Moral Graph Research Articles

Topology Learning of Linear Dynamical Systems With Latent Nodes Using Matrix Decomposition

In this article, we present a novel approach to reconstruct the topology of networked linear dynamical systems with latent nodes. The network is allowed to have directed loops and bi-directed edges. The main approach relies on the unique decomposition of the inverse of power spectral density matrix (IPSDM) obtained from observed nodes as a sum of sparse and low-rank matrices. We provide conditions and methods for decomposing the IPSDM into sparse and low-rank components. The sparse component yields moral graph (MG) associated with the observed nodes, and the low-rank component retrieves parents, children and spouses (the Markov Blanket) of the hidden nodes. The article provides necessary and sufficient conditions for the unique decomposition of a given skew symmetric matrix into sum of a sparse skew symmetric and a low-rank skew symmetric matrices. For a large class of systems, the unique decomposition of imaginary part of the IPSDM of observed nodes, a skew symmetric matrix, into the sparse and the low-rank components is sufficient to identify the MG of the observed nodes as well as the Markov Blanket of latent nodes. For a large class of systems, all spurious links in the MG formed by the observed nodes can be identified. Assuming conditions on hidden nodes required for identifiability, links between hidden and observed nodes can be reconstructed, thus retrieving the exact topology of the network from the IPSDM. Moreover, for finite data, we provide bounds on entry-wise distance between the true and the estimated IPSDMs.

The difficulty of being moral

A family of Markov blankets, if learned purely as a set of subsets of variables, may not be consistent with any Bayesian network. This inconsistency may have a negative impact on Markov blanket-based structure learning. In this paper, we propose checking Markov blanket consistency using graph morality. We define an alternative concept of moral graph – weak recursive simpliciality – without relying on Bayesian networks. Although it is NP-complete to decide if an undirected graph in general is moral, we propose linear and quadratic time algorithms for deciding morality for maximum degree 3 and 4 graphs respectively. In addition, we prove that the problem remains NP-complete for graphs with maximum degree higher than 4, hence there are no remaining unknown complexities for this kind of problem.

Learning Moral Graphs in Construction of High-Dimensional Bayesian Networks for Mixed Data.

Bayesian networks have been widely used in many scientific fields for describing the conditional independence relationships for a large set of random variables. This letter proposes a novel algorithm, the so-called p-learning algorithm, for learning moral graphs for high-dimensional Bayesian networks. The moral graph is a Markov network representation of the Bayesian network and also the key to construction of the Bayesian network for constraint-based algorithms. The consistency of the p-learning algorithm is justified under the small-n, large-p scenario. The numerical results indicate that the p-learning algorithm significantly outperforms the existing ones, such as the PC, grow-shrink, incremental association, semi-interleaved hiton, hill-climbing, and max-min hill-climbing. Under the sparsity assumption, the p-learning algorithm has a computational complexity of O(p2) even in the worst case, while the existing algorithms have a computational complexity of O(p3) in the worst case.

Superdiffusive quantum stochastic walk definable on arbitrary directed graph

In this paper we define a quantum stochastic walk on arbitrary directed graph with super-diffusive propagation on a line graph. Our model is based on global environment interaction QSW, which is known to have ballistic propagation. However we discovered, that in this case additional amplitude transitions occur, hence graph topology is changed into moral graph. Because of that we call the effect a spontaneous moralization. We propose a general correction scheme, which is proved to remove unnecessary transition and thus to preserve the graph topology. In the end we numerically show, that superdiffusive propagation is preserved. Because of that our new model may be applied as effective evolution on arbitrary directed graph.

Optimistic Decision Making using an Approximate Graphical Model

Min-based qualitative possibilistic networks are one of the effective tools for a compact representation of decision problems under uncertainty. The exact approaches for computing decision based on possibilistic networks are limited by the size of the possibility distributions. Generally, these approaches are based on possibilistic propagation algorithms. An important step in the computation of the decision is the transformation of the DAG (Direct Acyclic Graph) into a secondary structure, known as the junction trees (JT). This transformation is known to be costly and represents a difficult problem. We propose in this paper a new approximate approach for the computation of decision under uncertainty within possibilistic networks. The computing of the optimal optimistic decision no longer goes through the junction tree construction step. Instead, it is performed by calculating the degree of normalization in the moral graph resulting from the merging of the possibilistic network codifying knowledge of the agent and that codifying its preferences.

Triangulation of Moral Graph Using Bayesian Optimization Algorithm

Finding the optimistic triangulation in Bayesian network, is NP hard. Bayesian Optimization Algorithm is a new kind of evolutionary algorithm estimation of distribution algorithms (EDAs). An improved BOA is proposed to get approximate optimistic triangulation in this paper. We carry out four EDAs including our method, on four standard Bayesian networks. Comparing with other Estimation of distribution algorithms, our method displays better experimental results and robustness.

A Decomposition Algorithm for Learning Bayesian Networks Based on Scoring Function

Learning Bayesian network (BN) structure from data is a typical NP‐hard problem. But almost existing algorithms have the very high complexity when the number of variables is large. In order to solve this problem(s), we present an algorithm that integrates with a decomposition‐based approach and a scoring‐function‐based approach for learning BN structures. Firstly, the proposed algorithm decomposes the moral graph of BN into its maximal prime subgraphs. Then it orientates the local edges in each subgraph by the K2‐scoring greedy searching. The last step is combining directed subgraphs to obtain final BN structure. The theoretical and experimental results show that our algorithm can efficiently and accurately identify complex network structures from small data set.

Die moralischen Vorfahren: Adjustierungsmengen in Kausaldiagrammen schnell und einfach berechnen

This article is concerned with the application of causal diagrams (also called DAGs) to the following 2 tasks, which are often faced in epidemiology: (1) a posteriori verification of the adjustment performed in an empirical study; (2) a priori identification of appropriate covariate sets for adjustment during study design. Causal diagram theory provides several methods for solving both of these tasks. However, some of these methods are computationally highly demanding, and thus cannot be carried out by hand and even pose problems for fast modern computers. In order to ease everyday work with causal diagrams, we discuss here the most efficient method known to date for performing the stated tasks. This method is based on the so-called "ancestor moral graph" construction by Lauritzen et al. and enables epidemiologists to solve at ease even large causal diagrams with dozens of variables and associations. Moreover, the presented method is well-suited for implementation in computer software, like it has been done in the DAG program and its graphical counterpart DAGitty.

Structural learning for Bayesian networks by testing complete separators in prime blocks

In this paper, we consider how to recover the structure of a Bayesian network from a moral graph. We present a more accurate characterization of moral edges, based on which a complete subset (i.e., a separator) contained in the neighbor set of one vertex of the putative moral edge in some prime block of the moral graph can be chosen. This results in a set of separators needing to be searched generally smaller than the sets required by some existing algorithms. A so-called structure-finder algorithm is proposed for structural learning. The complexity analysis of the proposed algorithm is discussed and compared with those for several existing algorithms. We also demonstrate how to construct the moral graph locally from, separately, the Markov blanket, domain knowledge and d -separation trees. Simulation studies are used to evaluate the performances of various strategies for structural learning. We also analyze a gene expression data set by using the structure-finder algorithm.

A note on minimal d-separation trees for structural learning

Structural learning of a Bayesian network is often decomposed into problems related to its subgraphs, although many approaches without decomposition were proposed. In 2006, Xie, Geng and Zhao proposed using a d-separation tree to improve the power of conditional independence tests and the efficiency of structural learning. In our research note, we study a minimal d-separation tree under a partial ordering, by which the maximal efficiency can be obtained. Our results demonstrate that a minimal d-separation tree of a directed acyclic graph (DAG) can be constructed by searching for the clique tree of a minimal triangulation of the moral graph for the DAG.

Triangulation of Bayesian networks with recursive estimation of distribution algorithms

Bayesian networks can be used as a model to make inferences in domains with intrinsic uncertainty, that is, to determine the probability distribution of a set of variables given the instantiation of another set. The inference is an NP-hard problem. There are several algorithms to make exact and approximate inference. One of the most popular, and that is also an exact method, is the evidence propagation algorithm of Lauritzen and Spiegelhalter [S.L. Lauritzen, D.J. Spiegelhalter, Local computations with probabilities on graphical structures and their application on expert systems, Journal of the Royal Statistical Society B 50 (2) (1988) 157–224], improved later by Jensen et al. [F.V. Jensen, S.L. Lauritzen, K.G. Olesen, Bayesian updating in causal probalistic networks by local computations, In Computational Statistics Quaterly 4 (1990) 269–282]. This algorithm needs an ordering of the variables in order to make the triangulation of the moral graph associated with the original Bayesian network structure. The effectiveness of the inference depends on the variable ordering. In this paper, we will use a new paradigm for evolutionary computation, the estimation of distribution algorithms (EDAs), to get the optimal ordering of the variables to obtain the most efficient triangulation. We will also present a new type of evolutionary algorithm, the recursive EDAs (REDAs). We will prove that REDAs improve the behaviour of EDAs in this particular problem, and that their results are competitive with other triangulation techniques.

Belief updating in Bayesian networks by using a criterion of minimum time

Variable elimination (VE) and clustering algorithms (CAs) are two widely used algorithms for exact inference in Bayesian networks. Both the problem of finding an optimal variable elimination ordering in VE and the problem of finding an optimal graph triangulation in CAs are NP -complete, although greedy algorithms work well in practice. Usually, VE selects the next variable to be eliminated such that a new potential of minimum size is generated during the elimination process. CAs create a variable elimination sequence in order to triangulate the moral graph; usually, the next variable to be eliminated is selected such that a new clique of minimum size is created during the elimination process. This paper presents an approach which makes use of a criterion of minimum time (CMT) for the selection of the next variable to be eliminated in VE or in CAs, and compares its performance with that of the traditional approaches using a criterion of minimum size. The results show that, in general, the CMT introduced in this paper allows inference time to be reduced. Results regarding memory requirements are also reported.

A Note on Collapsibility in DAG Models of Contingency Tables

Abstract. Necessary and sufficient conditions for collapsibility of a directed acyclic graph (DAG) model for a contingency table are derived. By applying the conditions, we can easily check collapsibility over any variable in a given model either by using the joint probability distribution or by using the graph of the model structure. It is shown that collapsibility over a set of variables can be checked in a sequential manner. Furthermore, a DAG is compared with its moral graph in the context of collapsibility.

Maximal prime subgraph decomposition of Bayesian networks

The authors present a method for decomposition of Bayesian networks into their maximal prime subgraphs. The correctness of the method is proven and results relating the maximal prime subgraph decomposition (MPD) to the maximal complete subgraphs of the moral graph of the original Bayesian network are presented. The maximal prime subgraphs of a Bayesian network can be organized as a tree which can be used as the computational structure for LAZY propagation. We also identify a number of tasks performed on Bayesian networks that can benefit from MPD. These tasks are: divide and conquer triangulation, hybrid propagation algorithms combining exact and approximative inference techniques, and incremental construction of junction trees. We compare the proposed algorithm with standard algorithms for decomposition of undirected graphs into their maximal prime subgraphs. The discussion shows that the proposed algorithm is simpler, more easy to comprehend, and it has the same complexity as the standard algorithms.

Bayes linear analysis for graphical models: The geometric approach to local computation and interpretive graphics

This paper concerns the geometric treatment of graphical models using Bayes linear methods. We introduce Bayes linear separation as a second order generalised conditional independence relation, and Bayes linear graphical models are constructed using this property. A system of interpretive and diagnostic shadings are given, which summarise the analysis over the associated moral graph. Principles of local computation are outlined for the graphical models, and an algorithm for implementing such computation over the junction tree is described. The approach is illustrated with two examples. The first concerns sales forecasting using a multivariate dynamic linear model. The second concerns inference for the error variance matrices of the model for sales, and illustrates the generality of our geometric approach by treating the matrices directly as random objects. The examples are implemented using a freely available set of object-oriented programming tools for Bayes linear local computation and graphical diagnostic display.

Decomposing Bayesian networks: triangulation of the moral graph with genetic algorithms

In this paper we consider the optimal decomposition of Bayesian networks. More concretely, we examine empirically the applicability of genetic algorithms to the problem of the triangulation of moral graphs. This problem constitutes the only difficult step in the evidence propagation algorithm of Lauritzen and Spiegelhalter (1988) and is known to be NP-hard (Wen, 1991). We carry out experiments with distinct crossover and mutation operators and with different population sizes, mutation rates and selection biasses. The results are analysed statistically. They turn out to improve the results obtained with most other known triangulation methods (Kj\sgmaelig;rulff, 1990) and are comparable to results obtained with simulated annealing (Kj\sgmaelig;rulff, 1990; Kj\sgmaelig;rulff, 1992).

A pedigree-based algorithm for finding efficient peeling sequences

The computational cost, in terms of both storage requirements and calculation required, of performing an elimination ordering on a graph is considered as a function of the order in which the vertices of the moral graph are eliminated. Useful properties of the moral graph of a pedigree with respect to vertex elimination are observed and these properties extended to define a k-pedigree as a graph permitting allocation of one of k sexes to each vertex of the graph, such that the subgraph induced by vertices of a single six contains no cycles. Properties of k-pedigrees include an upper bound of 2k on clique size. A novel algorithm, SEXY, based upon these properties is proposed and its performance compared with other algorithms used to generate elimination sequences. It is found to give a widely dispersed range of of sequences, including some sequences requiring under a quarter of the storage and under a half of the computational time than had previously been found using standard methods.