  In this work we introduce a novel weighted message-passing algorithm based on
the cavity method to estimate volume-related properties of random polytopes,
properties which are relevant in various research fields ranging from metabolic
networks, to neural networks, to compressed sensing. Unlike the usual approach
consisting in approximating the real-valued cavity marginal distributions by a
few parameters, we propose an algorithm to faithfully represent the entire
marginal distribution. We explain various alternatives to implement the
algorithm and benchmark the theoretical findings by showing concrete
applications to random polytopes. The results obtained with our approach are
found to be in very good agreement with the estimates produced by the
Hit-and-Run algorithm, known to produce uniform sampling.
