Models that are constructed from conditionally specified distributions are often applied to data sets that possess a spatial structure, even data sets with complex dependence structures. These conditionally specified distributions specify the distribution of a value at a location given the values at all other locations. If the value is dependent only on values at a subset of locations, called a neighborhood, then the resulting joint probability measure is referred to as a Markov random field (MRF) model. When the conditionally specified distributions are exponential family distributions, several results are available; hence, there has been much interest in Markov random field models that have been constructed with Gaussian, Poisson, and binomial distributions specified as the conditionally specified distributions. One exponential family distribution that has not been subject to much interest in the area of MRF models as of yet is the multinomial distribution, even though the multinomial distribution is an extension of the binomial distribution. Consequently, in this paper, we construct a MRF model with multinomial conditional distributions and then study the behavior of this model regarding, for example, symmetry of the model, variances and covariances of the conditional expectations and marginal variances and covariances. Finally, this model is applied to a data set with spatial structure.