Algebraic geometry of Poisson regression

Designing experiments for generalized linear models is difficultbecause optimal designs depend on unknown parameters.  Here weinvestigate local optimality.  We propose to study for a given designits region of optimality in parameter space.  Often these regions aresemi-algebraic and feature interesting symmetries.  We demonstratethis with the Rasch Poisson counts model.  For any given interactionorder between the explanatory variables we give a characterization ofthe regions of optimality of a special saturated design. This extendsknown results from the case of no interaction.  We also give analgebraic and geometric perspective on optimality of experimentaldesigns for the Rasch Poisson counts model using polyhedral andspectrahedral geometry.