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Consistent model selection of discrete Bayesian networks from incomplete data. (English) Zbl 1336.62087
Summary: A maximum likelihood based model selection of discrete Bayesian networks is considered. The structure learning is performed by employing a scoring function $$S$$, which, for a given network $$G$$ and $$n$$-sample $$D_{n}$$, is defined as the maximum marginal log-likelihood $$l$$ minus a penalization term $$\lambda_{n}h$$ proportional to network complexity $$h(G)$$, $S(G|D_{n})=l(G|D_{n})-\lambda_{n}h(G).$ An available case analysis is developed with the standard log-likelihood replaced by the sum of sample average node log-likelihoods. The approach utilizes partially missing data records and allows for comparison of models fitted to different samples. In missing completely at random settings the estimation is shown to be consistent if and only if the sequence $$\lambda_{n}$$ converges to zero at a slower than $$n^{-{1/2}}$$ rate. In particular, the BIC model selection $$(\lambda_{n}=0.5\log(n)/n)$$ applied to the node-average log-likelihood is shown to be inconsistent in general. This is in contrast to the complete data case when BIC is known to be consistent. The conclusions are confirmed by numerical experiments.
##### MSC:
 62F12 Asymptotic properties of parametric estimators 62H12 Estimation in multivariate analysis 62F15 Bayesian inference 62-09 Graphical methods in statistics (MSC2010)
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##### References:
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