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

The adaptive interpolation method: A simple scheme to prove replica formulas in Bayesian inference