Adaptive Inferential Method for Monotone Graph Invariants

TL;DR

This paper introduces a new inferential framework for estimating and testing monotone graph invariants in undirected graphical models, requiring weaker assumptions than full graph recovery.

Contribution

It proposes a skip-down algorithm for hypothesis testing and confidence intervals of graph invariants, with proven optimality and adaptivity.

Findings

The method provides valid confidence intervals for graph invariants.

Intervals are proven to be optimal and adaptive to signal strength.

Numerical results demonstrate effectiveness on synthetic and real data.

Abstract

We consider the problem of undirected graphical model inference. In many applications, instead of perfectly recovering the unknown graph structure, a more realistic goal is to infer some graph invariants (e.g., the maximum degree, the number of connected subgraphs, the number of isolated nodes). In this paper, we propose a new inferential framework for testing nested multiple hypotheses and constructing confidence intervals of the unknown graph invariants under undirected graphical models. Compared to perfect graph recovery, our methods require significantly weaker conditions. This paper makes two major contributions: (i) Methodologically, for testing nested multiple hypotheses, we propose a skip-down algorithm on the whole family of monotone graph invariants (The invariants which are non-decreasing under addition of edges). We further show that the same skip-down algorithm also provides valid confidence intervals for the targeted graph invariants. (ii) Theoretically, we prove that the length of the obtained confidence intervals are optimal and adaptive to the unknown signal strength. We also prove generic lower bounds for the confidence interval length for various invariants. Numerical results on both synthetic simulations and a brain imaging dataset are provided to illustrate the usefulness of the proposed method.