Two objections, at the very least:

1. Running (many) chains in parallel reflects on the distribution of the starting values as we cannot be sure to "reach" stationarity for all of them in a finite number of steps. Hence a bias.

2. Weighting MCMC values by their likelihood means the likelihood is counted twice (as a power of two!), since the values are approximately distributed from the posterior, i.e., the prior x the likelihood. Hence another bias.

Now importance sampling may be associated with MCMC, as we recently demonstrated.

Two objections, at the very least:

1. Running (many) chains in parallel reflects on the distribution of the starting values as we cannot be sure to "reach" stationarity for all of them in a finite number of steps. Hence a bias.

2. Weighting MCMC values by their likelihood means the likelihood is counted twice (as a power of two!), since the values are approximately distributed from the posterior, i.e., the prior x the likelihood. (The harmonic mean approximation to the evidence is actually relying on this representation, by averaging the inverse likelihoods.) Hence another bias.

Now importance sampling may be associated with MCMC, as we recently demonstrated.