  The so-called block-term decomposition (BTD) tensor model, especially in its
rank-$(L_r,L_r,1)$ version, has been recently receiving increasing attention
due to its enhanced ability of representing systems and signals that are
composed of \emph{blocks} of rank higher than one, a scenario encountered in
numerous and diverse applications. Uniqueness conditions and fitting methods
have thus been thoroughly studied. Nevertheless, the challenging problem of
estimating the BTD model structure, namely the number of block terms, $R$, and
their individual ranks, $L_r$, has only recently started to attract significant
attention, mainly through regularization-based approaches which entail the need
to tune the regularization parameter(s). In this work, we build on ideas of
sparse Bayesian learning (SBL) and put forward a fully automated Bayesian
approach. Through a suitably crafted multi-level \emph{hierarchical}
probabilistic model, which gives rise to heavy-tailed prior distributions for
the BTD factors, structured sparsity is \emph{jointly} imposed. Ranks are then
estimated from the numbers of blocks ($R$) and columns ($L_r$) of
non-negligible energy. Approximate posterior inference is implemented, within
the variational inference framework. The resulting iterative algorithm
completely avoids hyperparameter tuning, which is a significant defect of
regularization-based methods. Alternative probabilistic models are also
explored and the connections with their regularization-based counterparts are
brought to light with the aid of the associated maximum a-posteriori (MAP)
estimators. We report simulation results with both synthetic and real-word
data, which demonstrate the merits of the proposed method in terms of both rank
estimation and model fitting as compared to state-of-the-art relevant methods.
