  We consider models of Bayesian inference of signals with vectorial components
of finite dimensionality. We show that, under a proper perturbation, these
models are replica symmetric in the sense that the overlap matrix concentrates.
The overlap matrix is the order parameter in these models and is directly
related to error metrics such as minimum mean-square errors. Our proof is valid
in the optimal Bayesian inference setting. This means that it relies on the
assumption that the model and all its hyper-parameters are known so that the
posterior distribution can be written exactly. Examples of important problems
in high-dimensional inference and learning to which our results apply are
low-rank tensor factorization, the committee machine neural network with a
finite number of hidden neurons in the teacher-student scenario, or multi-layer
versions of the generalized linear model.
