  We formulate the problem of numerical analytic continuation in a way that
lets us draw meaningful conclusions about properties of the spectral function
based solely on the input data. Apart from ensuring consistency with the input
data (within their error bars) and the {\it a priori} and {\it a posteriori}
(conditional) constraints, it is crucial to reliably characterize the
accuracy---or even ambiguity---of the output. We explain how these challenges
can be met with two approaches: stochastic optimization with consistent
constraints and the modified maximum entropy method. We perform illustrative
tests for spectra with a double-peak structure, where we critically examine
which spectral properties are accessible and which ones are lost. For an
important practical example, we apply our protocol to the Fermi polaron
problem.
