Abstract
We study the conjugacy approximation models in the context of Bayesian ranking and selection with unknown correlations. Under the assumption of normal-inverse-Wishart prior distribution, the posterior distribution remains a normal-inverse-Wishart distribution thanks to the conjugacy property when all alternatives are sampled at each step. However, this conjugacy property no longer holds if only one alternative is sampled at a time, an appropriate setting when there is a limited budget on the number of samples. We propose two new conjugacy approximation models based on the idea of moment matching. Both of them yield closed-form Bayesian prior updating formulas. We apply these updating formulas in Bayesian ranking and selection using the knowledge gradient method and show the superiority of the proposed conjugacy approximation models in applications of wind farm placement and computer model calibration.
- Y. Chen and I. O. Ryzhov. 2016. Approximate bayesian inference as a form of stochastic approximation: A new consistency theory with applications. In Proceedings of the 2016 Winter Simulation Conference. IEEE Press, 534--544. Google ScholarCross Ref
- S. Chick. 2006. Bayesian ideas and discrete event simulation: Why, what and how. In Proceedings of the Winter Simulation Conference, L. Perrone, F. Wieland, J. Liu, B. Lawson, D. Nicol, and R. Fujimoto (Eds.). 96--105. Google ScholarCross Ref
- S. Chick and P. Frazier. 2012. Sequential sampling for selection with economics of selection procedures. Manage. Sci. 58, 3 (2012), 550--569. Google ScholarDigital Library
- B. A. Cosgrove, D. Lohmann, K. E. Mitchell, P. R. Houser, E. F. Wood, J. C. Schaake, A. Robock, and others. 2003. Real-time and retrospective forcing in the North American land data assimilation system (NLDAS) project. J. Geophys. Res. 108, D22 (2003), 8842--8853. Google ScholarCross Ref
- M. H. DeGroot. 2004. Optimal Statistical Decisions. John Wiley 8 Sons.Google Scholar
- W. Fan, L. J. Hong, and B. L. Nelson. 2016. Indifference-zone-free selection of the best. Operat. Res. 64, 6 (2016), 1499--1514. Google ScholarDigital Library
- P. I. Frazier, W. B. Powell, and S. Dayanik. 2009. The knowledge-gradient policy for correlated normal rewards. INFORMS J. Comput. 21, 4 (2009), 599--613. Google ScholarCross Ref
- G. H. Golub and C. F. Van Loan. 1996. Matrix Computations. Johns Hopkins Studies in the Mathematical Science.Google Scholar
- A. K. Gupta and D. K. Nagar. 2000. Matrix Variate Distributions. Chapman 8 Hall.Google Scholar
- L. J. Hong and B. L. Nelson. 2009. A brief introduction to optimization via simulation. In Proceedings of the Winter Simulation Conference, M. D. Rosetti, R. R. Hill, B. Johansson, A. Dunkin, and R. G. Ingalls (Eds.). 75--85. Google ScholarCross Ref
- S.-H. Kim and B. L. Nelson. 2001. A fully sequential procedure for indifference-zone selection in simulation. ACM Trans. Model. Comput. Simul. 11, 3 (2001), 251--273. Google ScholarDigital Library
- S.-H. Kim and B. L. Nelson. 2006. On the asymptotic validity of fully sequential selection procedures for steady-state simulation. Operat. Res. 54, 3 (2006), 475--488. Google ScholarDigital Library
- S.-H. Kim and B. L. Nelson. 2007. Recent advances in ranking and selection. In Proceedings of the Winter Simulation Conference, S. G. Henderson, B. Biller, M.-H. Hsieh, J. Shortle, J. D. Tew, and R. R. Barton (Eds.). 162--172.Google Scholar
- T. P. Minka. 2001a. Expectation propagation for approximate Bayesian inference. In Proceedings of the 17th Conference on Uncertainty in Artificial Intelligence. Morgan Kaufmann, 362--369.Google Scholar
- T. P. Minka. 2001b. A Family of Algorithms for Approximate Bayesian Inference. Ph.D. Dissertation. Massachusetts Institute of Technology.Google ScholarDigital Library
- M. D. Morris, T. J. Mitchell, and D. Ylvisaker. 1993. Bayesian design and analysis of computer experiments: Use of derivatives in surface prediction. Technometrics 35 (1993), 243--255. Google Scholar
- W. B. Powell and I. O. Ryzhov. 2012. Optimal Learning. John Wiley 8 Sons.Google Scholar
- P. Z. G. Qian. 2012. Sliced Latin hypercube designs. J. Am. Stat. Assoc. 107, 497 (2012), 393--399. Google ScholarCross Ref
- H. Qu, I. O. Ryzhov, M. C. Fu, and Z. Ding. 2015. Sequential selection with unknown correlation structures. Operat. Res. 63, 4 (2015), 931--948. Google ScholarDigital Library
- W. Scott, P. Frazier, and W. B. Powell. 2011. The correlated knowledge gradient for simulation optimization of continuous parameters using gaussian process regression. SIAM J. Optimiz. 21, 3 (2011), 996--1026. Google ScholarCross Ref
- R. Tuo and J. C. F. Wu. 2015. Efficient calibration for imperfect computer models. Ann. Stat. 43, 6 (2015), 2331--2352. Google ScholarCross Ref
- Raymond K. W. Wong, Curtis B. Storlie, and Thomas Lee. 2017. A frequentist approach to computer model calibration. J. Roy. Stat. Soc. B 79, 2 (2017), 635--648. Google ScholarCross Ref
- J. Xie and P. Frazier. 2013. Sequential bayes-optimal policies for multiple comparisons with a known standard. Operat. Res. 61, 3 (2013), 1174--1189. Google ScholarCross Ref
- Jianming Ye. 1998. On measuring and correcting the effects of data mining and model selection. J. Am. Stat. Ass. 93, 441 (1998), 120--131. Google ScholarCross Ref
- Q. Zhang and Y. Song. 2015. Simulation selection for empirical model comparison. In Proceedings of the 2015 Winter Simulation Conference. Google ScholarCross Ref
Index Terms
- Moment-Matching-Based Conjugacy Approximation for Bayesian Ranking and Selection
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