I am trying to get some information about identifiability with Bayesian inference and especially with mixture models. I have found a paper on this topic (Markov Chain Monte Carlo Methods and the Label Switching Problem in Bayesian Mixture Modeling) but I feel like I am not any smarter than before.
Jasra, Holmes and Stephens write:
“One of the main challenges of a Bayesian analysis
using mixtures is the nonidentifiability of the components.
That is, if exchangeable priors are placed upon
the parameters of a mixture model, then the resulting
posterior distribution will be invariant to permutations
in the labelling of the parameters. As a result,
the marginal posterior distributions for the parameters
will be identical for each mixture component. Therefore,
during MCMC simulation, the sampler encounters
the symmetries of the posterior distribution and the
interpretation of the labels switches. It is then meaningless
to draw inference directly from MCMC output
using ergodic averaging. Label switching significantly
increases the effort required to produce a satisfactory
Bayesian analysis of the data, but is a prerequisite of
convergence of an MCMC sampler and therefore must
Is there any way to explain this in a more simple way? Any help would be appreciated!