  An important problem in shape analysis is to match configurations of points
in space filtering out some geometrical transformation. In this paper we
introduce hierarchical models for such tasks, in which the points in the
configurations are either unlabelled, or have at most a partial labelling
constraining the matching, and in which some points may only appear in one of
the configurations. We derive procedures for simultaneous inference about the
matching and the transformation, using a Bayesian approach. Our model is based
on a Poisson process for hidden true point locations; this leads to
considerable mathematical simplification and efficiency of implementation. We
find a novel use for classic distributions from directional statistics in a
conditionally conjugate specification for the case where the geometrical
transformation includes an unknown rotation. Throughout, we focus on the case
of affine or rigid motion transformations. Under a broad parametric family of
loss functions, an optimal Bayesian point estimate of the matching matrix can
be constructed, that depends only on a single parameter of the family.
  Our methods are illustrated by two applications from bioinformatics. The
first problem is of matching protein gels in 2 dimensions, and the second
consists of aligning active sites of proteins in 3 dimensions. In the latter
case, we also use information related to the grouping of the amino acids. We
discuss some open problems and suggest directions for future work.
