Collaborative likelihood-ratio estimation over graphs

Assuming we have i.i.d observations from two unknown probability density functions (pdfs), p and p′, the likelihood-ratio estimation (LRE) is an approach to compare two pdfs without knowing its functional form explicitly. In this talk, we will introduce a graph-based extension of the problem. We will suppose each node v of a fixed graph has access to observations coming from two unknown node-specific pdfs, p_v and p′_v; the goal is then to compare the respective p_v and p′_v of each node by integrating information provided by the graph structure. This setting is interesting when the graph conveys some sort of `similarity' between the node-wise estimation tasks, which suggests that the nodes can collaborate to solve more efficiently their individual tasks. Our main contribution is a distributed non-parametric framework for graph-based LRE, called GRULSIF, that incorporates in a novel way elements from f-divergence functionals, Kernel methods, and Multitask Learning. Among the applications of LRE, we choose the two-sample hypothesis testing to develop a proof of concept for our graph-based learning framework. Our experiments compare favorably the performance of our approach against state-of-the-art non-parametric statistical tests that apply at each node independently, and thus disregard the graph structure.