Consistent Model Selection of Discrete Bayesian Networks from Incomplete Data

A maximum likelihood based model selection of discrete Bayesian networks is considered. The model selection is performed through scoring function $S$, which, for a given network $G$ and $n$-sample $D_n$, is defined to be the maximum log-likelihood $l$ minus a penalization term ${\lambda }_{n}h$ proportional to network complexity $h(G)$, $$S(G\mathrm{\mid }{D}_n)=l(G\mathrm{\mid }{D}_n)-{\lambda }_{n}h(G)\mathrm{.}$$ The data is allowed to have missing values at random that has prompted, to improve the efficiency of estimation, a replacement of the standard log-likelihood with the sum of sample average node log-likelihoods. The latter avoids the exclusion of most partially missing data records and allows the comparison of models fitted to different samples. Provided that a discrete Bayesian network is identifiable for a given missing data distribution, we show that if the sequence ${\lambda }_n$ converges to zero at a slower rate than $n^{-1\mathrm{/}2}$ then the estimation is consistent. Moreover, we establish that BIC model selection (${\lambda }_n=0.5\log (n)\mathrm{/}n$) applied to the node-average log-likelihood is in general not consistent. This is in contrast to the complete data case where BIC is known to be consistent. The conclusions are confirmed by numerical examples.