao.˜ Em segundo lugar, eles nao˜ podem ser considerados preditivos, uma vez que exigem que conhec¸amos a priori o resultado de um problema de decisao˜ para definir manualmente os parametrosˆ quanticosˆ que servem para explicar os resultados paradoxais. E em terceiro lugar, a forma como se pode definir esses parametrosˆ quanticosˆ e´ um campo inexplorado e ainda e´ uma questao˜ de investigac¸ao˜ aberta na literatura modelos cognitivos quanticos.ˆ Este trabalho centra-se em modelos probabil´ısticos graficos´ quanticos,ˆ consistindo num levanta- mento dos aspectos mais importantes da teoria da probabilidade classica,´ modelos quanticosˆ aplica- dos a` tomada de decisao˜ humana e em modelos probabil´ısticos graficos´ classicos.´ Tambem´ propomos uma rede Bayesiana quanticaˆ que pode escalar facilmente para cenarios´ de decisao˜ mais complexos devido a` sua estrutura de rede. De forma a abordar o problema de atribuic¸ao˜ de valores a um numero´ exponencial de parametrosˆ quanticos,ˆ tambem´ propomos func¸oes˜ heur´ısticas que podem definir um conjunto exponencial de parametrosˆ quanticosˆ 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 nao˜ podem ser generalizados para cenarios´ de tomada de decisao˜ mais complexos e que so´ podem fornecer uma natureza explicativa para os paradoxos observados. Palavras-chave: Cognic¸ao˜ Quantica,ˆ Redes Bayesianas Quanticas,ˆ Probabilidade Quantica,ˆ Efeitos de Interferenciaˆ Quantica,ˆ Modelos Quanticosˆ vii viii T´ıtulo Modelos Graficos´ Probabil´ısticos Quanticosˆ para Cognic¸ao˜ e Decisao˜ Nome Catarina Alexandra Pinto Moreira Doutoramento em Engenharia Informatica´ e de Computadores Orientador Doutor Andreas Miroslaus Wichert Resumo Extendido A cognic¸ao˜ quanticaˆ e´ uma area´ de investigac¸ao˜ que visa usar os princ´ıpios matematicos´ da mecanicaˆ quanticaˆ para modelar sistemas cognitivos para a tomada de decisoes˜ humanas. Dado que a teoria da probabilidade classica´ e´ muito r´ıgida no sentido de que ela apresenta muitas restric¸oes˜ e pressupostos (princ´ıpio da trajetoria´ unica,´ obedece a teoria dos conjuntos, etc.), torna-se muito limitado (ou mesmo imposs´ıvel) desenvolver modelos simples que possam capturar julgamentos humanos e decisoes,˜ uma vez que as pessoas podem violar as leis da logica´ e da teoria da probabilidade [33, 37,6]. A teoria da probabilidade quanticaˆ beneficia de muitas vantagens relativamente a` teoria classica.´ Pode representar eventos em espac¸os vectoriais. Consequentemente, pode levar em considerac¸ao˜ o problema da ordem dos efeitos [202, 188] e representar as amplitudes dos resultados experimentais ao mesmo tempo atraves´ de numa superposic¸ao.˜ Psicologicamente, o efeito de superposic¸ao˜ pode estar relacionado ao sentimento de confusao,˜ incerteza ou ambiguidade. Ou seja, pode representar a noc¸ao˜ de crenc¸a como um estado indefinido [34]. Alem´ disso, esta representac¸ao˜ do espac¸o vectorial nao˜ obedece ao axioma distributivo da logica´ booleana e nem a` lei da probabilidade total. Isso permite a construc¸ao˜ de modelos mais gerais que podem explicar matematicamente fenomenos´ cognitivos, como erros de conjunc¸ao/disjunc¸˜ ao˜ [40, 73] ou violac¸oes˜ do Princ´ıpio da Certeza [164, 110], que e´ o foco principal deste trabalho. Um problema dos actuais sistemas probabil´ısticos e´ o facto de nao˜ podem fazer previsoes˜ pre- cisas em situac¸oes˜ em que as leis da probabilidade classica´ sao˜ violadas. Estas situac¸oes˜ ocorrem frequentemente em sistemas que tentam modelar decisoes˜ humanas em cenarios´ onde o princ´ıpio da Certeza [170] e´ violado. Este princ´ıpio e´ fundamental na teoria da probabilidade classica´ e afirma que se alguem´ preferir a acc¸ao˜ A relativamente a` acc¸ao˜ B no estado do mundo X, e se alguem´ tambem´ preferir a acc¸ao˜ A relativamente a B sob o estado complementar do Mundo :X, entao˜ subentende- se que se deve preferir sempre a acc¸ao˜ A relativamente a B mesmo quando o estado do mundo nao˜ e´ conhecido. Violac¸oes˜ ao Princ´ıpio da Certeza implicam violac¸oes˜ a` lei da probabilidade total classica´ [193, 196, 198,9, 26]. Desta forma, neste trabalho, e´ proposta uma Rede Bayesiana quantica,ˆ inspirada nos formalismos de Integrais de caminho de Feynman [72]. Uma rede Bayesiana pode ser entendida como um grafico´ ac´ıclico direcionado, no qual cada no´ representa uma variavel´ aleatoria´ e cada uma das arestas representa uma influenciaˆ direta do no´ de origem para o no´ alvo (dependenciaˆ condicional). Por sua vez, os integrais do caminho de Feynman representam todos os caminhos poss´ıveis que uma part´ıcula pode percorrer para alcanc¸ar um ponto de ix destino, levando em considerac¸ao˜ que todos
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