An empirical study of the maximum pseudo-likelihood for discrete Markov random fields.
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In this text we will look at two parameter estimation methods for Markov random fields on a lattice. They are maximum pseudo-likelihood estimation and maximum general pseudo-likelihood estimation, which we abbreviate MPLE and MGPLE. The idea behind them is that by maximizing an approximation of the likelihood function, we avoid computing cumbersome normalising constants. In MPLE we maximize the product of the conditional distributions for each variable given all the other variables. In MGPLE we use a compromise between pseudo-likelihood and the likelihood function as the approximation. We evaluate and compare the performance of MPLE and MGPLE on three different spatial models, which we have generated observations of. We are specially interested to see what happens with the quality of the estimates when the number of observations increases. The models we use are the Ising model, the extended Ising model and the Sisim model. All the random variables in the models have two possible states, black or white. For the Ising and extended Ising model we have one and three parameters respectively. For Sisim we have $13$ parameters. The quality of both methods get better when the number of observations grow, and MGPLE gives better results than MPLE. However certain parameter combinations of the extended Ising model give worse results.