  This article proposes a Bayesian approach to regression with a scalar
response against vector and tensor covariates. Tensor covariates are commonly
vectorized prior to analysis, failing to exploit the structure of the tensor,
and resulting in poor estimation and predictive performance. We develop a novel
class of multiway shrinkage priors for the coefficients in tensor regression
models. Properties are described, including posterior consistency under mild
conditions, and an efficient Markov chain Monte Carlo algorithm is developed
for posterior computation. Simulation studies illustrate substantial gains over
vectorizing or using existing tensor regression methods in terms of estimation
and parameter inference. The approach is further illustrated in a neuroimaging
application.
  Keywords: Dimension reduction; multiway shrinkage prior; magnetic resonance
imaging (MRI); parafac decomposition; posterior consistency; tensor regression
