  Polycrystalline materials, such as metals, are comprised of heterogeneously
oriented crystals. Observed crystal orientations are modelled as a sample from
an orientation distribution function (ODF), which determines a variety of
material properties and is therefore of great interest to practitioners.
Observations consist of quaternions, 4-dimensional unit vectors reflecting both
orientation and rotation of a single crystal. Thus, an ODF must account for
known crystal symmetries as well as satisfy the unit length constraint. A
popular method for estimating ODFs non-parametrically is symmetrized kernel
density estimation. However, disadvantages of this approach include difficulty
in interpreting results quantitatively, as well as in quantifying uncertainty
in the ODF. We propose to use a mixture of symmetric Bingham distributions as a
flexible parametric ODF model, inferring the number of mixture components, the
mixture weights, and scale and location parameters based on crystal orientation
data. Furthermore, our Bayesian approach allows for structured uncertainty
quantification of the parameters of interest. We discuss details of the
sampling methodology and conclude with analyses of various orientation
datasets, interpretations of parameters of interest, and comparison with kernel
density estimation methods.
