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