Bayesian alignment using hierarchical models,
with applications in protein bioinformatics
Peter J. Green (Bristol) & Kanti Mardia (Leeds)
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.
Keywords: bioinformatics,
Markov chain Monte Carlo,
matching,
Poisson process,
protein gels,
protein structure,
shape analysis,
von Mises-Fisher distribution.
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