  For diffusive stochastic dynamics, the probability to observe any individual
trajectory is vanishingly small, making it unclear how to experimentally
validate theoretical results for ratios of path probabilities. We provide the
missing link between theory and experiment, by establishing a protocol to
extract ratios of path probabilities from measured time series. For experiments
on a single colloidal particle in a microchannel, we extract both ratios of
path probabilities, and the most probable path for a barrier crossing, and find
excellent agreement with independently calculated predictions based on the
Onsager-Machlup stochastic action. Our experimental results at room temperature
are found to be inconsistent with the low-noise Freidlin-Wentzell stochastic
action, and we discuss under which circumstances the latter is expected to
describe the most probable path. Furthermore, while the experimentally
accessible ratio of path probabilities is uniquely determined, the formal
path-integral action is known to depend on the time-discretization scheme used
for deriving it; we reconcile these two seemingly contradictory facts by
careful analysis of the time-slicing derivation of the path integral. Our
experimental protocol enables us to probe probability distributions on path
space, and allows us to relate theoretical single-trajectory results to
measurement.
