  At least two, different approaches to define and solve statistical models for
the analysis of economic systems exist: the typical, econometric one,
interpreting the Gravity Model specification as the expected link weight of an
arbitrary probability distribution, and the one rooted into statistical
physics, constructing maximum-entropy distributions constrained to satisfy
certain network properties. In a couple of recent, companion papers they have
been successfully integrated within the framework induced by the constrained
minimisation of the Kullback-Leibler divergence: specifically, two, broad
classes of models have been devised, i.e. the integrated and the conditional
ones, defined by different, probabilistic rules to place links, load them with
weights and turn them into proper, econometric prescriptions. Still, the
recipes adopted by the two approaches to estimate the parameters entering into
the definition of each model differ. In econometrics, a likelihood that
decouples the binary and weighted parts of a model, treating a network as
deterministic, is typically maximised; to restore its random character, two
alternatives exist: either solving the likelihood maximisation on each
configuration of the ensemble and taking the average of the parameters
afterwards or taking the average of the likelihood function and maximising the
latter one. The difference between these approaches lies in the order in which
the operations of averaging and maximisation are taken - a difference that is
reminiscent of the quenched and annealed ways of averaging out the disorder in
spin glasses. The results of the present contribution, devoted to comparing
these recipes in the case of continuous, conditional network models, indicate
that the annealed estimation recipe represents the best alternative to the
deterministic one.
