Parameter Space Compression Underlies Emergent Theories and Predictive Models

Information Physics Multiparameter models, which can emerge in biology and other disciplines, are often sensitive to only a small number of parameters and robust to changes in the rest; approaches from information theory can be used to distinguish between the two parameter groups. In physics, on the other hand, one does not need to know the details at smaller length and time scales in order to understand the behavior on large scales. This hierarchy has been recognized for a long time and formalized within the renormalization group (RG) approach. Machta et al. (p. 604) explored the connection between two scales by using an information-theoretical approach based on the Fisher Information Matrix to analyze two commonly used physics models—diffusion in one dimension and the Ising model of magnetism—as the time and length scales, respectively, were progressively coarsened. The expected “stiff” parameters emerged, in agreement with RG intuition. An information-theoretical approach is used to distinguish the important parameters in two archetypical physics models. The microscopically complicated real world exhibits behavior that often yields to simple yet quantitatively accurate descriptions. Predictions are possible despite large uncertainties in microscopic parameters, both in physics and in multiparameter models in other areas of science. We connect the two by analyzing parameter sensitivities in a prototypical continuum theory (diffusion) and at a self-similar critical point (the Ising model). We trace the emergence of an effective theory for long-scale observables to a compression of the parameter space quantified by the eigenvalues of the Fisher Information Matrix. A similar compression appears ubiquitously in models taken from diverse areas of science, suggesting that the parameter space structure underlying effective continuum and universal theories in physics also permits predictive modeling more generally.