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On Model Selection, Bayesian Networks, and the Fisher Information Integral

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Abstract

We study BIC-like model selection criteria and in particular, their refinements that include a constant term involving the Fisher information matrix. We perform numerical simulations that enable increasingly accurate approximation of this constant in the case of Bayesian networks. We observe that for complex Bayesian network models, the constant term is a negative number with a very large absolute value that dominates the other terms for small and moderate sample sizes. For networks with a fixed number of parameters, d, the leading term in the complexity penalty, which is proportional to d, is the same. However, as we show, the constant term can vary significantly depending on the network structure even if the number of parameters is fixed. Based on our experiments, we conjecture that the distribution of the nodes’ outdegree is a key factor. Furthermore, we demonstrate that the constant term can have a dramatic effect on model selection performance for small sample sizes.

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Notes

  1. We denote the binary (base-2) logarithm by \(\log \) and the natural logarithm by \(\ln \).

  2. Based on the observations in Sect. 4, which make it clear that Bayesian networks with a fixed number of parameters can have large differences in FII values, we evaluate the constants for individual networks instead of using the same complexity penalty for all networks with a fixed number of parameters.

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Acknowledgements

An earlier version of this paper was presented at the Second Workshop on Advanced Methodologies for Bayesian Networks (AMBN 2015) in Yokohama. The authors thank the anonymous reviewers for insightful comments and suggestions and the organizers of AMBN-2015 for their invitation to submit this work to this special issue. This work was funded in part by the Academy of Finland (Centre-of-Excellence COIN).

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Authors and Affiliations

  1. Helsinki Institute for Information Technology HIIT, Department of Computer Science, University of Helsinki, Gustaf Hällströmin katu 2b, 00014, Helsinki, Finland

    Yuan Zou & Teemu Roos

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  1. Yuan Zou
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  2. Teemu Roos
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Correspondence to Yuan Zou.

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Zou, Y., Roos, T. On Model Selection, Bayesian Networks, and the Fisher Information Integral. New Gener. Comput. 35, 5–27 (2017). https://doi.org/10.1007/s00354-016-0002-y

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