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 $h(G)$, $$ S(G|D_n) = l(G|D_n) - \lambda_n h(G). $$ 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 $n^{-{1/2}}$ then the estimation is consistent. Moreover, we establish that BIC model selection ($\lambda_n=0.5\log(n)/n$) 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. 
Nikolay H. Balov
math.ST stat.TH