  Ultrafast dynamical processes in photoexcited molecules can be observed with
pump-probe measurements, in which information about the dynamics is obtained
from the transient signal associated with the excited state. Background signals
provoked by pump and/or probe pulses alone often obscure these excited state
signals. Simple subtraction of pump-only and/or probe-only measurements from
the pump-probe measurement, as commonly applied, results in a degradation of
the signal-to-noise ratio and, in the case of coincidence detection, the danger
of overrated background subtraction. Coincidence measurements additionally
suffer from false coincidences. Here we present a probabilistic approach based
on Bayesian probability theory that overcomes these problems. For a pump-probe
experiment with photoelectron-photoion coincidence detection we reconstruct the
interesting excited-state spectrum from pump-probe and pump-only measurements.
This approach allows to treat background and false coincidences consistently
and on the same footing. We demonstrate that the Bayesian formalism has the
following advantages over simple signal subtraction: (i) the signal-to-noise
ratio is significantly increased, (ii) the pump-only contribution is not
overestimated, (iii) false coincidences are excluded, (iv) prior knowledge,
such as positivity, is consistently incorporated, (v) confidence intervals are
provided for the reconstructed spectrum, and (vi) it is applicable to any
experimental situation and noise statistics. Most importantly, by accounting
for false coincidences, the Bayesian approach allows to run experiments at
higher ionization rates, resulting in a significant reduction of data
acquisition times. The application to pump-probe coincidence measurements on
acetone molecules enables novel quantitative interpretations about the
molecular decay dynamics and fragmentation behavior.
