  We study the instantaneous inference of an unbounded planar flow from sparse
noisy pressure measurements. The true flow field comprises one or more
regularized point vortices of various strength and size. We interpret the true
flow's measurements with a vortex estimator, also consisting of regularized
vortices, and attempt to infer the positions and strengths of this estimator
assuming little prior knowledge. The problem often has several possible
solutions, many due to a variety of symmetries. To deal with this ill-posedness
and to quantify the uncertainty, we develop the vortex estimator in a Bayesian
setting. We use Markov-chain Monte Carlo and a Gaussian mixture model to sample
and categorize the probable vortex states in the posterior distribution,
tailoring the prior to avoid spurious solutions. Through experiments with one
or more true vortices, we reveal many aspects of the vortex inference problem.
With fewer sensors than states, the estimator infers a manifold of
equally-possible states. Using one more sensor than states ensures that no
cases of rank deficiency arise. Uncertainty grows rapidly with distance when a
vortex lies outside of the vicinity of the sensors. Vortex size cannot be
reliably inferred, but the position and strength of a larger vortex can be
estimated with a much smaller one. In estimates of multiple vortices their
individual signs are discernible because of the non-linear coupling in the
pressure. When the true vortex state is inferred from an estimator of fewer
vortices, the estimate approximately aggregates the true vortices where
possible.
