  We consider Bayesian inference of signals with vector-valued entries.
Extending concentration techniques from the mathematical physics of spin
glasses, we show that the matrix-valued minimum mean-square error concentrates
when the size of the problem increases. Such results are often crucial for
proving single-letter formulas for the mutual information when they exist. Our
proof is valid in the optimal Bayesian inference setting, meaning that it
relies on the assumption that the model and all its hyper-parameters are known.
Examples of inference and learning problems covered by our results are spiked
matrix and tensor models, the committee machine neural network with few hidden
neurons in the teacher-student scenario, or multi-layers generalized linear
models.
