  Bayesian analysis of functions and curves is considered, where warping and
other geometrical transformations are often required for meaningful
comparisons. We focus on two applications involving the classification of mouse
vertebrae shape outlines and the alignment of mass spectrometry data in
proteomics. The functions and curves of interest are represented using the
recently introduced square root velocity function, which enables a warping
invariant elastic distance to be calculated in a straightforward manner. We
distinguish between various spaces of interest: the original space, the ambient
space after standardizing, and the quotient space after removing a group of
transformations. Using Gaussian process models in the ambient space and
Dirichlet priors for the warping functions, we explore Bayesian inference for
curves and functions. Markov chain Monte Carlo algorithms are introduced for
simulating from the posterior, including simulated tempering for multimodal
posteriors. We also compare ambient and quotient space estimators for mean
shape, and explain their frequent similarity in many practical problems using a
Laplace approximation. A simulation study is carried out, as well as shape
classification of the mouse vertebra outlines and practical alignment of the
mass spectrometry functions.
