  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.
