Various embodiments are generally directed to techniques for producing statistically correct and efficient combinations of multiple simulated posterior samples from MCMC and related Bayesian sampling schemes are described. One or more chains from a Bayesian posterior distribution of values may be generated. It may be determine whether the one or more chains have reached stationarity through parallel processing on a plurality of processing nodes. Based upon the determination, each of the one or more chains that have reached stationarity through parallel processing on the plurality of processing nodes may be sorted. The one or more sorted chains may be resampled through parallel processing on the plurality of processing nodes. The one or more resampled chains may be combined. Other embodiments are described and claimed.
Please help me write a proper abstract based on the patent claims. CLAIM: 1. An article of manufacture comprising a non-transitory computer-readable storage medium comprising instructions that, when executed, cause a system to: generate one or more chains from a Bayesian posterior distribution of values; determine whether the one or more chains have reached stationarity through parallel processing on a plurality of processing nodes; based on the determination, sort each of the one or more chains that have reached stationarity through parallel processing on the plurality of processing nodes; resample the one or more sorted chains through parallel processing on the plurality of processing nodes; and combine the one or more resampled chains. 2. The article of claim 1, further comprising instructions that, when executed, cause the system to utilize a Markov Chain Monte Carlo (MCMC) algorithm to generate the one or more chains from the posterior distribution. 3. The article of claim 2, further comprising instructions that, when executed, cause the system to generate multiple chains from the Bayesian posterior distribution of values by processing multiple MCMC algorithms in parallel chains. 4. The article of claim 1, further comprising instructions that, when executed, cause the system to generate an approximate sample of a posterior distribution with the combined resampled chains. 5. The article of claim 4, further comprising instructions that, when executed, cause the system to determine a degree of approximation of the approximate sample of the posterior distribution as a function of a number of samples within each chain and a number of chains executed in parallel. 6. The article of claim 4, further comprising instructions that, when executed, cause the system to perform a Bayesian analysis on the approximate sample of the posterior distribution. 7. The article of claim 1, further comprising instructions that, when executed, cause the system to utilize multiple processing units or multiple threads within a processing unit to generate a plurality of simulated posterior samples. 8. The article of claim 1, wherein the combining utilizes a resampling algorithm with weights to represent the posterior distribution. 9. The article of claim 1, wherein each of the generated one or more chains represent a portion of the Bayesian posterior distribution. 10. The article of claim 1, the determining includes a determination of whether the one or more chains have reached convergence into local regions. 11. The article of claim 1, further comprising instructions that, when executed, cause the system to utilize a Geweke test to determine whether the one or more chains have reached stationarity. 12. The article of claim 11, further comprising instructions that, when executed, cause the system to utilize the Geweke test to check stationarity by comparing local means. 13. The article of claim 1, further comprising instructions that, when executed, cause the system to utilize a Heidelberger-Welch test to determine whether the one or more chains have reached stationarity. 14. The article of claim 13, further comprising instructions that, when executed, cause the system to utilize the Heidelberger-Welch test to determine covariance stationarity. 15. The article of claim 1, the one or more chains from the posterior distribution are selected from a plurality of regions of the posterior distribution. 16. The article of claim 1, further comprising instructions that, when executed, cause the system to assign weights to each chain proportional to the posterior distribution during resampling. 17. The article of claim 1, further comprising instructions that, when executed, cause the system to output the combined one or more resampled chains to a storage device. 18. A computer-implemented method, comprising: generating, by circuitry, one or more chains from a Bayesian posterior distribution of values; determining whether the one or more chains have reached stationarity through parallel processing on a plurality of processing nodes; based on the determination, sorting each of the one or more chains that have reached stationarity through parallel processing on the plurality of processing nodes; resampling the one or more sorted chains through parallel processing on the plurality of processing nodes; and combining the one or more resampled chains. 19. The computer-implemented method of claim 18, further comprising utilizing a Markov Chain Monte Carlo (MCMC) algorithm to generate the one or more chains from the posterior distribution. 20. The computer-implemented method of claim 18, the determining includes a determination of whether the one or more chains have reached convergence into local regions. 21. The computer-implemented method of claim 18, the further comprising utilizing a Geweke test to determine whether the one or more chains have reached stationarity. 22. The computer-implemented method of claim 21, further comprising utilizing the Geweke test to check stationarity by comparing local means. 23. The computer-implemented method of claim 18, further comprising utilizing a Heidelberger-Welch test to determine whether the one or more chains have reached stationarity. 24. The computer-implemented method of claim 23, further comprising utilizing the Heidelberger-Welch test to determine covariance stationarity. 25. The computer-implemented method of claim 18, the one or more chains from the posterior distribution are generated from a plurality of regions of the posterior distribution. 26. The computer-implemented method of claim 18, the resampling to assign weights to each chain proportional to the posterior distribution. 27. The computer-implemented method of claim 18, further comprising outputting the combined one or more resampled chains to a storage device. 28. An apparatus, comprising: processor circuitry; at least one memory unit communicatively coupled to the processor circuitry, the at least one memory unit configured to store: a sampling component that, when executed by the processor circuitry, is operative to receive one or more generated chains from a Bayesian posterior distribution from a selection component and determine whether the one or more chains have reached stationarity; a sorting component that, when executed by the processor circuitry, is operative to, based on the determination, sort each of the one or more chains that have reached stationarity; a resampling component that, when executed by the processor circuitry, is operative to resample the one or more sorted chains and provide the one or more resampled chains to a combining component. 29. The apparatus of claim 28, the selection component is operative to utilize a Markov Chain Monte Carlo (MCMC) algorithm to select the one or more chains from the posterior distribution. 30. The apparatus of claim 28, the determining includes a determination of whether the one or more chains have reached convergence into local regions. 31. The apparatus of claim 28, the sampling component, when executed by the processor circuitry, is operative to utilize a Geweke test to determine whether the one or more chains have reached stationarity. 32. The apparatus of claim 31, the sampling component, when executed by the processor circuitry, is operative to utilize the Geweke test to check stationarity by comparing local means. 33. The apparatus of claim 28, the sampling component, when executed by the processor circuitry, is operative to utilize a Heidelberger-Welch test to determine whether the one or more chains have reached stationarity. 34. The apparatus of claim 33, the sampling component, when executed by the processor circuitry, is operative to utilize the Heidelberger-Welch test to determine covariance stationarity. 35. The apparatus of claim 28, one or more chains from the posterior distribution are generated from a plurality of regions of the posterior distribution. 36. The apparatus of claim 28, the resampling component, when executed by the processor circuitry, is operative to assign weights to each chain proportional to the posterior distribution during resampling. 37. The apparatus of claim 28, the combining component is operative to output the combined one or more resampled chains to a storage device.