  We prove the large deviation principle (LDP) for posterior distributions
arising from subfamilies of full exponential families, allowing
misspecification of the model. Moreover, motivated by the so-called inverse
Sanov Theorem (see e.g. Ganesh and O'Connell 1999 and 2000), we prove the LDP
for the corresponding maximum likelihood estimator, and we study the
relationship between rate functions. In our setting, even in the non
misspecified case, it is not true in general that the rate functions for
posterior distributions and for maximum likelihood estimators are
Kullback-Leibler divergences with exchanged arguments.
