Visualizing theory space: Isometric embedding of probabilistic predictions, from the Ising model to the cosmic microwave background

Abstract

We develop an intensive embedding for visualizing the space of all predictions for probabalistic models, using replica theory. Our embedding is isometric (preserves the distinguishability between models) and faithful (yields low-dimensional visualizations of models with simple emergent behavior). We apply our intensive embedding to the Ising model of statistical mechanics and the $\Lambda$CDM model applied to cosmic microwave background radiation. It provides an intuitive, quantitative visualization applicable to renormalization-group calculations and optimal experimental design.