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The Gibbs sampler is a simple form of Markov Chain Monte Carlo simulation, widely used in Bayesian statistics, based on sampling from full conditional distributions for each variable or group of variables. The name comes from the method being first used on Gibbs random fields modeling of images by Geman and Geman (1984).

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Averaging Across Gibbs Sampling Runs with Reduced Dimensions

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. … At the end, average the estimates that were generated in the runs of Gibbs Sampling (so that there is only 1 estimate per parameter). …
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