  Recently a new adaptive path interpolation method has been developed as a
simple and versatile scheme to calculate exactly the asymptotic mutual
information of Bayesian inference problems defined on dense factor graphs.
These include random linear and generalized estimation, sparse superposition
codes, or low-rank matrix and tensor estimation. For all these systems, the
adaptive interpolation method directly proves that the replica symmetric
prediction is exact, in a simple and unified manner. When the underlying factor
graph of the inference problem is sparse the replica prediction is considerably
more complicated, and rigorous results are often lacking or obtained by rather
complicated methods. In this work we show how to extend the adaptive path
interpolation method to sparse systems. We concentrate on a Censored Block
Model, where hidden variables are measured through a binary erasure channel,
for which we fully prove the replica prediction.
