On the criticality of inferred models

TL;DR

This paper explores how advanced inference methods tend to produce models near phase transitions, highlighting the relationship between model susceptibility, stability, and criticality in high-dimensional data analysis.

Contribution

It demonstrates that inferred models often cluster near critical points due to the properties of Fisher Information and susceptibility, with practical illustration in financial data inference.

Findings

Models inferred from high-dimensional data tend to be near phase transitions.

Susceptibility diverges at critical points, affecting model stability.

Choice of observation time-scale influences proximity to criticality.

Abstract

Advanced inference techniques allow one to reconstruct the pattern of interaction from high dimensional data sets. We focus here on the statistical properties of inferred models and argue that inference procedures are likely to yield models which are close to a phase transition. On one side, we show that the reparameterization invariant metrics in the space of probability distributions of these models (the Fisher Information) is directly related to the model's susceptibility. As a result, distinguishable models tend to accumulate close to critical points, where the susceptibility diverges in infinite systems. On the other, this region is the one where the estimate of inferred parameters is most stable. In order to illustrate these points, we discuss inference of interacting point processes with application to financial data and show that sensible choices of observation time-scales naturally yield models which are close to criticality.