  The Jarzynski equality and the fluctuation theorem relate equilibrium free
energy differences to non-equilibrium measurements of the work. These relations
extend to single-molecule experiments that have probed the finite-time
thermodynamics of proteins and nucleic acids. The effects of experimental error
and instrument noise have not previously been considered. Here, we present a
Bayesian formalism for estimating free-energy changes from non-equilibrium work
measurements that compensates for instrument noise and combines data from
multiple driving protocols. We reanalyze a recent set of experiments in which a
single RNA hairpin is unfolded and refolded using optical tweezers at three
different rates. Interestingly, the fastest and farthest-from-equilibrium
measurements contain the least instrumental noise, and therefore provide a more
accurate estimate of the free energies than a few slow, more noisy,
near-equilibrium measurements. The methods we propose here will extend the
scope of single-molecule experiments; they can be used in the analysis of data
from measurements with AFM, optical, and magnetic tweezers.
