Variational principle and almost quasilocality for renormalized measures

We study the variational principle for some non-Gibbsian measures. We give a necessary and sufficient condition for the validity of the implication zero relative entropy density implies common version of conditional probabilities (so-called second part of the variational principle). Applying this to noisy decimations of the low-temperature phases of the Ising model, we obtain almost sure quasilocality for these measures and the second part of the variational principle. For the projection of low temperature Ising phases on a one-dimensional layer, we also obtain the second part of the variational principle.