Consider the complete regular binary tree of depth M oriented from the root to the leaves. To each node we associate a random variable and those variables are assumed to be independent. Under the null hypothesis, these random variables have the standard normal distribution while under the alternative, there is a path from the root to a leaf along which the nodes have the normal distribution with mean A and variance 1, and the standard normal distribution away from the path. We show that, as M increases, the hypotheses become separable if, and only if, A is larger than the square root of 2 ln 2. We obtain corresponding results for other graphs and other distributions. The concept of predictability profile plays a crucial role in our analysis.
Joint work with Emmanuel Candes, Hannes Helgason and Ofer Zeitouni.