We extend Andersson-Madigan-Perlman chain graphs by (i) relaxing the semidirected acyclity constraint so that only directed cycles are forbidden, and (ii) allowing up to two edges between any pair of nodes. We introduce global, and ordered local and pairwise Markov properties for the new models. We show the equivalence of these properties for strictly positive probability distributions. We also show that when the random variables are continuous, the new models can be interpreted as systems of structural equations with correlated errors. This enables us to adapt Pearl's do-calculus to them. Finally, we describe an exact algorithm for learning the new models from observational and interventional data via answer set programming.

An intervention may have an effect on units other than those to which the intervention was administered. This phenomenon is called interference and it usually goes unmodeled. In this paper, we propose to combine Lauritzen-Wermuth-Frydenberg and Andersson-Madigan-Perlman chain graphs to create a new class of causal models that can represent interference relationships. Specifically, we define the new class of models, introduce global and local and pairwise Markov properties for them, and prove their equivalence.

In this paper we discuss four problems regarding Markov equivalences for subclasses of loopless mixed graphs. We classify these four problems as finding conditions for internal Markov equivalence, which is Markov equivalence within a subclass, for external Markov equivalence, which is Markov equivalence between subclasses, for representational Markov equivalence, which is the possibility of a graph from a subclass being Markov equivalent to a graph from another subclass, and finding algorithms to generate a graph from a certain subclass that is Markov equivalent to a given graph. We particularly focus on the class of maximal ancestral graphs and its subclasses, namely regression graphs, bidirected graphs, undirected graphs, and directed acyclic graphs, and present novel results for representational Markov equivalence and algorithms.

Ali, Ayesha R., Richardson, Thomas S.

Ancestral graphs are a class of graphs that encode conditional independence relations arising in DAG models with latent and selection variables, corresponding to marginalization and conditioning. However, for any ancestral graph, there may be several other graphs to which it is Markov equivalent. We introduce a simple representation of a Markov equivalence class of ancestral graphs, thereby facilitating model search. \ More specifically, we define a join operation on ancestral graphs which will associate a unique graph with a Markov equivalence class. We also extend the separation criterion for ancestral graphs (which is an extension of d-separation) and provide a proof of the pairwise Markov property for joined ancestral graphs.

The paper formalizes constraint-based structure learning of the "true" causal graph from observed data when unobserved variables are also existent. We define a "generic" structure learning algorithm, which provides conditions that, under the faithfulness assumption, the output of all known exact algorithms in the literature must satisfy, and which outputs graphs that are Markov equivalent to the causal graph. More importantly, we provide clear assumptions, weaker than faithfulness, under which the same generic algorithm outputs Markov equivalent graphs to the causal graph. We provide the theory for the general class of models under the assumption that the distribution is Markovian to the true causal graph, and we specialize the definitions and results for structural causal models.