A Useful Algebraic System of Statistical Models

Abstract: This paper proposes a single form for statistical models that accommodates a broad range of models, from ordinary least squares to agent-based microsimulations. The definition makes it almost trivial to define morphisms to transform and combine existing models to produce new models. It offers a unified means of expressing and implementing methods that are typically given disparate treatment in the literature, including transformations via differentiable functions, Bayesian updating, multi-level and other types of composed models, Markov chain Monte Carlo, and several other common procedures. It especially offers benefit to simulation-type models, because of the value in being able to build complex models from simple parts, easily calculate robustness measures for simulation statistics and, where appropriate, test hypotheses. Running examples will be given using Apophenia, an open-source software library based on the model form and transformations described here.