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.