  We study the relative entropy density for generalized Gibbs measures. We
first show its existence and obtain a familiar expression in terms of entropy
and relative energy for a class of ``almost Gibbsian measures'' (almost sure
continuity of conditional probabilities). For quasilocal measures, we obtain a
full variational principle. For the joint measures of the random field Ising
model, we show that the weak Gibbs property holds, with an almost surely
rapidly decaying translation-invariant potential. For these measures we show
that the variational principle fails as soon as the measures lose the almost
Gibbs property. These examples suggest that the class of weakly Gibbsian
measures is too broad from the perspective of a reasonable thermodynamic
formalism.
