  Due to the interconnectedness of financial entities, estimating certain key
properties of a complex financial system (e.g. the implied level of systemic
risk) requires detailed information about the structure of the underlying
network. However, since data about financial linkages are typically subject to
confidentiality, network reconstruction techniques become necessary to infer
both the presence of connections and their intensity. Recently, several "horse
races" have been conducted to compare the performance of the available
financial network reconstruction methods. These comparisons, however, were
based on arbitrarily-chosen similarity metrics between the real and the
reconstructed network. Here we establish a generalised maximum-likelihood
approach to rigorously define and compare weighted reconstruction methods. Our
generalization maximizes the conditional entropy to solve the problem
represented by the fact that the density-dependent constraints required to
reliably reconstruct the network are typically unobserved. The resulting
approach admits as input any reconstruction method for the purely binary
topology and, conditionally on the latter, exploits the available partial
information to infer link weights. We find that the most reliable method is
obtained by "dressing" the best-performing binary method with an exponential
distribution of link weights having a properly density-corrected and
link-specific mean value and propose two unbiased (in the sense of maximum
conditional entropy) variants of it. While the one named CReMA is perfectly
general (as a particular case, it can place optimal weights on a network whose
topology is known), the one named CReMB is recommended both in case of full
uncertainty about the network topology and if the existence of some links is
certain. In these cases, the CReMB is faster and reproduces empirical networks
with highest generalised likelihood.
