Bayesian Analysis

Optimal Robustness Results for Relative Belief Inferences and the Relationship to Prior-Data Conflict

Luai Al Labadi and Michael Evans

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The robustness to the prior of Bayesian inference procedures based on a measure of statistical evidence is considered. These inferences are shown to have optimal properties with respect to robustness. Furthermore, a connection between robustness and prior-data conflict is established. In particular, the inferences are shown to be effectively robust when the choice of prior does not lead to prior-data conflict. When there is prior-data conflict, however, robustness may fail to hold.

Article information

Bayesian Anal., Volume 12, Number 3 (2017), 705-728.

First available in Project Euclid: 7 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference
Secondary: 62F35: Robustness and adaptive procedures

relative belief robustness prior-data conflict

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Al Labadi, Luai; Evans, Michael. Optimal Robustness Results for Relative Belief Inferences and the Relationship to Prior-Data Conflict. Bayesian Anal. 12 (2017), no. 3, 705--728. doi:10.1214/16-BA1024.

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  • Baskurt, Z. and Evans, M. (2013). “Hypothesis assessment and inequalities for Bayes factors and relative belief ratios.” Bayesian Analysis, 8(3): 569–590.
  • Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis. Springer, second edition.
  • Berger, J. O. (1990). “Robust Bayesian analysis: sensitivity to the prior.” Journal of Statistical Planning and Inference, 25: 303–328.
  • Berger, J. O. (1994). “An overview of robust Bayesian analysis (with discussion).” Test, 3: 5–124.
  • Berger, J. O., Liseo, B., and Wolpert, R. L. (1999). “Integrated likelihood methods for eliminating nuisance parameters.” Statistical Science, 14(1): 1–28.
  • Dahl, F. A., Gasemyr, J., and Natvig, B. (2007). “A robust conflict measure of inconsistencies in Bayesian hierarchical models.” Scandinavian Journal of Statistics, 34: 816–828.
  • de la Horra, J. and Fernandez, C. (1994). “Bayesian analysis under $\epsilon$-contaminated priors: a trade-off between robustness and precision.” Journal of Statistical Planning and Inference, 38: 13–30.
  • Dey, D. K. and Birmiwall, L. R. (1994). “Robust Bayesian analysis using divergence measures.” Statistics and Probability Letters, 20: 287–294.
  • Evans, M. (2015). Measuring Statistical Evidence Using Relative Belief. CRC Press, Taylor & Francis Group.
  • Evans, M., Guttman, I., and Swartz, T. (2006). “Optimality and computations for relative surprise inferences.” Canadian Journal of Statistics, 34(1): 113–129.
  • Evans, M. and Jang, G. H. (2011a). “Inferences from prior-based loss functions.” Technical Report, arXiv.1104.3258.
  • Evans, M. and Jang, G. H. (2011b). “A limit result for the prior predictive.” Statistics and Probability Letters, 81: 1034–1038.
  • Evans, M. and Jang, G. H. (2011c). “Weak informativity and the information in one prior relative to another.” Statistical Science, 26(3): 423–439.
  • Evans, M. and Moshonov, H. (2006). “Checking for prior-data conflict.” Bayesian Analysis, 1(4): 893–914.
  • Evans, M. and Shakhatreh, M. (2008). “Optimal properties of some Bayesian inferences.” Electronic Journal of Statistics, 2: 1268–1280.
  • Evans, M. and Zou, T. (2001). “Robustness of relative surprise inferences to choice of prior.” In Chaubey, Y. (ed.), Recent Advances in Statistical Methods, Proceedings of Statistics 2001 Canada: The 4th Conference in Applied Statistics, 90–115. Imperial College Press.
  • Federer, S. G. (1969). Geometric Measure Theory. Berlin: Springer-Verlag.
  • Hirsch, M. W. (1976). Differential Topology. New York: Springer-Verlag.
  • Huber, P. J. (1973). “The use of Choquet capacities in statistics.” Bulletin of the International Statistical Institute, 45: 181–191.
  • Krantz, S. G. and Parks, H. R. (2008). Geometric Integration Theory. Boston: Birkhäuser.
  • Liseo, B. (1996). “A note on the concept of robust likelihoods.” Metron, LIV: 25–38.
  • Marshall, E. C. and Spiegelhalter, D. J. (2007). “Identifying outliers in Bayesian hierarchical models: a simulation-based approach.” Bayesian Analysis, 2: 409–444.
  • O’Hagan, A. (2003). “HSS model criticism (with discussion).” In Highly Structured Stochastic Systems. Oxford University Press.
  • Piccinato, L. (1984). “A Bayesian property of likelihood sets.” Statistica, 2: 197–204.
  • Presanis, A. M., Ohlssen, D., Spiegelhalter, D. J., and Angelis, D. D. (2013). “Conflict diagnostics in directed acyclic graphs, with applications in Bayesian evidence synthesis.” Statistical Science, 28: 376–397.
  • Rios Insua, D. and Ruggeri, F. (2000). Robust Bayesian Analysis. Springer-Verlag.
  • Royall, R. M. and Tsou, T. (2003). “Interpreting statistical evidence by using imperfect models: robust adjusted likelihood functions.” Journal of the Royal Statistical Society. Series B., 65(2): 391–404.
  • Rudin, W. (1974). Real and Complex Analysis, New York: McGraw-Hill, second edition.
  • Ruggeri, F. and Wasserman, L. (1993). “Infinitesimal sensitivity of posterior distributions.” The Canadian Journal of Statistics, 21(2): 195–203.
  • Scheel, I., Green, P. J., and Rougier, J. C. (2011). “A graphical diagnostic for identifying influential model choices in Bayesian hierarchical models.” Scandinavian Journal of Statistics, 38(3): 529–550.
  • Sivaganesan, S., Berliner, M., and Berger, J. O. (1993). “Optimal robust credible sets for contaminated priors.” Statistics and Probability Letters, 18(5): 383–388.
  • Tjur, T. (1974). Conditional Probability Distributions. Institute of Mathematical Statistics, University of Copenhagen.
  • Wasserman, L. (1989). “A robust Bayesian interpretation of likelihood regions.” Annals of Statistics, 17(3): 1387–1393.