Stability threshold and phase transition of generalized censored block models

The generalized censored block model considers the problem of inferring hidden binary variables from observations that are outputs of pairwise measurements from a symmetric channel. We give an exact formula for the stability threshold of density evolution by using an analysis of the potential functional of the model. In this model the phase transition is continuous so that this threshold is also the one for partial recovery of hidden variables. The formula is valid for all symmetric channels and generalizes the one already known for binary symmetric channels. We also give a bound on the finite slope of the Bhattacharyya parameter at the stability threshold. Finally, we briefly discuss implications for a heuristic derivation of the replica formula for the conditional entropy of the model.