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.