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 elegant approach to compare the two pdfs just by relying on the available data, and without knowing the pdfs explicitly. In this paper we introduce a graph-based extension of this problem: 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 also 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, while on the other hand trying to limit the data sharing among them. Our main contribution is a distributed non-parametric framework for graph-based LRE, called GRULSIF, that incorporates in a novel way elements from f-divengence functionals, Kernel methods, and Multitask Learning. Among the several 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.