  Multiway data often naturally occurs in a tensorial format which can be
approximately represented by a low-rank tensor decomposition. This is useful
because complexity can be significantly reduced and the treatment of
large-scale data sets can be facilitated. In this paper, we find a low-rank
representation for a given tensor by solving a Bayesian inference problem. This
is achieved by dividing the overall inference problem into sub-problems where
we sequentially infer the posterior distribution of one tensor decomposition
component at a time. This leads to a probabilistic interpretation of the
well-known iterative algorithm alternating linear scheme (ALS). In this way,
the consideration of measurement noise is enabled, as well as the incorporation
of application-specific prior knowledge and the uncertainty quantification of
the low-rank tensor estimate. To compute the low-rank tensor estimate from the
posterior distributions of the tensor decomposition components, we present an
algorithm that performs the unscented transform in tensor train format.
