On the Differential Privacy of Bayesian Inference

We study how to communicate findings of Bayesian inference to third parties, while preserving the strong guarantee of differential privacy. Our main contributions are four different algorithms for private Bayesian inference on proba-bilistic graphical models. These include two mechanisms for adding noise to the Bayesian updates, either directly to the posterior parameters, or to their Fourier transform so as to preserve update consistency. We also utilise a recently introduced posterior sampling mechanism, for which we prove bounds for the specific but general case of discrete Bayesian networks; and we introduce a maximum-a-posteriori private mechanism. Our analysis includes utility and privacy bounds, with a novel focus on the influence of graph structure on privacy. Worked examples and experiments with Bayesian na{\"i}ve Bayes and Bayesian linear regression illustrate the application of our mechanisms.Comment: AAAI 2016, Feb 2016, Phoenix, Arizona, United State

Machine learning for modelling urban dynamics

We live in the age of cities. More than half of the world’s population live in cities and this urbanisation trend is only forecasted to continue. To understand cities now and in the foreseeable future, we need to take seriously the idea that it is not enough to study cities as sets of locations as we have done in the past. Instead, we need to switch our traditional focus from locations to interactions and in doing so, invoke novel approaches to modelling cities. Cities are becoming “smart” recording their daily interactions via various sensors and yielding up their secrets in large databases. We are faced with an unprecedented opportunity to reason about them directly from such secondary data. In this thesis, we propose model-based machine learning as a flexible framework for reasoning about cities at micro and macro scales. We use model-based machine learning to encode our knowledge about cities and then to automatically learn about them from urban tracking data. Driven by questions about urban dynamics, we develop novel Bayesian inference algorithms that improve our ability to learn from highly complex, temporal data feeds, such as tracks of vehicles in cities. Overall, the thesis proposes a novel machine learning toolkit, which, when applied to urban data, can challenge how we can think about cities now and about how to make them ”smarter”

A Brief Introduction to Machine Learning for Engineers

This monograph aims at providing an introduction to key concepts, algorithms, and theoretical results in machine learning. The treatment concentrates on probabilistic models for supervised and unsupervised learning problems. It introduces fundamental concepts and algorithms by building on first principles, while also exposing the reader to more advanced topics with extensive pointers to the literature, within a unified notation and mathematical framework. The material is organized according to clearly defined categories, such as discriminative and generative models, frequentist and Bayesian approaches, exact and approximate inference, as well as directed and undirected models. This monograph is meant as an entry point for researchers with a background in probability and linear algebra.Comment: This is an expanded and improved version of the original posting. Feedback is welcom

Bayesian inference in probabilistic graphical models

This thesis consists of four papers studying structure learning and Bayesian inference in probabilistic graphical models for both undirected and directed acyclic graphs (DAGs). Paper A presents a novel algorithm, called the Christmas tree algorithm (CTA), that incrementally construct junction trees for decomposable graphs by adding one node at a time to the underlying graph. We prove that CTA with positive probability is able to generate all junction trees of any given number of underlying nodes. Importantly for practical applications, we show that the transition probability of the CTA kernel has a computationally tractable expression. Applications of the CTA transition kernel are demonstrated in a sequential Monte Carlo (SMC) setting for counting the number of decomposable graphs. Paper B presents the SMC scheme in a more general setting specifically designed for approximating distributions over decomposable graphs. The transition kernel from CTA from Paper A is incorporated as proposal kernel. To improve the traditional SMC algorithm, a particle Gibbs sampler with a systematic refreshment step is further proposed. A simulation study is performed for approximate graph posterior inference within both log-linear and decomposable Gaussian graphical models showing efficiency of the suggested methodology in both cases. Paper C explores the particle Gibbs sampling scheme of Paper B for approximate posterior computations in the Bayesian predictive classification framework. Specifically, Bayesian model averaging (BMA) based on the posterior exploration of the class-specific model is incorporated into the predictive classifier to take full account of the model uncertainty. For each class, the dependence structure underlying the observed features is represented by a distribution over the space of decomposable graphs. Due to the intractability of explicit expression, averaging over the approximated graph posterior is performed. The proposed BMA classifier reveals superior performance compared to the ordinary Bayesian predictive classifier that does not account for the model uncertainty, as well as to a number of out-of-the-box classifiers. Paper D develops a novel prior distribution over DAGs with the ability to express prior knowledge in terms of graph layerings. In conjunction with the prior, a stochastic optimization algorithm based on the layering property of DAGs is developed for performing structure learning in Bayesian networks. A simulation study shows that the algorithm along with the prior has superior performance compared with existing priors when used for learning graph with a clearly layered structure.QC 20170915</p

Bayesian inference in probabilistic graphical models

This thesis consists of four papers studying structure learning and Bayesian inference in probabilistic graphical models for both undirected and directed acyclic graphs (DAGs). Paper A presents a novel algorithm, called the Christmas tree algorithm (CTA), that incrementally construct junction trees for decomposable graphs by adding one node at a time to the underlying graph. We prove that CTA with positive probability is able to generate all junction trees of any given number of underlying nodes. Importantly for practical applications, we show that the transition probability of the CTA kernel has a computationally tractable expression. Applications of the CTA transition kernel are demonstrated in a sequential Monte Carlo (SMC) setting for counting the number of decomposable graphs. Paper B presents the SMC scheme in a more general setting specifically designed for approximating distributions over decomposable graphs. The transition kernel from CTA from Paper A is incorporated as proposal kernel. To improve the traditional SMC algorithm, a particle Gibbs sampler with a systematic refreshment step is further proposed. A simulation study is performed for approximate graph posterior inference within both log-linear and decomposable Gaussian graphical models showing efficiency of the suggested methodology in both cases. Paper C explores the particle Gibbs sampling scheme of Paper B for approximate posterior computations in the Bayesian predictive classification framework. Specifically, Bayesian model averaging (BMA) based on the posterior exploration of the class-specific model is incorporated into the predictive classifier to take full account of the model uncertainty. For each class, the dependence structure underlying the observed features is represented by a distribution over the space of decomposable graphs. Due to the intractability of explicit expression, averaging over the approximated graph posterior is performed. The proposed BMA classifier reveals superior performance compared to the ordinary Bayesian predictive classifier that does not account for the model uncertainty, as well as to a number of out-of-the-box classifiers. Paper D develops a novel prior distribution over DAGs with the ability to express prior knowledge in terms of graph layerings. In conjunction with the prior, a stochastic optimization algorithm based on the layering property of DAGs is developed for performing structure learning in Bayesian networks. A simulation study shows that the algorithm along with the prior has superior performance compared with existing priors when used for learning graph with a clearly layered structure.QC 20170915</p

ReactiveMP.jl: A Julia Package for Reactive Message Passing-based Bayesian Inference

ReactiveMP.jl is a native Julia implementation of reactive message passing-based Bayesian inference in probabilistic graphical models with Factor Graphs. The package does Constrained Bethe Free Energy minimisation and supports both exact and variational Bayesian inference, provides a convenient syntax for model specification and allows for extra factorisation and form constraints specification of the variational family of distributions. In addition, ReactiveMP.jl includes a large range of standard probabilistic models and can easily be extended to custom novel nodes and message update rules. In contrast to non-reactive (imperatively coded) Bayesian inference packages, ReactiveMP.jl scales easily to support inference on a standard laptop for large conjugate models with tens of thousands of variables and millions of nodes

4D Tomographic Image Reconstruction and Parametric Maps Estimation: a model-based strategy for algorithm design using Bayesian inference in Probabilistic Graphical Models

This work is inspired by the search for an answer to two grand challenges affecting 4D emission tomography, namely the solution of the inverse problem of computing the rate of emission in the imaging volume in case of an extreme photon-limited regime, and the estimation of maps of pharmacokinetic parameters. The strategy to tackle these issues proposed in this thesis is based on the idea that a unified and synergistic approach to the estimation of both dynamic activity time series and parametric maps could provide mutual benefits, by integrating the lack of measured information with predictions made by the chosen model. Framing emission tomography imaging in the Bayesian framework via probabilistic graphical models, we are able to define a model-based approach to the design of integrated inference algorithms. From one side, this modeling approach has shown itself able to encompass traditional literature about emission tomography image reconstruction. From another, it provides a flexible tool to describe causal relationships between variables, and a straightforward strategy to derive inference algorithms from such a combination of graphical and probabilistic representations. A number of different models are proposed, justified and discussed, in the light of the model-based inference framework proposed in this thesis. A comprehensive description of the phenomenon of image formation allows us to devise unified inference approaches to tackle at once and in a synergistic way the solution of multiple problems that traditionally are dealt with in a sequential way. At the deeper level, pharmacokinetic models can be used to concisely describe in a mathematical way the physiological interactions between tissues and tracer; these interactions are responsible for determining how the injected radiotracer distributes within tissues, in space, and thus of what we eventually see in the form of images; lastly, the spatial location of radioactive molecules is the source of the measured coincidence photons on which we base our inference. The formulations presented in this thesis are unifying in several ways, combining in a single model information from multiple domains, and attempting to unify reconstruction and kinetic modeling, tasks usually addressed with a sequential approach. Moreover, this modeling approach is able to abstract over details that are specific of a certain imaging modality in such a way that the inference strategies developed for PET can be (quite) easily adapted to other imaging modalities that may face similar challenges (like the case of DCE-MRI discussed in this work), requiring just minor changes of the assumptions made during model-design