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Bayesian Statistics provide us with a powerful approach to model real-world phenomena and quantify the uncertainty therein by treating unknown factors as random. The Bayesian paradigm prescribes quantifying the prior knowledge about some state of the world, and after having obtained new information updating that knowledge in order to update the prior beliefs and propose posterior knowledge. A central approach to Bayesian Statistics is modelling, i.e. to represent a data generating process using statistical models equipped with some parameters which are to be estimated via Bayesian inference. Bayesian Nonparametric modelling comes with great flexibility as it provides infnitely many parameters. Bayesian Nonparametric models are data adaptive because when given a finite set of data, the size of the finite set of parameters to be used, adapts to the complexity of the observed data. Bayesian Nonparametric models have been used in many applications of machine learning such as density estimation, clustering, latent feature models, survival analysis and function approximation. One of the contributions of this thesis to Bayesian Nonparametrics is the formulation of priors on random graphs to model networks. Our motivation for networks sources from the fact that networks are found in numerous areas of modern society reflecting the patterns of connection across a wide range of physical and social phenomena. Our research focus is on modelling two types of networks; undirected static networks and directed networks with temporal connections. In both cases our objective is to overcome the limitations of the current literature and propose network methodology that captures real-world network properties. Precisely, the property of graph sparsity is particularly important. However, traditional Bayesian network models that assume the desirable property of exchangeability are necessarily not sparse. The first random graph to allow sparsity as well as exchangeability was recently introduced by Caron and Fox [2017] whose framework we follow. Our objective is to propose models with communities in their latent structure, and we therefore propose a generalisation of existing undirected network models with community structure to the sparse regime. Regarding temporal interaction network data, our goal is to capture reciprocation in interactions and therefore define a family of network models that generalises existing classes of reciprocating models to the sparse graph case. The key building blocks of our models as well as for various other popular Bayesian Nonparametric constructions, are infnite-activity completely random measures. The use of these random measures offers flexibility, but is also accompanied by computational complexity. The third objective of this thesis is a methodological and theoretical contribution to Bayesian Nonparametrics consisting of a novel framework to form approximations of infinite-activity completely random measures. By providing finite approximations to these infinite data structures we offer practicality and efficiency. Our constructions can be used to develop efficient posterior inference algorithms and shed light on current Bayesian Nonparametric issues of computational complexity. |
