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We present a general method to derive continuity estimates for conditionalprobabilities of general (possibly continuous) spin models sub jected to localtransformations. Such systems arise in the study of a stochastic time-evolutionof Gibbs measures or as noisy observations. We exhibit the minimal necessarystructure for such double-layer systems. Assuming no a priori metric on thelocal state spaces, we define the posterior metric on the local image space. Weshow that it allows in a natural way to divide the local part of the continuityestimates from the spatial part (which is treated by Dobrushin uniquenesshere). We show in the concrete example of the time evolution of rotators on theq-1 dimensional sphere how this method can be used to obtain estimates in termsof the familiar Euclidean metric.

The Posterior metric and the Goodness of Gibbsianness for transforms of Gibbs measures