  Thermodynamic quantities, such as heat and work, are not functions of state,
but rather of the process undergone by a physical system. Assessing them can
therefore be difficult, since it requires probing the system at least twice.
This is even more so when these quantities are to be assessed at the stochastic
level. In this letter we show how to obtain optimal estimates of thermodynamic
quantities solely from indirect measurement unravellings of an auxiliary
system. The method always yields the true average. And the mean-squared error
of the prediction is directly proportional to how well the method estimates the
variance. As an application, we study energy fluctuations in a driven system,
and in an avoided crossing work protocol.
