  Assuming we have iid observations from two unknown probability density
functions (pdfs), $p$ and $q$, the likelihood-ratio estimation (LRE) is an
elegant approach to compare the two pdfs only by relying on the available data.
In this paper, we introduce the first -to the best of our knowledge-graph-based
extension of this problem, which reads as follows: Suppose each node $v$ of a
fixed graph has access to observations coming from two unknown node-specific
pdfs, $p_v$ and $q_v$, and the goal is to estimate for each node the
likelihood-ratio between both pdfs by also taking into account the information
provided by the graph structure. The node-level estimation tasks are supposed
to exhibit similarities conveyed by the graph, which suggests that the nodes
could collaborate to solve them more efficiently. We develop this idea in a
concrete non-parametric method that we call Graph-based Relative Unconstrained
Least-squares Importance Fitting (GRULSIF). We derive convergence rates for our
collaborative approach that highlights the role played by variables such as the
number of available observations per node, the size of the graph, and how
accurately the graph structure encodes the similarity between tasks. These
theoretical results explicit the situations where collaborative estimation
effectively leads to an improvement in performance compared to solving each
problem independently. Finally, in a series of experiments, we illustrate how
GRULSIF infers the likelihood-ratios at the nodes of the graph more accurately
compared to state-of-the art LRE methods, which would operate independently at
each node, and we also verify that the behavior of GRULSIF is aligned with our
previous theoretical analysis.
