  As a Bayesian approach to fitting motorway traffic flow models remains rare
in the literature, we explore empirically the sampling challenges this approach
offers which have to do with the strong correlations and multi-modality of the
posterior distribution. In particular, we provide a unified statistical model
to estimate using motorway data both boundary conditions and fundamental
diagram parameters in LWR, a well known motorway traffic flow model. This
allows us to provide a traffic flow density estimation method that is shown to
be superior to two methods found in the traffic flow literature. To sample from
this challenging posterior distribution we use a state-of-the-art gradient-free
function space sampler augmented with parallel tempering.
