  In studying structural inter-connections in the human brain, it is common to
first estimate fiber bundles connecting different regions of the brain relying
on diffusion MRI. These fiber bundles act as highways for neural activity and
communication, snaking through the brain and connecting different regions.
Current statistical methods for analyzing these fibers reduce the rich
information into an adjacency matrix, with the elements containing a count of
the number of fibers or a mean diffusion feature (such as fractional
anisotropy) along the fibers. The goal of this article is to avoid discarding
the rich functional data on the shape, size and orientation of fibers,
developing flexible models for characterizing the population distribution of
fibers between brain regions of interest within and across different
individuals. We start by decomposing each fiber in each individual's brain into
a corresponding rotation matrix, shape and translation from a global reference
curve. These components can be viewed as data lying on a product space composed
of different Euclidean spaces and manifolds. To non-parametrically model the
distribution within and across individuals, we rely on a hierarchical mixture
of product kernels specific to the component spaces. Taking a Bayesian approach
to inference, we develop an efficient method for posterior sampling. The
approach automatically produces clusters of fibers within and across
individuals, and yields interesting new insights into variation in fiber
curves, while providing a useful starting point for more elaborate models
relating fibers to covariates and neuropsychiatric traits.
