## Electronic Journal of Statistics

- Electron. J. Statist.
- Volume 7 (2013), 1047-1077.

### Consistent model selection of discrete Bayesian networks from incomplete data

#### Abstract

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.

#### Article information

**Source**

Electron. J. Statist., Volume 7 (2013), 1047-1077.

**Dates**

First available in Project Euclid: 15 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ejs/1366031050

**Digital Object Identifier**

doi:10.1214/13-EJS802

**Mathematical Reviews number (MathSciNet)**

MR3044509

**Zentralblatt MATH identifier**

1336.62087

**Subjects**

Primary: 62F12: Asymptotic properties of estimators

Secondary: 62H12: Estimation

**Keywords**

Bayesian networks categorical data model selection penalized maximum likelihood missing completely at random

#### Citation

Balov, Nikolay. Consistent model selection of discrete Bayesian networks from incomplete data. Electron. J. Statist. 7 (2013), 1047--1077. doi:10.1214/13-EJS802. https://projecteuclid.org/euclid.ejs/1366031050