On the criticality of inferred models

Abstract

Advanced inference techniques allow one to reconstruct a pattern of interaction from highdimensional data sets, from probing simultaneously thousands of units of extendedsystems—such as cells, neural tissues and financial markets. We focus here on thestatistical properties of inferred models and argue that inference procedures are likely toyield models which are close to singular values of parameters, akin to critical points inphysics where phase transitions occur. These are points where the response of physicalsystems to external perturbations, as measured by the susceptibility, is very largeand diverges in the limit of infinite size. We show that the reparameterizationinvariant metrics in the space of probability distributions of these models (the Fisherinformation) are directly related to the susceptibility of the inferred model. As a result,distinguishable models tend to accumulate close to critical points, where the susceptibilitydiverges in infinite systems. This region is the one where the estimate of inferredparameters is most stable. In order to illustrate these points, we discuss inferenceof interacting point processes with application to financial data and show thatsensible choices of observation time scales naturally yield models which are close tocriticality.