Identifying Nonlinear 1-Step Causal Influences in Presence of Latent Variables

We propose an approach for learning the causal structure in stochastic dynamical systems with a $1$-step functional dependency in the presence of latent variables. We propose an information-theoretic approach that allows us to recover the causal relations among the observed variables as long as the latent variables evolve without exogenous noise. We further propose an efficient learning method based on linear regression for the special sub-case when the dynamics are restricted to be linear. We validate the performance of our approach via numerical simulations

Latent tree models

Latent tree models are graphical models defined on trees, in which only a subset of variables is observed. They were first discussed by Judea Pearl as tree-decomposable distributions to generalise star-decomposable distributions such as the latent class model. Latent tree models, or their submodels, are widely used in: phylogenetic analysis, network tomography, computer vision, causal modeling, and data clustering. They also contain other well-known classes of models like hidden Markov models, Brownian motion tree model, the Ising model on a tree, and many popular models used in phylogenetics. This article offers a concise introduction to the theory of latent tree models. We emphasise the role of tree metrics in the structural description of this model class, in designing learning algorithms, and in understanding fundamental limits of what and when can be learned

Structured Prediction of Sequences and Trees using Infinite Contexts

Linguistic structures exhibit a rich array of global phenomena, however commonly used Markov models are unable to adequately describe these phenomena due to their strong locality assumptions. We propose a novel hierarchical model for structured prediction over sequences and trees which exploits global context by conditioning each generation decision on an unbounded context of prior decisions. This builds on the success of Markov models but without imposing a fixed bound in order to better represent global phenomena. To facilitate learning of this large and unbounded model, we use a hierarchical Pitman-Yor process prior which provides a recursive form of smoothing. We propose prediction algorithms based on A* and Markov Chain Monte Carlo sampling. Empirical results demonstrate the potential of our model compared to baseline finite-context Markov models on part-of-speech tagging and syntactic parsing

Learning Topologies of Acyclic Networks with Tree Structures

Network topology identification is known as the process of revealing the interconnections of a network where each node is representative of an atomic entity in a complex system. This procedure is an important topic in the study of dynamic networks since it has broad applications spanning different scientific fields. Furthermore, the study of tree structured networks is deemed significant since a large amount of scientific work is devoted to them and the techniques targeting trees can often be further extended to study more general structures. This dissertation considers the problem of learning the unknown structure of a network when the underlying topology is a directed tree, namely, it does not contain any cycles.The first result of this dissertation is an algorithm that consistently learns a tree structure when only a subset of the nodes is observed, given that the unobserved nodes satisfy certain degree conditions. This method makes use of an additive metric and statistics of the observed data only up to the second order. As it is shown, an additive metric can always be defined for networks with special dynamics, for example when the dynamics is linear. However, in the case of generic networks, additive metrics cannot always be defined. Thus, we derive a second result that solves the same problem, but requires the statistics of the observed data up to the third order, as well as stronger degree conditions for the unobserved nodes. Moreover, for both cases, it is shown that the same degree conditions are also necessary for a consistent reconstruction, achieving the fundamental limitations. The third result of this dissertation provides a technique to approximate a complex network via a simpler one when the assumption of linearity is exploited. The goal of this approximation is to highlight the most significant connections which could potentially reveal more information about the network. In order to show the reliability of this method, we consider high frequency financial data and show how well the businesses are clustered together according to their sector

A survey of statistical network models

Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.Comment: 96 pages, 14 figures, 333 reference