  Quantum error correction allows to actively correct errors occurring in a
quantum computation when the noise is weak enough. To make this error
correction competitive information about the specific noise is required.
Traditionally, this information is obtained by benchmarking the device before
operation. We address the question of what can be learned from only the
measurements done during decoding. Such estimation of noise models was proposed
for surface codes, exploiting their special structure, and in the limit of low
error rates also for other codes. However, so far it has been unclear under
what general conditions noise models can be estimated from the syndrome
measurements. In this work, we derive a general condition for identifiability
of the error rates. For general stabilizer codes, we prove identifiability
under the assumption that the rates are small enough. Without this assumption
we prove a result for perfect codes. Finally, we propose a practical estimation
method with linear runtime for concatenated codes. We demonstrate that it
outperforms other recently proposed methods and that the estimation is optimal
in the sense that it reaches the Cram\'{e}r-Rao Bound. Our method paves the way
for practical calibration of error corrected quantum devices during operation.
