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Faster algorithms for Markov equivalence
Z.Hu, R.Evans
Accepted at the Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI-2020), 2020
Maximal ancestral graphs (MAGs) have many desirable properties; in particular they can fully describe conditional independences from directed acyclic graphs (DAGs) in the presence of latent and selection variables. However, different MAGs may encode the same conditional independences, and are said to be \emph{Markov equivalent}. Thus identifying necessary and sufficient conditions for equivalence is essential for structure learning. Several criteria for this already exist, but in this paper we give a new non-parametric characterization in terms of the heads and tails that arise in the parameterization for discrete models. We also provide a polynomial time algorithm (O(ne2), where n and e are the number of vertices and edges respectively) to verify equivalence. Moreover, we extend our criterion to ADMGs and summary graphs and propose an algorithm that converts an ADMG or summary graph to an equivalent MAG in polynomial time (O(n2e)). Hence by combining both algorithms, we can also verify equivalence between two summary graphs or ADMGs.
Towards standard imsets for maximal ancestral graphs
Z.Hu, R.Evans
submitted to Bernoulli, 2022
The imsets of \citet{studeny2006probabilistic} are an algebraic method for representing conditional independence models. They have many attractive properties when applied to such models, and they are particularly nice for working with directed acyclic graph (DAG) models. In particular, the standard imset for a DAG is in one-to-one correspondence with the independences it induces, and hence is a label for its Markov equivalence class. We present a proposed extension to standard imsets for maximal ancestral graph (MAG) models, using the parameterizing set representation of \citet{hu2020faster}. By construction, our imset also represents the Markov equivalence class of the MAG. We show that for many such graphs our proposed imset is \emph{perfectly Markovian} with respect to the graph thus providing a scoring criteria by measuring the discrepancy for a list of independences that define the model; this gives an alternative to the usual BIC score. Unfortunately, for some models the representation does not work, and in certain cases does not represent any independences at all. We prove that it does work for \emph{simple} MAGs where there are only heads of size less than three, as well as for a large class of purely bidirected models. We also show that of independence models that do represent the MAG, the one we give is the simplest possible, in a manner we make precise. Further we refine the ordered local Markov property, which relates to finding the best imsets representing general MAGs.
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Teaching experience 1
Undergraduate course, University 1, Department, 2014
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Teaching experience 2
Workshop, University 1, Department, 2015
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