  The choice of free parameters in network models is subjective, since it
depends on what topological properties are being monitored. However, we show
that the Maximum Likelihood (ML) principle indicates a unique, statistically
rigorous parameter choice, associated to a well defined topological feature. We
then find that, if the ML condition is incompatible with the built-in parameter
choice, network models turn out to be intrinsically ill-defined or biased. To
overcome this problem, we construct a class of safely unbiased models. We also
propose an extension of these results that leads to the fascinating possibility
to extract, only from topological data, the `hidden variables' underlying
network organization, making them `no more hidden'. We test our method on the
World Trade Web data, where we recover the empirical Gross Domestic Product
using only topological information.
