We propose a 'subjective' way of defining similarity between probability distributions.Our measure is parameterized by a collection H of subsets of the domain over which the probability distributions are defined. Intuitively speaking, H is the collection of 'subsets of interest' with respect to theproperties of the distributions that one wishes to analyze. The motivation behind the introduction of that measure comes from real life scenarios in which one cares only about certain distribution changes. In contrast with more traditional notions of distribution similarity (such as the L1 Norm) our measure can be reliably estimated from a pair of finite samples drawn from the two distributions. We have demonstrated the usefulness of the new measure in several areas of applications, including change detection in streaming data, the analysis of sensor network data and domain adaptation learning.