  Fluctuation theorems have become an important tool in single molecule
biophysics to measure free energy differences from non-equilibrium experiments.
When significant coarse-graining or noise affect the measurements, the
determination of the free energies becomes challenging. In order to address
this thermodynamic inference problem, we propose improved estimators of free
energy differences based on fluctuation theorems, which we test on a number of
examples. The effect of the noise can be described by an effective temperature,
which only depends on the signal to noise ratio, when the work is Gaussian
distributed and uncorrelated with the error made on the work. The notion of
effective temperature appears less useful for non-Gaussian work distributions
or when the error is correlated with the work, but nevertheless, as we show,
improved estimators can still be constructed for such cases. As an example of
non-trivial correlations between the error and the work, we also consider
measurements with delay, as described by linear Langevin equations.
