I can derive an MCMC algorithm for sampling from the posterior distribution of a parameter vector of interest, but only starting with a dataset that has no missing values. The actual dataset that I want to use for inference has substantial missingness in its covariates.
One approach would be to build a more complex MCMC algorithm that, for example, first fills in the missing data with draws from the missing values' posterior predictive distribution. However, this feels intractably hard.
What I'd rather do is use an off-the-shelf method to generate multiple imputations of the dataset (such as a MICE package), then run my existing MCMC algorithm on each imputed, complete dataset and then recombine into final estimates of (for example) a posterior expectation or a posterior interval for a parameter of interest.
Is there a body of literature that attempts to solve problems in this way? Or is there a much better way to do this? Or is this approach wrong-headed or infeasible? Any pointers would be helpful.