  It is argued that there is a sensible way to define conditional probabilities
in quantum mechanics, assuming only Bayes's theorem and standard quantum
theory. These probabilities are equivalent to the ``weak measurement''
predictions due to Aharonov {\it et al.}, and hence describe the outcomes of
real measurements made on subensembles. In particular, this approach is used to
address the question of the history of a particle which has tunnelled across a
barrier. A {\it gedankenexperiment} is presented to demonstrate the physically
testable implications of the results of these calculations, along with graphs
of the time-evolution of the conditional probability distribution for a
tunneling particle and for one undergoing allowed transmission. Numerical
results are also presented for the effects of loss in a bandgap medium on
transmission and on reflection, as a function of the position of the lossy
region; such loss should provide a feasible, though indirect, test of the
present conclusions. It is argued that the effects of loss on the pulse {\it
delay time} are related to the imaginary value of the momentum of a tunneling
particle, and it is suggested that this might help explain a small discrepancy
in an earlier experiment.
