  We present mathematical and conceptual foundations for the task of robust
amplitude estimation using engineered likelihood functions (ELFs), a framework
introduced in Wang et al. [PRX Quantum 2, 010346 (2021)] that uses Bayesian
inference to enhance the rate of information gain in quantum sampling. These
ELFs, which are obtained by choosing tunable parameters in a parametrized
quantum circuit to minimize the expected posterior variance of an estimated
parameter, play an important role in estimating the expectation values of
quantum observables. We give a thorough characterization and analysis of
likelihood functions arising from certain classes of quantum circuits and
combine this with the tools of Bayesian inference to give a procedure for
picking optimal ELF tunable parameters. Finally, we present numerical results
to demonstrate the performance of ELFs.
