Information topology identifies emergent model classes

Abstract

We develop a language for describing the relationship among observations, mathematical models, and the underlying principles from which they are derived. Using Information Geometry, we consider geometric properties of statistical models for different observations. As observations are varied, the model manifold may be stretched, compressed, or even collapsed. Observations that preserve the structural identifiability of the parameters also preserve certain topological features (such as edges and corners) that characterize the model's underlying physical principles. We introduce Information Topology in analogy with information geometry as characterizing the "abstract model" of which statistical models are realizations. Observations that change the topology, i.e., "manifold collapse," require a modification of the abstract model in order to construct identifiable statistical models. Often, the essential topological feature is a hierarchical structure of boundaries (faces, edges, corners, etc.) which we represent as a hierarchical graph known as a Hasse diagram. Low-dimensional elements of this diagram are simple models that describe the dominant behavioral modes, what we call emergent model classes. Observations that preserve the Hasse diagram are diffeomorphically related and form a group, the collection of which form a partially ordered set. All possible observations have a semi-group structure. For hierarchical models, we consider how the topology of simple models is embedded in that of larger models. When emergent model classes are unstable to the introduction of new parameters, we classify the new parameters as relevant. Conversely, the emergent model classes are stable to the introduction of irrelevant parameters. In this way, information topology provides a general language for exploring representations of physical systems and their relationships to observations.