Inference via low-dimensional couplings

We investigate the low-dimensional structure of deterministic transformations between random variables, i.e., transport maps between probability measures. In the context of statistics and machine learning, these transformations can be used to couple a tractable "reference" measure (e.g., a standard Gaussian) with a target measure of interest. Direct simulation from the desired measure can then be achieved by pushing forward reference samples through the map. Yet characterizing such a map---e.g., representing and evaluating it---grows challenging in high dimensions. The central contribution of this paper is to establish a link between the Markov properties of the target measure and the existence of low-dimensional couplings, induced by transport maps that are sparse and/or decomposable. Our analysis not only facilitates the construction of transformations in high-dimensional settings, but also suggests new inference methodologies for continuous non-Gaussian graphical models. For instance, in the context of nonlinear state-space models, we describe new variational algorithms for filtering, smoothing, and sequential parameter inference. These algorithms can be understood as the natural generalization---to the non-Gaussian case---of the square-root Rauch-Tung-Striebel Gaussian smoother.Comment: 78 pages, 25 figure

Topics in social network analysis and network science

This chapter introduces statistical methods used in the analysis of social networks and in the rapidly evolving parallel-field of network science. Although several instances of social network analysis in health services research have appeared recently, the majority involve only the most basic methods and thus scratch the surface of what might be accomplished. Cutting-edge methods using relevant examples and illustrations in health services research are provided

Networks and the epidemiology of infectious disease

The science of networks has revolutionised research into the dynamics of interacting elements. It could be argued that epidemiology in particular has embraced the potential of network theory more than any other discipline. Here we review the growing body of research concerning the spread of infectious diseases on networks, focusing on the interplay between network theory and epidemiology. The review is split into four main sections, which examine: the types of network relevant to epidemiology; the multitude of ways these networks can be characterised; the statistical methods that can be applied to infer the epidemiological parameters on a realised network; and finally simulation and analytical methods to determine epidemic dynamics on a given network. Given the breadth of areas covered and the ever-expanding number of publications, a comprehensive review of all work is impossible. Instead, we provide a personalised overview into the areas of network epidemiology that have seen the greatest progress in recent years or have the greatest potential to provide novel insights. As such, considerable importance is placed on analytical approaches and statistical methods which are both rapidly expanding fields. Throughout this review we restrict our attention to epidemiological issues

Randomly-connected Non-Local Conditional Random Fields

Structural data modeling is an important field of research. Structural data are the combination of latent variables being related to each other. The incorporation of these relations in modeling and taking advantage of those to have a robust estimation is an open field of research. There are several approaches that involve these relations such as Markov chain models or random field frameworks. Random fields specify the relations among random variables in the context of probability distributions. Markov random fields are generative models used to represent the prior distribution among random variables. On the other hand, conditional random fields (CRFs) are known as discriminative models computing the posterior probability of random variables given observations directly. CRFs are one of the most powerful frameworks in image modeling. However practical CRFs typically have edges only between nearby nodes. Utilizing more interactions and expressive relations among nodes make these methods impractical for large-scale applications, due to the high computational complexity. Nevertheless, studies have demonstrated that obtaining long-range interactions in the modeling improves the modeling accuracy and addresses the short-boundary bias problem to some extent. Recent work has shown that fully connected CRFs can be tractable by defining specific potential functions. Although the proposed frameworks present algorithms to efficiently manage the fully connected interactions/relatively dense random fields, there exists the unanswered question that fully connected interactions are usually useful in modeling. To the best of our knowledge, no research has been conducted to answer this question and the focus of research was to introduce a tractable approach to utilize all connectivity interactions. This research aims to analyze this question and attempts to provide an answer. It demonstrates that how long-range of connections might be useful. Motivated by the answer of this question, a novel framework to tackle the computational complexity of a fully connected random fields without requiring specific potential functions is proposed. Inspired by random graph theory and sampling methods, this thesis introduces a new clique structure called stochastic cliques. The stochastic cliques specify the range of effective connections dynamically which converts a conditional random field (CRF) to a randomly-connected CRF. The randomly-connected CRF (RCRF) is a marriage between random graphs and random fields, benefiting from the advantages of fully connected graphs while maintaining computational tractability. To address the limitations of RCRF, the proposed stochastic clique structure is utilized in a deep structural approach (deep structure randomly-connected conditional random field (DRCRF)) where various range of connectivities are obtained in a hierarchical framework to maintain the computational complexity while utilizing long-range interactions. In this thesis the concept of randomly-connected non-local conditional random fields is explored to address the smoothness issues of local random fields. To demonstrate the effectiveness of the proposed approaches, they are compared with state-of-the-art methods on interactive image segmentation problem. A comprehensive analysis is done via different datasets with noiseless and noisy situations. The results shows that the proposed method can compete with state-of-the-art algorithms on the interactive image segmentation problem

Noisy Hamiltonian Monte Carlo for doubly-intractable distributions

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the statistician's toolbox as an alternative sampling method in settings when standard Metropolis-Hastings is inefficient. HMC generates a Markov chain on an augmented state space with transitions based on a deterministic differential flow derived from Hamiltonian mechanics. In practice, the evolution of Hamiltonian systems cannot be solved analytically, requiring numerical integration schemes. Under numerical integration, the resulting approximate solution no longer preserves the measure of the target distribution, therefore an accept-reject step is used to correct the bias. For doubly-intractable distributions -- such as posterior distributions based on Gibbs random fields -- HMC suffers from some computational difficulties: computation of gradients in the differential flow and computation of the accept-reject proposals poses difficulty. In this paper, we study the behaviour of HMC when these quantities are replaced by Monte Carlo estimates