Bayesian Inference for a New Class of Distributions on Equivalence Classes of 3-D Orientations With Applications to Materials Science

Experiments in materials science investigating cubic crystalline structures often collect data which are in truth <i>equivalence classes</i> of crystallographically symmetric orientations. These intend to represent how lattice structures of particles are orientated relative to a reference coordinate system. Motivated by a materials science application, we formulate parametric probability models for “unlabeled orientation data.” This amounts to developing models on equivalence classes of 3-D rotations. We use a flexible existing model class for random rotations (called uniform-axis-random-spin models) to induce probability distributions on the equivalence classes of rotations. We develop one-sample Bayesian inference for the parameters in these models, and compare this methodology to some likelihood-based approaches. We also contrast the new parametric analysis of unlabeled orientation data with other analyses that proceed as if the data have been preprocessed into honest orientation data. On-line supplementary materials are also available, providing additional computational materials.