Some observations on the ambivalent role of symmetries in Bayesian inference problems

We collect in this note some observations on the role of symmetries in Bayesian inference problems, that can be useful or detrimental depending on the way they act on the signal and on the observations. We emphasize in particular the need to gauge away unobservable invariances in the definition of a distance between a signal and its estimator, and the consequences this implies for the statistical mechanics treatment of such models, taking as a motivating example the extensive rank matrix factorization problem.