  The PARAFAC2 is a multimodal factor analysis model suitable for analyzing
multi-way data when one of the modes has incomparable observation units, for
example because of differences in signal sampling or batch sizes. A fully
probabilistic treatment of the PARAFAC2 is desirable in order to improve
robustness to noise and provide a well founded principle for determining the
number of factors, but challenging because the factor loadings are constrained
to be orthogonal. We develop two probabilistic formulations of the PARAFAC2
along with variational procedures for inference: In the one approach, the mean
values of the factor loadings are orthogonal leading to closed form variational
updates, and in the other, the factor loadings themselves are orthogonal using
a matrix Von Mises-Fisher distribution. We contrast our probabilistic
formulation to the conventional direct fitting algorithm based on maximum
likelihood. On simulated data and real fluorescence spectroscopy and gas
chromatography-mass spectrometry data, we compare our approach to the
conventional PARAFAC2 model estimation and find that the probabilistic
formulation is more robust to noise and model order misspecification. The
probabilistic PARAFAC2 thus forms a promising framework for modeling multi-way
data accounting for uncertainty.
