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Exact Bayesian Inference in Graphical Models : Tree-structured Network Inference and Segmentation

Abstract : In this dissertation we investigate the problem of network inference. The statistical frame- work tailored to this task is that of graphical models, in which the (in)dependence relation- ships satis ed by a multivariate distribution are represented through a graph. We consider the problem from a Bayesian perspective and focus on a subset of graphs making structure inference possible in an exact and e cient manner, namely spanning trees. Indeed, the integration of a function de ned on spanning trees can be performed with cubic complexity with respect to number of variables under some factorisation assumption on the edges, in spite of the super-exponential cardinality of this set. A careful choice of prior distributions on both graphs and distribution parameters allows to use this result for network inference in tree-structured graphical models, for which we provide a complete and formal framework.We also consider the situation in which observations are organised in a multivariate time- series. We assume that the underlying graph describing the dependence structure of the distribution is a ected by an unknown number of abrupt changes throughout time. Our goal is then to retrieve the number and locations of these change-points, therefore dealing with a segmentation problem. Using spanning trees and assuming that segments are inde- pendent from one another, we show that this can be achieved with polynomial complexity with respect to both the number of variables and the length of the series.
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Submitted on : Wednesday, November 23, 2016 - 1:31:10 PM
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Loïc Schwaller. Exact Bayesian Inference in Graphical Models : Tree-structured Network Inference and Segmentation. Statistics [math.ST]. Université Paris Saclay (COmUE), 2016. English. ⟨NNT : 2016SACLS210⟩. ⟨tel-01401458⟩

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