# How can I pool posterior means and credible intervals after multiple imputation?

I have used multiple imputation to obtain a number of completed datasets.

I have used Bayesian methods on each of the completed datasets to obtain posterior distributions for a parameter (a random effect).

How can I combine / pool the results for this parameter ?

More context:

My model is hierarchical in the sense of individual pupils (one observation per pupil) clustered in schools. I have done multiple imputations (using MICEin R) on my data where I included school as one of the predictors for the missing data - to try to incorporate the data hierarchy into the imputations.

I have fitted a simple random slope model to each of the completed datasets (using MCMCglmm in R). The outcome is binary.

I have found that the posterior densities of the random slope variance are "well behaved" in the sense that they look something like this:

How can I combine/pool the posterior means and credible intervals from each imputed dataset, for this random effect ?

Update1:

From what I understand so far, I could apply Rubin's rules to the posterior mean, to give a multiply imputed posterior mean - are there any problems with doing this ? But I have no idea how I can combine the 95% credible intervals. Also, since I have an actual posterior density sample for each imputation - could I somehow combine these ?

Update2:

As per @cyan's suggestion in the comments, I very much like the idea of simply combining the samples from the posterior distributions obtained from each complete dataset from multiple imputation. However, I should like to know the theoretical justification for doing this.

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If the missingness of any given datum is independent of associated outcome value, it's correct to just throw all the posterior samples from the different imputed data sets together and take the mean and 95% credible intervals of the combined posterior samples. –  Cyan Aug 6 '12 at 1:30
@Cyan is that the same as saying that the missingness mechanism is either "missing at random" or "missing completely at random" but not "missing not at random" (the usual assumptions I learned about for performing MI)? Do you know any reference where this "throwing together" is justified formally ? –  Joe King Aug 6 '12 at 6:02
Multiple imputation IS a Bayesian procedure at its heart. If you use Bayesian methods for estimation (MCMC and such), you should just throw simluation of the missing data as an additional MCMC sampling step for a fully Bayesian model, and won't bother trying to come up with an interface between these approaches. –  StasK Aug 9 '12 at 10:40
@StasK thank you for your comment. I will try to use that approach on my next project but unfortunately I don't have time to change the model now. I already ran the imputations and the Bayesian model on each imputed dataset - it took nearly 3 weeks to run. Do you think it is invalid for me to combine the posterior samples ? –  Joe King Aug 9 '12 at 11:36
Rubin's rules apply to moments only. I don't know if you can apply them to a distribution in a meaningful way. May be, may be not. It may well be that the best you can do is to say that the MCMC run produced the point estimates (posterior means) and standard errors (posterior variances), and then use Rubin's rules to get the overall point and variance estimates. You know how tragic the losses of d.f.s in hierarchical model can be, and how dangerous it is to pool the data: if you have 5 imputed complete data sets and 1M MCMC samples on each, it means you have 5 clusters, not 5M iid MCMC points. –  StasK Aug 9 '12 at 11:47
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