Graphical Models

By Michael I. Jordan. 2020

. Abstract Working on different problems related to graph theory combined with statistics is the main purpose of this thesis. In paper I, we worked on the distance between trees and defined another definition for the interleaving distance that was already introduced for determining the distance between merge trees. The new definition was based on only one map from one of the trees to another one. Therefore, we could gain fixed-parameter tractable algorithms for finding the interleaving distance between merge trees with some conditions. In paper II, we worked on the clustering coefficient of the networks. The clustering coefficient indicates the tendency of the vertices of the network to form a triangle. We introduced another clustering coefficient, which we called it Relative clustering coefficient. Finally, the importance of the relative clustering coefficient. In paper III, we worked on the financial relationship between companies in Sweden, and we used two methods (the Pearson correlation coefficient (PCC) and the generalized variance decomposition (GVD). We then applied these methods to the financial data consisting of the daily returns on the 28 stocks included in the computation of the OMX index (the index of the Swedish capital market). Gaussian Graphical Model is the main subject of paper IV. In this paper, we consider three types of precision matrices, and corresponding to each type of precision matrix, we develop the exact test theory. Finally, the new approaches are compared to the benchmark method via an extensive simulation study.

Subfields: Probabilistic & Bayesian Methods

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Citations in the corpus

The Composition Theorem for Differential Privacy (2013, background): “…node b from the rest of the bayesian network, it follows immediately from the Markov property of this Bayesian network that (17) is true (cf. [Lau96]). b R 6 Proof of Theorem 3.4 We need to provide an outer bound on the privacy region achieved by ˜ X 0 and ˜ X 1 deﬁned in (12) and (13) under…”
Towards Causal Representation Learning (2021, background): “This distribution has structural properties inherited from the graph [147, 183]: it”
Causal Models for Dynamical Systems (2020, background): “(4) In the i.i.d. setting, Markov conditions connect properties of the graph, such as d -separation [Pearl, 2009], with properties of the joint distribution, such as conditional independence [Lauritzen, 1996].”
Causality for Machine Learning (2019, background): “This distribution has structural properties inherited from the graph (Pearl, 2009a; Lauritzen, 1996): it satisfies the causal Markov condition stating that conditioned on its parents, earch Xj is independent of its non-descendants.”
The hardness of conditional independence testing and the generalised covariance measure (2018, background): “During the last few decades, conditional independence relations have played an increasingly important role in computational statistics, too, since they are the building blocks of graphical models [36, 33, 41].”
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The Problem of Infra-marginality in Outcome Tests for Discrimination (2016, neutral): “We note that our results are robust to the exact specification of priors.5 Figure 2 shows this process represented as a graphical model (Jordan, 2004).”
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Graphical models of protein–protein interaction specificity from correlated mutations and interaction data (2009, unknown): “Using the standard semantics for an undirected graphical model,32 this joint probability is computed as Here, deg( (cid:2) ) indicates the degree of the vertex and Z nor-malizes the products into a probability measure (the equation for Z is given in the Supporting information).”
Cross-Validation for Correlated Data (2019, methodology): “Other examples of this setting that are commonly analyzed and represented using graphical probabilistic models tools are hidden Markov models and mixture models (Jordan 2004).”
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Graphical models for extremes (2018, background): “…this factorization can be rewritten in terms of marginal densities fX.x/= ∏ C∈C fC.xC/∏ D∈D fD.xD/ , x ∈X , .15/ where D is a multiset containing intersections between the cliques called separator sets; see Lauritzen (1996) and Appendix A for the definition of decomposability and separator sets.”