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Bias and Variance Approximation in Value Function Estimates

Author

Listed:
  • Shie Mannor

    (Department of Electrical and Computer Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7)

  • Duncan Simester

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

  • Peng Sun

    (Fuqua School of Business, Duke University, Durham, North Carolina 27708)

  • John N. Tsitsiklis

    (Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139)

Abstract

We consider a finite-state, finite-action, infinite-horizon, discounted reward Markov decision process and study the bias and variance in the value function estimates that result from empirical estimates of the model parameters. We provide closed-form approximations for the bias and variance, which can then be used to derive confidence intervals around the value function estimates. We illustrate and validate our findings using a large database describing the transaction and mailing histories for customers of a mail-order catalog firm.

Suggested Citation

  • Shie Mannor & Duncan Simester & Peng Sun & John N. Tsitsiklis, 2007. "Bias and Variance Approximation in Value Function Estimates," Management Science, INFORMS, vol. 53(2), pages 308-322, February.
    • Handle: RePEc:inm:ormnsc:v:53:y:2007:i:2:p:308-322
    DOI: 10.1287/mnsc.1060.0614
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    References listed on IDEAS

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    1. Rust, John, 1987. "Optimal Replacement of GMC Bus Engines: An Empirical Model of Harold Zurcher," Econometrica, Econometric Society, vol. 55(5), pages 999-1033, September.
    2. Duncan I. Simester & Peng Sun & John N. Tsitsiklis, 2006. "Dynamic Catalog Mailing Policies," Management Science, INFORMS, vol. 52(5), pages 683-696, May.
    3. Keane, Michael P & Wolpin, Kenneth I, 1994. "The Solution and Estimation of Discrete Choice Dynamic Programming Models by Simulation and Interpolation: Monte Carlo Evidence," The Review of Economics and Statistics, MIT Press, vol. 76(4), pages 648-672, November.
    4. Yihong Xia, 2001. "Learning about Predictability: The Effects of Parameter Uncertainty on Dynamic Asset Allocation," Journal of Finance, American Finance Association, vol. 56(1), pages 205-246, February.
    5. Charles E. Clark, 1961. "The Greatest of a Finite Set of Random Variables," Operations Research, INFORMS, vol. 9(2), pages 145-162, April.
    6. Jeffrey I. McGill & Garrett J. van Ryzin, 1999. "Revenue Management: Research Overview and Prospects," Transportation Science, INFORMS, vol. 33(2), pages 233-256, May.
    7. Jerzy A. Filar & L. C. M. Kallenberg & Huey-Miin Lee, 1989. "Variance-Penalized Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 14(1), pages 147-161, February.
    8. Gabriel R. Bitran & Susana V. Mondschein, 1996. "Mailing Decisions in the Catalog Sales Industry," Management Science, INFORMS, vol. 42(9), pages 1364-1381, September.
    9. Gregory A. Godfrey & Warren B. Powell, 2002. "An Adaptive Dynamic Programming Algorithm for Dynamic Fleet Management, I: Single Period Travel Times," Transportation Science, INFORMS, vol. 36(1), pages 21-39, February.
    10. Füsun Gönül & Meng Ze Shi, 1998. "Optimal Mailing of Catalogs: A New Methodology Using Estimable Structural Dynamic Programming Models," Management Science, INFORMS, vol. 44(9), pages 1249-1262, September.
    11. Jan Roelf Bult & Tom Wansbeek, 1995. "Optimal Selection for Direct Mail," Marketing Science, INFORMS, vol. 14(4), pages 378-394.
    12. Nicholas Barberis, 2000. "Investing for the Long Run when Returns Are Predictable," Journal of Finance, American Finance Association, vol. 55(1), pages 225-264, February.
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    3. Wolfram Wiesemann & Daniel Kuhn & Berç Rustem, 2010. "Robust Markov Decision Processes," Working Papers 034, COMISEF.
    4. David L. Kaufman & Andrew J. Schaefer, 2013. "Robust Modified Policy Iteration," INFORMS Journal on Computing, INFORMS, vol. 25(3), pages 396-410, August.
    5. Raghuram Iyengar & Kamel Jedidi & Skander Essegaier & Peter J. Danaher, 2011. "The Impact of Tariff Structure on Customer Retention, Usage, and Profitability of Access Services," Marketing Science, INFORMS, vol. 30(5), pages 820-836, September.
    6. V Varagapriya & Vikas Vikram Singh & Abdel Lisser, 2023. "Joint chance-constrained Markov decision processes," Annals of Operations Research, Springer, vol. 322(2), pages 1013-1035, March.
    7. Huan Xu & Shie Mannor, 2012. "Distributionally Robust Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 37(2), pages 288-300, May.
    8. Ricardo Montoya & Oded Netzer & Kamel Jedidi, 2010. "Dynamic Allocation of Pharmaceutical Detailing and Sampling for Long-Term Profitability," Marketing Science, INFORMS, vol. 29(5), pages 909-924, 09-10.
    9. Yuanhui Zhang & Haipeng Wu & Brian T. Denton & James R. Wilson & Jennifer M. Lobo, 2019. "Probabilistic sensitivity analysis on Markov models with uncertain transition probabilities: an application in evaluating treatment decisions for type 2 diabetes," Health Care Management Science, Springer, vol. 22(1), pages 34-52, March.
    10. Bren, Austin & Saghafian, Soroush, 2018. "Data-Driven Percentile Optimization for Multi-Class Queueing Systems with Model Ambiguity: Theory and Application," Working Paper Series rwp18-008, Harvard University, John F. Kennedy School of Government.
    11. Erick Delage & Shie Mannor, 2010. "Percentile Optimization for Markov Decision Processes with Parameter Uncertainty," Operations Research, INFORMS, vol. 58(1), pages 203-213, February.
    12. Maximilian Blesch & Philipp Eisenhauer, 2023. "Robust Decision-Making under Risk and Ambiguity," Rationality and Competition Discussion Paper Series 463, CRC TRR 190 Rationality and Competition.
    13. Shie Mannor & Ofir Mebel & Huan Xu, 2016. "Robust MDPs with k -Rectangular Uncertainty," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1484-1509, November.
    14. Shiau Hong Lim & Huan Xu & Shie Mannor, 2016. "Reinforcement Learning in Robust Markov Decision Processes," Mathematics of Operations Research, INFORMS, vol. 41(4), pages 1325-1353, November.
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