On Testing for Parameters in Ising Models

Abstract

We consider testing for the parameters of Ferromagnetic Ising models. While testing for the presence of possibly sparse magnetizations, we provide a general lower bound of minimax separation rates which yields sharp results in high temperature regimes. Our matching upper bounds are adaptive over both underlying dependence graph and temperature parameter. Moreover our results include the nearest neighbor model on lattices, the sparse Erdös-Rényi random graphs, and regular rooted trees -right up to the critical parameter in the high temperature regime. We also provide parallel results for the entire low temperature regime in nearest neighbor model on latticeshowever in the plus boundary pure phase. Our results for the nearest neighbor model crucially depends on finite volume analogues of correlation decay property for both high and low temperature regimes -the derivation of which borrows crucial ideas from FK-percolation theory and might be of independent interest. Finally, we also derive lower bounds for estimation and testing rates in two parameter Ising models -which turn out to be optimal according to several recent results in this area.

Detecting Structured Signals in Ising Models

“…We note that at the critical point β = β c (d, L) we do not expect Theorem 8 to hold, and the detection boundary to be lower (see the discussion in Mukherjee and Ray (2019) for heuristics in this regard).…”

Fluctuations in Mean-Field Ising models

“…Here, part (a) follows by invoking [13, Lemma 3.2] and (b) can be obtained by making minor adjustments in the proof of [32,Lemma 1].…”

“…The Ising model is a discrete Markov random field which was initially introduced as a mathematical model of ferromagnetism in statistical physics, and has received extensive attention in Probability and Statistics (c.f. [1,4,5,10,14,16,17,20,21,23,24,25,26,27,29,31,32,34,35] and references therein). The model can be described by the following probability mass function in σ := (σ 1 ,…”

Sharp Signal Detection Under Ferromagnetic Ising Models

“…As an immediate interesting question pertains to the Ising model on lattices and figuring out the exact detection thresholds at the critical temperature to complete the narrative of precise benefit of critical dependence in this model. As was discussed in [41] this might require new ideas. Moreover, even for non-critical temperatures it remains to explore the multi-scale procedures for adaptive testing of thick clusters for Ising models over lattices (see e.g.…”

“…where the last line uses the bound max sB 2 z 2 , sB|z| 3 + sz 4 ≤ sB|z| for all |z| ≤ δ. We now bound each of the terms in the RHS of (41). The denominator in the RHS of (41) by a change of variable equals…”