Use importance sampling to post processing the posterior result of the MCMC chains

Question

In one of the studies, I once found the following heuristics to perform the calibration,

Step 1: Running MCMC to get model parameters, with K chains Step 2: Compute weight for these K chains, the weight is based on the likelihood based on the parameters corresponding to each chain Step 3: The posterior prediction is then an ensemble prediction based on these K chains where the weight is computed in step 2.

It is more like an importance sampling approach.

Is this approach quite common in posterior prediction.

Answer 1

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.