This paper constructs perfect (exact) samplers for the posterior distributions of finite mixtures of exponential-family densities under conjugate priors — MCMC schemes that draw exactly from the target rather than only asymptotically, sidestepping the burn-in / convergence problem. Building on coupling-from-the-past, the authors first develop a perfect slice sampler that relies on a marginalization akin to Rao-Blackwellization and the Diebolt-Robert duality principle, embedding the finite-support latent-allocation distribution inside an easier-to-simulate continuous-support distribution. They then show this embedding approximation can be very poor, and propose a better alternative: a perfect sampler based on a single backward chain that handles much larger sample sizes.
"Perfect sampling has been developed in recent years to take advantage of Markov chain Monte Carlo algorithms without suffering from the drawback of MCMC methods, namely that the distribution of interest is only the asymptotic distribution of the Markov chain generated."
The paper is a useful reality check on perfect sampling: the coupling-from-the-past idea is beautiful — exact draws, no convergence diagnostics — but the honest lesson here is how hard it is to realize for a model as ordinary as a finite mixture, and how a natural-looking continuous embedding can quietly misrepresent the discrete target. The reusable ideas for the wiki are two: marginalizing the latent allocations (the same Rao-Blackwellization that stabilizes ordinary mixture Gibbs samplers) is what makes the state space tractable, and the retreat to a single backward chain shows that scalability, not just correctness, decides which perfect scheme is usable. In practice perfect sampling never displaced diagnostic-based MCMC for mixtures, which is itself part of the story: exactness is worth a great deal only when it is affordable.