# Averaging Across Gibbs Sampling Runs with Reduced Dimensions

I need help thinking through my approach to Gibbs Sampling of many parameters and I'd like to know if there is literature on this topic:

I have a dataset with 3 dimensions: state, year, code. With the terms and interaction terms, there are $$50 + 5 + 5 + 50*5 + 5*5 + 50*5*5 = 1,585$$ parameters that I need to estimate. I am using Gibbs Sampling (JAGS via R) to fit a model to the data. I am running 3 chains (each on a separate processor). With 1,585 parameters to estimate, JAGS takes about 12 hours to run.

Now, I want to swap the code dimension for another dimension: type. (Note that type and code are separate fields and type is not a refinement of code.) I am also removing the state dimension. This results in $$5 + 750 + 5*750 = 4,505$$ parameters to estimate.

From testing, it appears that each added parameter takes exponentially more time to fit. So this model cannot complete within a week.

I have access to several servers (about 100 cores in total), so I want to take advantage of that. The idea: instead of doing 1 run of Gibbs Sampling with all parameters, do $n$ runs of Gibbs Sampling in parallel on subsets of parameters.

Process:

1. Perform stratified sampling to select a subset of values from type (sample 100 values, instead of all 750 at once).
2. All unselected data is aggregated to create an 'all other' bucket. I am doing this so that the population means across all runs are similar.
3. Run the model 200 times (each time estimating 605 parameters which were selected prior to the run).
4. At the end, average the estimates that were generated in the runs of Gibbs Sampling (so that there is only 1 estimate per parameter).

This makes me think of Random Forests in modeling decision trees (where you sample from the dataset, run the model on a subset of data and average the results), but that is in the context of empirical models. Does similar logic work for Bayesian hierarchical models? Do you see a flaw in this approach? And is there literature on this topic? Thanks!