On the Information-theoretic Limits of Graphical Model Selection for Gaussian Time Series

TL;DR

This paper establishes fundamental information-theoretic limits on the sample size needed to accurately infer the conditional independence graph of Gaussian time series, highlighting the minimal data requirements for reliable graphical model selection.

Contribution

It derives a universal lower bound on the error probability for CIG inference and shows that this bound is achievable for sparse processes with smooth spectral densities.

Findings

Lower bound on sample size for reliable CIG selection

Achievability of bounds for sparse processes with smooth SDM

No parametric assumptions, only spectral density smoothness

Abstract

We consider the problem of inferring the conditional independence graph (CIG) of a multivariate stationary dicrete-time Gaussian random process based on a finite length observation. Using information-theoretic methods, we derive a lower bound on the error probability of any learning scheme for the underlying process CIG. This bound, in turn, yields a minimum required sample-size which is necessary for any algorithm regardless of its computational complexity, to reliably select the true underlying CIG. Furthermore, by analysis of a simple selection scheme, we show that the information-theoretic limits can be achieved for a subclass of processes having sparse CIG. We do not assume a parametric model for the observed process, but require it to have a sufficiently smooth spectral density matrix (SDM).