Dag Sonntag

Approximate Counting of Graphical Models via MCMC Revisited

We apply Markov chain Monte Carlo (MCMC) sampling to approximately calculate some quantities, and discuss their implications for learning directed and acyclic graphs (DAGs) from data. Specifically, we calculate the approximate ratio of essential graphs (EGs) to DAGs for up to 31 nodes. Our ratios suggest that the average Markov equivalence class is small. We show that a large majority of the classes seem to have a size that is close to the average size. This suggests that one should not expect more than a moderate gain in efficiency when searching the space of EGs instead of the space of DAGs. We also calculate the approximate ratio of connected EGs to connected DAGs, of connected EGs to EGs, and of connected DAGs to DAGs. These new ratios are interesting because, as we will see, the DAG or EG learnt from some given data is likely to be connected. Furthermore, we prove that the latter ratio is asymptotically 1. Finally, we calculate the approximate ratio of EGs to largest chain graphs for up to 25 nodes. Our ratios suggest that Lauritzen–Wermuth–Frydenberg chain graphs are considerably more expressive than DAGs. We also report similar approximate ratios and conclusions for multivariate regression chain graphs.

Chain graph interpretations and their relations revisited

In this paper we study how different theoretical concepts of Bayesian networks have been extended to chain graphs. Today there exist mainly three different interpretations of chain graphs in the li ...

Chain graph interpretations and their relations

This paper deals with different chain graph interpretations and the relations between them in terms of representable independence models. Specifically, we study the Lauritzen-Wermuth-Frydenberg, Andersson-Madigan-Pearlman and multivariate regression interpretations and present the necessary and sufficient conditions for when a chain graph of one interpretation can be perfectly translated into a chain graph of another interpretation. Moreover we also present a feasible split for the Andersson-Madigan-Pearlman interpretation with similar features as the feasible splits presented for the other two interpretations.

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. We study the expressiveness of the different chain graph interpretations in terms of representable independence models.We show how an MCMC sampling algorithm can be used to sample chain graph models.We show that chain graphs can represent exponentially many more independence models compared to Bayesian networks.We show (numerically) that the average number of chain graphs per chain graph model appears to converge.We show that AMP and MVR chain graphs are more expressive than LWF chain graphs.

Chain Graphs and Gene Networks

Chain graphs are graphs with possibly directed and undirected edges, and no semidirected cycle. They have been extensively studied as a formalism to represent probabilistic independence models, because they can model symmetric and asymmetric relationships between random variables. This allows chain graphs to represent a wider range of systems than Bayesian networks. This in turn allows for a more correct representation of systems that may contain both causal and non-causal relationships between its variables, like for example biological systems. In this chapter we give an overview of how to use chain graphs and what research exists on them today. We also give examples on how chain graphs can be used to model advanced systems, that are not well understood, such as gene networks.

On Expressiveness of the AMP Chain Graph Interpretation

In this paper we study the expressiveness of the Andersson-Madigan-Perlman interpretation of chain graphs. It is well known that all independence models that can be represented by Bayesian networks also can be perfectly represented by chain graphs of the Andersson-Madigan-Perlman interpretation but it has so far not been studied how much more expressive this second class of models is. In this paper we calculate the exact number of representable independence models for the two classes, and the ratio between them, for up to five nodes. For more than five nodes the explosive growth of chain graph models does however make such enumeration infeasible. Hence we instead present, and prove the correctness of, a Markov chain Monte Carlo approach for sampling chain graph models uniformly for the Andersson-Madigan-Perlman interpretation. This allows us to approximate the ratio between the numbers of independence models representable by the two classes as well as the average number of chain graphs per chain graph model for up to 20 nodes. The results show that the ratio between the numbers of representable independence models for the two classes grows exponentially as the number of nodes increases. This indicates that only a very small fraction of all independence models representable by chain graphs of the Andersson-Madigan-Perlman interpretation also can be represented by Bayesian networks.