Abstract. We apply 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.
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 literature. These are the Lauritzen-Wermuth-Frydenberg, the Andersson-Madigan-Perlman and the multivariate regression interpretations. The different chain graph interpretations have been studied independently and over time different theoretical concepts have been extended from Bayesian networks to also work for the different chain graph interpretations. This has however led to confusion regarding what concepts exist for what interpretation.In this article we do therefore study some of these concepts and how they have been extended to chain graphs as well as what results have been achieved so far. More importantly we do also identify when the concepts have not been extended and contribute within these areas. Specifically we study the following theoretical concepts: Unique representations of independence models, the split and merging operators, the conditions for when an independence model representable by one chain graph interpretation can be represented by another chain graph interpretation and finally the extension of Meek's conjecture to chain graphs. With our new results we give a coherent overview of how each of these concepts is extended for each of the different chain graph interpretations.
Abstract. 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.
Probabilistic graphical models are today one of the most well used architectures for modelling and reasoning about knowledge with uncertainty. The most widely used subclass of these models is Bayesian networks that has found a wide range of applications both in industry and research. Bayesian networks do however have a major limitation which is that only asymmetric relationships, namely cause and effect relationships, can be modelled between its variables. A class of probabilistic graphical models that has tried to solve this shortcoming is chain graphs. It is achieved by including two types of edges in the models, representing both symmetric and asymmetric relationships between the connected variables. This allows for a wider range of independence models to be modelled. Depending on how the second edge is interpreted this has also given rise to different chain graph interpretations.Although chain graphs were first presented in the late eighties the field has been relatively dormant and most research has been focused on Bayesian networks. This was until recently when chain graphs got renewed interest. The research on chain graphs has thereafter extended many of the ideas from Bayesian networks and in this thesis we study what this new surge of research has been focused on and what results have been achieved. Moreover we do also discuss what areas that we think are most important to focus on in further research.
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