Changing Parameters by Partial Mappings

Changes between different sets of parameters are often needed in multivariate statistical modeling, such as transformations within linear regression or in exponential models. There may, for instance, be specific inference questions based on subject matter interpretations, alternative well-fitting constrained models, compatibility judgements of seemingly distinct constrained models, or different reference priors under alternative parameterizations. We introduce and discuss a partial mapping, called partial replication, and relate it to a more complex mapping, called partial inversion. Both operations are used to decompose matrix operations, to explain recursion relations among sets of linear parameters, to change between different types of linear models, to approximate maximum-likelihood estimates in exponential family models under independence constraints, and to switch partially between sets of canonical and moment parameters in exponential family distributions or between sets of corresponding maximum-likelihood estimates.