  We study the parameter estimation problem in mixture models with
observational nonidentifiability: the full model (also containing hidden
variables) is identifiable, but the marginal (observed) model is not. Hence
global maxima of the marginal likelihood are (infinitely) degenerate and
predictions of the marginal likelihood are not unique. We show how to
generalize the marginal likelihood by introducing an effective temperature, and
making it similar to the free energy. This generalization resolves the
observational nonidentifiability, since its maximization leads to unique
results that are better than a random selection of one degenerate maximum of
the marginal likelihood or the averaging over many such maxima. The generalized
likelihood inherits many features from the usual likelihood, e.g. it holds the
conditionality principle, and its local maximum can be searched for via
suitably modified expectation-maximization method. The maximization of the
generalized likelihood relates to entropy optimization.
