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**Concept : **Markov Equivalence in Bayesian Networkshttps://www.cs.ru.nl › P.Lucas › markoveq

**Counting Markov Equivalence Classes for DAG models on Trees(****arXiv****)**

**Author :** Adityanarayanan Radhakrishnan, Liam Solus, Caroline Uhler

**Abstract :**DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) G. Such models are used to model complex cause-effect systems across a variety of research fields. From observational data alone, a DAG model G is only recoverable up to Markov equivalence. Combinatorially, two DAGs are Markov equivalent if and only if they have the same underlying undirected graph (i.e. skeleton) and the same set of the induced subDAGs i→j←k, known as immoralities. Hence it is of interest to study the number and size of Markov equivalence classes (MECs). In a recent paper, the authors introduced a pair of generating functions that enumerate the number of MECs on a fixed skeleton by number of immoralities and by class size, and they studied the complexity of computing these functions. In this paper, we lay the foundation for studying these generating functions by analyzing their structure for trees and other closely related graphs. We describe these polynomials for some important families of graphs including paths, stars, cycles, spider graphs, caterpillars, and complete binary trees. In doing so, we recover important connections to independence polynomials, and extend some classical identities that hold for Fibonacci numbers. We also provide tight lower and upper bounds for the number and size of MECs on any tree. Finally, we use computational methods to show that the number and distribution of high degree nodes in a triangle-free graph dictates the number and size of MECs.

**2.Minimal enumeration of all possible total effects in a Markov equivalence class (****arXiv****)**

**Author : **F. Richard Guo, Emilija Perković

**Abstract : **In observational studies, when a total causal effect of interest is not identified, the set of all possible effects can be reported instead. This typically occurs when the underlying causal DAG is only known up to a Markov equivalence class, or a refinement thereof due to background knowledge. As such, the class of possible causal DAGs is represented by a maximally oriented partially directed acyclic graph (MPDAG), which contains both directed and undirected edges. We characterize the minimal additional edge orientations required to identify a given total effect. A recursive algorithm is then developed to enumerate subclasses of DAGs, such that the total effect in each subclass is identified as a distinct functional of the observed distribution. This resolves an issue with existing methods, which often report possible total effects with duplicates, namely those that are numerically distinct due to sampling variability but are in fact causally identical

**3.Counting Markov Equivalence Classes for DAG models on Trees (****arXiv****)**

**Author : **Adityanarayanan Radhakrishnan, Liam Solus, Caroline Uhler

**Abstract :** DAG models are statistical models satisfying a collection of conditional independence relations encoded by the nonedges of a directed acyclic graph (DAG) G. Such models are used to model complex cause-effect systems across a variety of research fields. From observational data alone, a DAG model G is only recoverable up to Markov equivalence. Combinatorially, two DAGs are Markov equivalent if and only if they have the same underlying undirected graph (i.e. skeleton) and the same set of the induced subDAGs i→j←k, known as immoralities. Hence it is of interest to study the number and size of Markov equivalence classes (MECs). In a recent paper, the authors introduced a pair of generating functions that enumerate the number of MECs on a fixed skeleton by number of immoralities and by class size, and they studied the complexity of computing these functions. In this paper, we lay the foundation for studying these generating functions by analyzing their structure for trees and other closely related graphs. We describe these polynomials for some important families of graphs including paths, stars, cycles, spider graphs, caterpillars, and complete binary trees. In doing so, we recover important connections to independence polynomials, and extend some classical identities that hold for Fibonacci numbers. We also provide tight lower and upper bounds for the number and size of MECs on any tree. Finally, we use computational methods to show that the number and distribution of high degree nodes in a triangle-free graph dictates the number and size of MECs.

**4. Supplement to “Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs”(****arXiv****)**

**Author :** Yangbo He, Jinzhu Jia, Bin Yu

**Abstract : **This supplementary material includes three parts: some preliminary results, four examples, an experiment, three new algorithms, and all proofs of the results in the paper “Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs”.

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