A graphical characterization of the largest chain graphs

The paper presents a graphical characterization of the largest chain graphs which serve as unique representatives of classes of Markov equivalent chain graphs. The characterization is a basis for an algorithm constructing, for a given chain graph, the largest chain graph equivalent to it. The algorithm was used to generate a catalog of the largest chain graphs with at most five vertices. Every item of the catalog contains the largest chain graph of a class of Markov equivalent chain graphs and an economical record of the induced independency model.     

Learning marginal AMP chain graphs under faithfulness revisited

Marginal AMP chain graphs are a recently introduced family of models that is based on graphs that may have undirected, directed and bidirected edges. They unify and generalize the AMP and the multivariate regression interpretations of chain graphs. In this paper, we present a constraint based algorithm for learning a marginal AMP chain graph from a probability distribution which is faithful to it. We show that the marginal AMP chain graph returned by our algorithm is a distinguished member of its Markov equivalence class. We also show that our algorithm performs well in practice. Finally, we show that the extension of Meek's conjecture to marginal AMP chain graphs does not hold, which compromises the development of efficient and correct score+search learning algorithms under assumptions weaker than faithfulness.     

Learning AMP chain graphs and some marginal models thereof under faithfulness

This paper deals with chain graphs under the Andersson–Madigan–Perlman (AMP) interpretation. In particular, we present a constraint based algorithm for learning an AMP chain graph a given probability distribution is faithful to. Moreover, we show that the extension of Meek's conjecture to AMP chain graphs does not hold, which compromises the development of efficient and correct score + search learning algorithms under assumptions weaker than faithfulness.We also study the problem of how to represent the result of marginalizing out some nodes in an AMP CG. We introduce a new family of graphical models that solves this problem partially. We name this new family maximal covariance–concentration graphs because it includes both covariance and concentration graphs as subfamilies.     

Enumeration of labelled chain graphs and labelled essential directed acyclic graphs

A chain graph is a digraph whose strong components are undirected graphs and a directed acyclic graph (ADG or DAG) G is essential if the Markov equivalence class of G consists of only one element. We provide recurrence relations for counting labelled chain graphs by the number of chain components and vertices; labelled essential DAGs by the number of vertices. The second one is a lower bound for the number of labelled essential graphs. The formula for labelled chain graphs can be extended in such a way, that allows us to count digraphs with two additional properties, which essential graphs have.     

Marginal AMP chain graphs

We present a new family of models that is based on graphs that may have undirected, directed and bidirected edges. We name these new models marginal AMP (MAMP) chain graphs because each of them is Markov equivalent to some AMP chain graph under marginalization of some of its nodes. However, MAMP chain graphs do not only subsume AMP chain graphs but also multivariate regression chain graphs. We describe global and pairwise Markov properties for MAMP chain graphs and prove their equivalence for compositional graphoids. We also characterize when two MAMP chain graphs are Markov equivalent.For Gaussian probability distributions, we also show that every MAMP chain graph is Markov equivalent to some directed and acyclic graph with deterministic nodes under marginalization and conditioning on some of its nodes. This is important because it implies that the independence model represented by a MAMP chain graph can be accounted for by some data generating process that is partially observed and has selection bias. Finally, we modify MAMP chain graphs so that they are closed under marginalization for Gaussian probability distributions. This is a desirable feature because it guarantees parsimonious models under marginalization.     

On expressiveness of the chain graph interpretations

In this article we study the expressiveness of the different chain graph interpretations. Chain graphs is a class of probabilistic graphical models that can contain two types of edges, representing different types of relationships between the variables in question. Chain graphs is also a superclass of directed acyclic graphs, i.e. Bayesian networks, and can thereby represent systems more accurately than this less expressive class of models. Today there do however exist several different ways of interpreting chain graphs and what conditional independences they encode, giving rise to different so-called chain graph interpretations. Previous research has approximated the number of representable independence models for the Lauritzen–Wermuth–Frydenberg and the multivariate regression chain graph interpretations using an MCMC based approach. In this article we use a similar approach to approximate the number of models representable by the latest chain graph interpretation in research, the Andersson–Madigan–Perlman interpretation. Moreover we summarize and compare the different chain graph interpretations with each other. Our results confirm previous results that directed acyclic graphs only can represent a small fraction of the models representable by chain graphs, even for a low number of nodes. The results also show that the Andersson–Madigan–Perlman and multivariate regression interpretations can represent about the same amount of models and twice the amount of models compared to the Lauritzen–Wermuth–Frydenberg interpretation. However, at the same time almost all models representable by the latter interpretation can only be represented by that interpretation while the former two have a large intersection in terms of representable models.     

Local uniformity properties for glauber dynamics on graph colorings

We investigate some local properties which hold with high probability for randomly selected colorings of a fixed graph with no short cycles. In a number of related works, establishing these particular properties has been a crucial step towards proving rapid convergence for the single‐site Glauber dynamics, a Markov chain for sampling colorings uniformly at random. For a large class of graphs, this approach yields the most efficient known algorithms for sampling random colorings. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Modelling and inference with Conditional Gaussian Probabilistic Decision Graphs

Probabilistic Decision Graphs (PDGs) are probabilistic graphical models that represent a factorisation of a discrete joint probability distribution using a “decision graph”-like structure over local marginal parameters. The structure of a PDG enables the model to capture some context specific independence relations that are not representable in the structure of more commonly used graphical models such as Bayesian networks and Markov networks. This sometimes makes operations in PDGs more efficient than in alternative models. PDGs have previously been defined only in the discrete case, assuming a multinomial joint distribution over the variables in the model. We extend PDGs to incorporate continuous variables, by assuming a Conditional Gaussian (CG) joint distribution. We also show how inference can be carried out in an efficient way.     

A Scoring Criterion For Learning Chain Graphs

A chain graph allows both directed and undirected edges, and contains the underlying mathematical properties of the two. An important method of learning graphical models is to use scoring criteria to measure how well the graph structures fit the data. In this paper, we present a scoring criterion for learning chain graphs based on the Kullback Leibler distance. It is score equivalent, that is, equivalent chain graphs obtain the same score, so it can be used to perform model selection and model averaging.     

A linear-time algorithm for classifying the states of a finite Markov chain

Classifying the states of a finite Markov chain requires the identification of all irreducible closed sets and the set of transient states. This paper presents an algorithm for identifying these states that executes in time O(MAX(|V|, |E|)) where number of states and |E| is the number of positive entries in the Markov matrix. The algorithm finds the closed strongly connected components of the transition graph using a depth-first search.     

Choosing a random spanning subtree: a case study

Broder⁽⁴⁾ has suggested a stochastic algorithm for generating a random spanning subtree of a graph. This paper studies this algorithm for a special class of graphs. A complete spectral decomposition of the associated Markov chain is given. The analysis available from this is compared to stopping-time techniques and purely geometric bounds on the second eigenvalue.     

一类Kn-同胚图色等价类的结构特征

用K（s，n）表示完全图Kn的一条边被长为s（s≥2）的路Ps＋1替代后得到的图．对n≥7，且n-2为素数，刻画了色等价类【K（s，n）]中图的结构特征，进一步，证明了任意任意n≥7，且n-2为素数，K（2，n），K（3，n）是色唯一的．     

Simulating tail asymptotics of a Markov chain

This paper develops a rare event simulation algorithm for a discrete-time Markov chain in the first orthant. The algorithm gives a very good estimate of the stationary distribution along one of the axes and it is shown to be efficient. A key idea is to study an associated time reversed Markov chain that starts at the rare event. We will apply the algorithm to a Markov chain related to a Jackson network with two stations.     

Efficient Bayesian Inference for Multivariate Probit Models With Sparse Inverse Correlation Matrices

We propose a Bayesian approach for inference in the multivariate probit model, taking into account the association structure between binary observations. We model the association through the correlation matrix of the latent Gaussian variables. Conditional independence is imposed by setting some off-diagonal elements of the inverse correlation matrix to zero and this sparsity structure is modeled using a decomposable graphical model. We propose an efficient Markov chain Monte Carlo algorithm relying on a parameter expansion scheme to sample from the resulting posterior distribution. This algorithm updates the correlation matrix within a simple Gibbs sampling framework and allows us to infer the correlation structure from the data, generalizing methods used for inference in decomposable Gaussian graphical models to multivariate binary observations. We demonstrate the performance of this model and of the Markov chain Monte Carlo algorithm on simulated and real datasets. This article has online supplementary materials.     

Random walks on graphs with a strong isoperimetric property

A random walk on a graph is a Markov chain whose state space consists of the vertices of the graph and where transitions are only allowed along the edges. We study (strongly) reversible random walks and characterize the class of graphs where then-step transition probabilities tend to zero exponentially fast (geometric ergodicity). These characterizations deal with an isoperimetric property, norm inequalities for certain associated operators, and eigenvalues of the Laplace operator. There is some (strong) similarity with the theory of (non)amenable groups.     

Markov chain mixing time on cycles

Mixing time quantifies the convergence speed of a Markov chain to the stationary distribution. It is an important quantity related to the performance of MCMC sampling. It is known that the mixing time of a reversible chain can be significantly improved by lifting, resulting in an irreversible chain, while changing the topology of the chain. We supplement this result by showing that if the connectivity graph of a Markov chain is a cycle, then there is an Ω(n²) lower bound for the mixing time. This is the same order of magnitude that is known for reversible chains on the cycle.     

Efficient algorithms to compute all articulation points of a permutation graph

Based on the geometric representation, an efficient algorithm is designed to find all articulation points of a permutation graph. The proposed algorithm takes onlyO(n logn) time andO(n) space, wheren represents the number of vertices. The proposed sequential algorithm can easily be implemented in parallel which takesO(logn) time andO(n) processors on an EREW PRAM. These are the first known algorithms for the problem on this class of graph.     

完全3-部图K_(1，10，n)的交叉数

在上世纪五十年代初,Zarankiewicz猜想完全2-部图Km,m(m≤n)的交叉数为[m/2][m-1/2][n/2][n-1/2](对任意实数x,[x]表示不超过x的最大整数),目前只证明了当m ≤ 6时,Zarankiewicz猜想是正确的.假定Zarankiewicz猜想对m=11的情形成立,本文确定完全3-部图K1,10,n的交叉数.     

Probabilistic compositional models: Solution of an equivalence problem

Probabilistic compositional models, similarly to graphical Markov models, are able to represent multidimensional probability distributions using factorization and closely related concept of conditional independence. Compositional models represent an algebraic alternative to the graphical models. The system of related conditional independencies is not encoded explicitly (e.g. using a graph) but it is hidden in a model structure itself. This paper provides answers to the question how to recognize whether two different compositional model structures are equivalent – i.e., whether they induce the same system of conditional independencies. Above that, it provides an easy way to convert one structure into an equivalent one in terms of some elementary operations on structures, closely related ability to generate all structures equivalent with a given one, and a unique representative of a class of equivalent structures.     

On the Markov equivalence of maximal ancestral graphs

Necessary and sufficient conditions for a maximal ancestral graph (MAnG) to be Markov equivalent to another MAnG and to a DAG are provided respectively. Also a polynomial-time algorithm for converting a MAnG into its equivalent DAG is given for the first time.