  Quantum sensors typically translate external fields into a periodic response
whose frequency is then determined by analyses performed in Fourier space. This
allows for a linear inference of the parameters that characterize external
signals. In practice, however, quantum sensors are able to detect fields only
in a narrow range of amplitudes and frequencies. A departure from this range,
as well as the presence of significant noise sources and short detection times,
lead to a loss of the linear relationship between the response of the sensor
and the target field, thus limiting the working regime of the sensor. Here we
address these challenges by means of a Bayesian inference approach that is
tolerant to strong deviations from desired periodic responses of the sensor and
is able to provide reliable estimates even with a very limited number of
measurements. We demonstrate our method for an $^{171}$Yb$^{+}$ trapped-ion
quantum sensor but stress the general applicability of this approach to
different systems.
