Markov chain Monte Carlo (MCMC) is a powerful methodology for the approximation of posterior distributions. However, the iterative nature of MCMC does not naturally facilitate its use with modern highly parallel computation on HPC and cloud environments. Another concern is the identification of the bias and Monte Carlo error of produced averages. The above have prompted the recent development of fully ('embarrassingly') parallel unbiased Monte Carlo methodology based on coupling of MCMC algorithms. A caveat is that formulation of effective coupling is typically not trivial and requires model-specific technical effort. We propose coupling of MCMC chains deriving from sequential Monte Carlo (SMC) by considering adaptive SMC methods in combination with recent advances in unbiased estimation for state-space models. Coupling is then achieved at the SMC level and is, in principle, not problem-specific. The resulting methodology enjoys desirable theoretical properties. A central motivation is to extend unbiased MCMC to more challenging targets compared to the ones typically considered in the relevant literature. We illustrate the effectiveness of the algorithm via application to two complex statistical models: (i) horseshoe regression; (ii) Gaussian graphical models.

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