Changing sets of parameters by partial mappings

Abstract: Changes between different sets of parameters are needed in multivariate statistical modeling. 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, and different reference priors under alternative parameterizations. We introduce and discuss a partial mapping, called partial replication, and derive from it a more complex mapping, called partial inversion. Both operations are shown to be tools for decomposing matrix operations, for explaining recursion relations among sets of linear parameters, for deriving effects of omitted variables in alternative model formulations, for changing between different types of linear chain graph models, for approximating maximum-likelihood estimates in exponential family models under independence constraints, and for switching partially between sets of canonical and moment parameters in distributions of the exponential family or between sets of corresponding maximum-likelihood estimates.