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Markov Chain Confidence Intervals and Biases

Author

Listed:
  • Yu Hang Jiang
  • Tong Liu
  • Zhiya Lou
  • Jeffrey S. Rosenthal
  • Shanshan Shangguan
  • Fei Wang
  • Zixuan Wu

Abstract

We derive explicit asymptotic confidence intervals for any Markov chain Monte Carlo (MCMC) algorithm with finite asymptotic variance, started at any initial state, without requiring a Central Limit Theorem nor reversibility nor geometric ergodicity nor any bias bound. We also derive explicit non-asymptotic confidence intervals assuming bounds on the bias or first moment, or alternatively that the chain starts in stationarity. We relate those non-asymptotic bounds to properties of MCMC bias, and show that polynomially ergodicity implies certain bias bounds. We also apply our results to several numerical examples. It is our hope that these results will provide simple and useful tools for estimating errors of MCMC algorithms when CLTs are not available.

Suggested Citation

  • Yu Hang Jiang & Tong Liu & Zhiya Lou & Jeffrey S. Rosenthal & Shanshan Shangguan & Fei Wang & Zixuan Wu, 2022. "Markov Chain Confidence Intervals and Biases," International Journal of Statistics and Probability, Canadian Center of Science and Education, vol. 11(1), pages 1-29, March.
    • Handle: RePEc:ibn:ijspjl:v:11:y:2022:i:1:p:29
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    References listed on IDEAS

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    1. Jones, Galin L. & Haran, Murali & Caffo, Brian S. & Neath, Ronald, 2006. "Fixed-Width Output Analysis for Markov Chain Monte Carlo," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1537-1547, December.
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    4. George S. Fishman, 1971. "Estimating Sample Size in Computing Simulation Experiments," Management Science, INFORMS, vol. 18(1), pages 21-38, September.
    5. Averill M. Law & W. David Kelton, 1984. "Confidence Intervals for Steady-State Simulations: I. A Survey of Fixed Sample Size Procedures," Operations Research, INFORMS, vol. 32(6), pages 1221-1239, December.
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    • R00 - Urban, Rural, Regional, Real Estate, and Transportation Economics - - General - - - General
    • Z0 - Other Special Topics - - General

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