Smoothness of Marginal Log-Linear Parameterizations

This chapter gives a detailed discussion of the smoothness of various parameterizations of discrete distributions. The concept of smoothness of a parameterization was introduced in the previous chapter. The exponential family framework to investigate smoothness is used, and log-linear, mean value, and mixed parameterizations are considered first, and smoothness is proved using the full rank design matrices developed in the previous chapter. The results are somewhat different but closely related for general and for probability distributions. The essential difference is that in the case of probability distributions, to obtain smoothness, the overall effect has to be omitted from among the parameters, but normalization needs to be applied. These smoothness results are then used to prove the smoothness of marginal log-linear parameterizations, through a series of reparameterizations. This prepares the discussion of marginal log-linear models in the next chapter.