Decomposable graphs and hypergraphs

Lauritzen, Steffen; Speed, T. P.; Vijayan, K.

doi:10.1017/s1446788700027300

Cited by 72 publications

(20 citation statements)

References 22 publications

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“…The graph G(H) associated with the hypergraph H is a graph with the same set of vertices as H and an edge between every vertex pair that lies in some hyperedge. Then the following four properties: (a), (b), (c) and (d) have shown to be equivalent to each other (see Beeri et al (1983) and Lauritzen et al (1984)). …”

Section: Regular Covermentioning

confidence: 95%

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

Doan

Natarajan

2015

Operations Research

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In this paper, we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses. For a Fréchet class of discrete distributions with overlapping marginals, we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming. To guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information, we make use of a graph theoretic property known as the running intersection property. Building on this property, we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst-case Conditional Value-at-Risk measure. Lastly, we use a data-driven approach with financial return data to identify the Fréchet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions. Numerical results in two different datasets show that the distributionally robust portfolio optimization model improves on the sample-based approach

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Section: Regular Covermentioning

confidence: 95%

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

Doan

Natarajan

2015

Operations Research

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“…We associate each clique C in the set of cliques C(G) to the |C|-dimensional random variable X C and each separator S in the set of separators S(G) to the |S|-dimensional random variable X S . A decomposable model (Lauritzen et al, 1984) is given by M = (G, P G ), where G is a decomposable graph, and P G is a set of probabilities associated to the cliques and the separators of G,…”

Section: Decomposable Modelsmentioning

confidence: 99%

“…were d S is the number of cliques that contain S minus one (Lauritzen et al, 1984). We call to d S degree of the separator S. We call the decomposable models with kDG and MkDG structures, k-order decomposable models and maximal k-order decomposable models, respectively.…”

Section: Decomposable Modelsmentioning

confidence: 99%

Efficient approximation of probability distributions with k-order decomposable models

Pérez

Inza

Lozano

2016

International Journal of Approximate Reasoning

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During the last decades several learning algorithms have been proposed to learn probability distributions based on decomposable models. Some of these algorithms can be used to search for a maximum likelihood decomposable model with a given maximum clique size, k. Unfortunately, the problem of learning a maximum likelihood decomposable model given a maximum clique size is NP-hard for k > 2. In this work, we propose the fractal tree family of algorithms which approximates this problem with a computational complexity of O(k 2 · n 2 · N ) in the worst case, where n is the number of implied random variables and N is the size of the training set.The fractal tree algorithms construct a sequence of maximal i-order decomposable graphs, for i = 2, ..., k, in k − 1 steps. At each step, the algorithms follow a divide-and-conquer strategy that decomposes the problem into a set of separator problems. Each separator problem is efficiently solved using the generalized Chow-Liu algorithm. Fractal trees can be considered a natural extension of the Chow-Liu algorithm, from k = 2 to arbitrary values of k, and they have shown a competitive behavior to deal with the maximum likelihood problem. Due to their competitive behavior, their low computational complexity and their modularity, which allow them to implement different parallelization strategies, the proposed procedures are especially advisable for modeling high dimensional domains.

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“…and other issues had a particularly simple solution, the combinatorial theory of these graphs being further studied in Lauritzen et al [19].…”

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confidence: 99%

Interaction Models

Lauritzen

2012

Selected Works of Terry Speed

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The articles in this bundle are all associated with the notion of interaction and represent the genesis of the subject of graphical models in its modern form, the origins of these being traceable back to Gibbs [11] and Wright [30] and earlier.Around 1976, Terry was fascinated by the notion of conditional independence, along the lines later published in Dawid [6,7]. In 1976, Terry invited me to Perth and we were running a daily research seminar with the theme of studying similarities and differences between Statistics and Statistical Mechanics. In particular, we wondered what the relations were between notions of interaction as represented in linear models, in multi-dimensional contingency tables, and in stochastic models for particle systems; in addition, the purpose was also to understand what was the relation between these concepts and conditional independence.As we discovered that these were all essentially the same concepts, the similarity being obscured by very different traditions of notation, the term graphical model was coined. Our findings, also obtained in collaboration with John Darroch, were collected in Darroch et al. Of these articles, Darroch et al.[5] rather quickly had a seminal impact and a small community of researchers in the area of graphical models gradually emerged. In a certain sense, the article does not contain much formally new material (if any at all), but for the first time a simple, visual description and interpretation of the class of log-linear models [12,13], which otherwise could seem obscure, was available. The interpretation of a subclass of the models in terms of conditional independence had an immediate intuitive appeal. In addition, the article identified and emphasized models represented by chordal or triangulated graphs as those where estimation

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Decomposable graphs and hypergraphs

Cited by 72 publications

References 22 publications

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

Efficient approximation of probability distributions with k-order decomposable models

Interaction Models

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