** by Jukka
Corander, Karin Dahmström & Per Dahmström**

** Research
Report 1998:8**

Department of Statistics, Stockholm University, S-106 91 Stockholm, Sweden

**Abstract**

Maximum likelihood estimation of Markov graph parameters is considered. An iterative technique using an expansion of the expected values of sufficient statistics in a Markov graph in terms of cumulants is introduced. Efficient starting values for parameter estimates are shown to be obtainable from the cumulants of graph statistics in uniform random graphs. A Markov chain Monte Carlo method is used to generate samples of Markov graphs with fixed parameter values at successive iteration steps. Complete enumeration is used in a small graph to show that the iterative estimation technique performs satisfactorily compared to the exact maximum likelihood estimates. Properties of the maximum likelihood estimators and pseudolikelihood estimators, suggested earlier in the statistical literature, are investigated in small and large graphs. Our results for an undirected graph with a univariate sufficient statistic suggest that in graphs with up to approximately 40 vertices the maximum likelihood estimator is uniformly better over such parts of the parameter space where the model behavior is not degenerate. In larger graphs with 40-100 nodes, the two estimators seem to be nearly equivalent. However, the pseudolikelihood estimators are shown to vary with different graphs with the same value of the sufficient statistic.

**Key words:** Cumulants,
Markov Chain Monte Carlo, Markov Graph, Maximum Likelihood Estimation,
Pseudolikelihood Estimation.

*Last update: 990916/CE*