Let \(x\) be the observed data, \(z=(z_1,\dots,z_n)\in \mathbb Z^n\), \(z_i\in(1,\dots,k)\) be the missing data, \(\eta=(\eta_1,\dots,\eta_k)\) be unknown parameters. The authors consider models with likelihoods of the form \[ f(x|\eta)=\sum_{z\in Z}f(x,z|\eta), \] where \(f(x,z|\eta)\) is a complete likelihood with the property that for any permutation \((t_1,\dots,t_k)\) of \((1,\dots,k)\) \[ f(x,(t_{z_1},\dots,t_{z_n})|\eta)=f(x,z|(\eta_{t_1},\dots,\eta_{t_k}). \] (Finite mixture models can be considered as an example). The prior distribution of \(\eta\) is assumed to be permutation invariant. In this case, the model exhibits a symmetric likelihood and symmetric posterior distribution.
The authors propose a modification of the Markov chain Monte Carlo (MCMC) algorithm for sampling from the posterior which is based on the splitting of \(\mathbb Z^n\) into \(k!\) classes of equivalence with respect to symmetry and sampling for only one representative from each class with random permutation of the result.
It is shown that the convergence rate of this equivalence classes representatives (ECR) algorithm is as good as the convergence of the random permutation sampler and not worse than the one of the original MCMC.