  We study the well-posedness of the Bayesian inverse problem for scalar
hyperbolic conservation laws where the statistical information about inputs
such as the initial datum and (possibly discontinuous) flux function are
inferred from noisy measurements. In particular, the Lipschitz continuity of
the measurement to posterior map as well as the stability of the posterior to
approximations, are established with respect to the Wasserstein distance.
Numerical experiments are presented to illustrate the derived estimates.
