  We discuss a method of parameter reduction in complex models known as the
Manifold Boundary Approximation Method (MBAM). This approach, based on a
geometric interpretation of statistics, maps the model reduction problem to a
geometric approximation problem. It operates iteratively, removing one
parameter at a time, by approximating a high-dimension, but thin manifold by
its boundary. Although the method makes no explicit assumption about the
functional form of the model, it does require that the model manifold exhibit a
hierarchy of boundaries, i.e., faces, edges, corners, hyper-corners, etc. We
empirically show that a variety of model classes have this curious feature,
making them amenable to MBAM. These model classes include models composed of
elementary functions (e.g., rational functions, exponentials, and partition
functions), a variety of dynamical system (e.g., chemical and biochemical
kinetics, Linear Time Invariant (LTI) systems, and compartment models), network
models (e.g., Bayesian networks, Markov chains, artificial neural networks, and
Markov random fields), log-linear probability distributions, and models with
symmetries. We discuss how MBAM recovers many common approximation methods for
each model class and discuss potential pitfalls and limitations.
