  In recent years important progress has been achieved towards proving the
validity of the replica predictions for the (asymptotic) mutual information (or
"free energy") in Bayesian inference problems. The proof techniques that have
emerged appear to be quite general, despite they have been worked out on a
case-by-case basis. Unfortunately, a common point between all these schemes is
their relatively high level of technicality. We present a new proof scheme that
is quite straightforward with respect to the previous ones. We call it the
adaptive interpolation method because it can be seen as an extension of the
interpolation method developped by Guerra and Toninelli in the context of spin
glasses, with an interpolation path that is adaptive. In order to illustrate
our method we show how to prove the replica formula for three non-trivial
inference problems. The first one is symmetric rank-one matrix estimation (or
factorisation), which is the simplest problem considered here and the one for
which the method is presented in full details. Then we generalize to symmetric
tensor estimation and random linear estimation. We believe that the present
method has a much wider range of applicability and also sheds new insights on
the reasons for the validity of replica formulas in Bayesian inference.
