## Section:
New Results2>
### Adaptive Metropolis with Online
Relabeling3>

In [23] we proposed a novel adaptive MCMC algorithm named AMOR (Adaptive Metropolis with Online Relabeling) for efficiently simulating from permutation-invariant targets occurring in, for example, Bayesian analysis of mixture models. An important feature of the algorithm is to tie the adaptation of the proposal distribution to the choice of a particular restriction of the target to a domain where label switching cannot occur. The algorithm relies on a stochastic approximation procedure for which we design a Lyapunov function that formally defines the criterion used for selecting the relabeling rule. This criterion reveals an interesting connection with the problem of optimal quantifier design in vector quantization which was only implicit in previous works on the label switching problem. In benchmark examples, the algorithm turns out to be fast-converging and efficient at selecting meaningful non-trivial relabeling rules to allow accurate parameter inference. In [24] the algorithm was applied to a synthetic mixture model inspired by the muonic water Cherenkov signal of the surface detectors in the Pierre Auger Experiment.