Electronic Journal of Statistics

Consistent model selection of discrete Bayesian networks from incomplete data

Nikolay Balov

Full-text: Open access

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


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