Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of `conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.Comment: 38 pages, lots of picture

Lower and Upper Conditioning in Quantum Bayesian Theory

Updating a probability distribution in the light of new evidence is a very basic operation in Bayesian probability theory. It is also known as state revision or simply as conditioning. This paper recalls how locally updating a joint state can equivalently be described via inference using the channel extracted from the state (via disintegration). This paper also investigates the quantum analogues of conditioning, and in particular the analogues of this equivalence between updating a joint state and inference. The main finding is that in order to obtain a similar equivalence, we have to distinguish two forms of quantum conditioning, which we call lower and upper conditioning. They are known from the literature, but the common framework in which we describe them and the equivalence result are new.Comment: In Proceedings QPL 2018, arXiv:1901.0947

On quantum bayesian networks

Dissertação de mestrado em Computer ScienceAs a compact representation of joint probability distributions over a dependence graph of random variables, and a tool for modeling and reasoning in the presence of uncertainty, Bayesian networks are becoming increasingly relevant both for natural and social sciences, for example, to combine domain knowledge, capture causal relationships, or learn from in complete datasets. Known as an NP- hard problem in a classical setting, Bayesian inference pops up as a class of algorithms worth to explore in a quantum framework. The present dissertation explores this research field and extends the previous algorithm by embedding them in decision-making processes. In this regard, several attempts were made in order to find new and enhanced ways to deal with these processes. In a first at tempt, the quantum device was considered to run a subprocess of the decision-making pro cess, resulting in a quadratic speed-up for that subprocess. Afterward, “decision-networks” were taken into account and allowed a fully quantum implementation of a decision-making process, benefiting from a quadratic speed-up during the whole process. Lastly, a solution was found. It differs from the existing ones by the judicious use of the utility function in an entangled configuration. This algorithm explores the structure of input data to efficiently compute a solution. In addition, for each one of the algorithms developed, their computa tional complexity was determined in order to provide the information necessary to choose the most efficient one for a concrete decision problem. A prototype implementation in Qiskit (a Python-based program development language for the IBM Q machines) was developed as a proof-of-concept. If Qiskit offered a simulation platform for the algorithm considered in this dissertation, string diagrams provided the verification framework for algorithmic proprieties. Further, string diagrams were studied with the intention to obtain formal proofs about the algorithms developed. This framework provided relevant examples and the proof that two different implementations for the same algorithm are equivalent.As redes Bayesianas tem-se tornado cada vez mais importantes no domínio das ciências naturais e sociais, na medida em que permitem inferir relações de causalidade entre variáveis e aprender através de conjuntos incompletos de dados. Trata-se de uma representação compacta de distribuição de probabilidade conjunta feita sobre um grafo que representa dependências entre variáveis. Num contexto clássico, inferência sobre estas estruturas é visto como um problema de complexidade NP destacando-se como uma das classes de algoritmos a explorar num enquadramento quântico. Esta dissertação explora este domínio de investigação e insere as redes Bayesianas num processo de tomada de decisão. Neste sentido, foram feitas várias tentativas para se encontrarem novas e melhores formas de lidar com estes processos. Numa primeira tentativa, considerou-se que o dispositivo quântico executava um subprocesso do processo de tomada de decisão, resultando numa aceleração quadrática do mesmo. Posteriormente, foram consideradas decision networks que permitiram uma implementação totalmente quântica de um processo de tomada de decisão. Através desta implementação foi possível obter uma aceleração quadrática durante todo o processo. Por fim, foi encontrada uma solução viável. Difere das já existentes pelo uso criterioso da função de utilidade num estado emaranhado. Este algoritmo explora a estrutura dos dados de entrada para calcular de forma eficaz uma solução. Além disso, para cada um dos algoritmos desenvolvidos, foi determinada a respetiva complexidade computacional de modo a que fossem conhecidas todas as informações necessárias para escolher o algoritmo mais eficiente para um determinado problema de decisão. Foi desenvolvida uma implementação inicial no Qiskit (um software que permite o desenvolvimento de programas baseados em Python para as máquinas IBM Q) como prova de conceito. Se o Qiskit ofereceu uma plataforma de simulação para o algoritmo considerado nesta dissertação, os string diagrams forneceram a estrutura de verificação para propriedades algorítmicas. Além disso, estes diagramas foram estudados com a intenção de se obter provas formais sobre os algoritmos desenvolvidos. Esta estrutura forneceu exemplos relevantes e a prova de que duas implementações diferentes para o mesmo algoritmo são equivalentes

Terminality implies non-signalling

A 'process theory' is any theory of systems and processes which admits sequential and parallel composition. `Terminality' unifies normalisation of pure states, trace-preservation of CP-maps, and adding up to identity of positive operators in quantum theory, and generalises this to arbitrary process theories. We show that terminality and non-signalling coincide in any process theory, provided one makes causal structure explicit. In fact, making causal structure explicit is necessary to even make sense of non-signalling in process theories. We conclude that because of its much simpler mathematical form, terminality should be taken to be a more fundamental notion than non-signalling.Comment: In Proceedings QPL 2014, arXiv:1412.810

Reformulation of quantum mechanics and strong complementarity from Bayesian inference requirements

This paper provides an epistemic reformulation of quantum mechanics (QM) in terms of inference consistency requirements of objective Bayesianism, which include the principle of maximum entropy under physical constraints. Physical constraints themselves are understood in terms of consistency requirements. The by-product of this approach is that QM must additionally be understood as providing the theory of theories. Strong complementarity - that different observers may "live" in separate Hilbert spaces - follows as a consequence, which resolves the firewall paradox. Other clues pointing to this reformulation are analyzed. The reformulation, with the addition of novel transition probability arithmetic, resolves the measurement problem completely, thereby eliminating subjectivity of measurements from quantum mechanics. An illusion of collapse comes from Bayesian updates by observer's continuous outcome data. Dark matter and dark energy pop up directly as entropic tug-of-war in the reformulation