Consider a graph with a set of vertices and oriented edges connecting pairs of vertices. Each vertex is associated with a random variable and these are assumed to be independent. In this setting, suppose we wish to solve the following hypothesis testing problem: under the null, the random variables have common distribution N (0, 1) while under the alternative, there is an unknown path along which random variables have distribution N (µ, 1), µ > 0, and distribution N (0, 1) away from it. For which values of the mean shift µ can one reliably detect and for which values is this impossible? This paper develops detection thresholds for two types of common graphs which exhibit a different behavior. The first is the usual regular lattice with vertices of the form {(i, j) : 0 ≤ i, −i ≤ j ≤ i and j has the parity of i} and oriented edges (i, j) → (i+1, j +s) where s = ±1. We show that for paths of length m starting at the origin, the hypotheses become distinguishable (in a minimax sense) if µ m ≫ √ log m, while they are not if µ m ≪ log m. We derive equivalent results in a Bayesian setting where one assumes that all paths are equally likely; there the asymptotic threshold is µ m ≈ m −1/4 . We obtain corresponding results for trees (where the threshold is of order 1 and independent of the size of the tree), for distributions other than the Gaussian, and for other graphs. The concept of predictability profile, first introduced by Benjamini, Pemantle and Peres, plays a crucial role in our analysis. Keywords. Detecting a chain of nodes in a network, minimax detection, Bayesian detection, predictability profile of a stochastic process, martingales, exponential families of random variables.
In REF , an unknown path corresponding to nonzero-mean variables needs to be detected out of zero-mean variables in a network with nodes connected in a graph.
Searching for a Trail of Evidence in a Maze
Annals of Statistics 2008, Vol. 36, No. 4, 1726-1757
Mathematics