  Functional data registration is a necessary processing step for many
applications. The observed data can be inherently noisy, often due to
measurement error or natural process uncertainty, which most functional
alignment methods cannot handle. A pair of functions can also have multiple
optimal alignment solutions, which is not addressed in current literature. In
this paper, a flexible Bayesian approach to functional alignment is presented,
which appropriately accounts for noise in the data without any pre-smoothing
required. Additionally, by running parallel MCMC chains, the method can account
for multiple optimal alignments via the multi-modal posterior distribution of
the warping functions. To most efficiently sample the warping functions, the
approach relies on a modification of the standard Hamiltonian Monte Carlo to be
well-defined on the infinite-dimensional Hilbert space. This flexible Bayesian
alignment method is applied to both simulated data and real data sets to show
its efficiency in handling noisy functions and successfully accounting for
multiple optimal alignments in the posterior; characterizing the uncertainty
surrounding the warping functions.
