Skip to main content

Linearization Techniques in Bayesian Robustness

  • Chapter
Robust Bayesian Analysis

Part of the book series: Lecture Notes in Statistics ((LNS,volume 152))

Abstract

This paper deals with techniques which permit one to obtain the range of a posterior expectation through a sequence of linear optimizations. In the context of Bayesian robustness, the linearization algorithm plays a fundamental role. Its mathematical aspects and its connections with fractional programming procedures are reviewed and a few instances of its broad applicability are listed. At the end, some alternative approaches are briefly discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Berger, J.O. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. New York: Springer-Verlag.

    MATH  Google Scholar 

  • Berger, J.O. (1994). An overview of robust Bayesian analysis. Test, 3, 5–59.

    Article  MathSciNet  MATH  Google Scholar 

  • Berger, J.O. Ando’hagan, A. (1988). Ranges of posterior probabilites for unimodal priors with specified quantiles. In Bayesian Statistics 3 (J.M. Bernardo, M.H. DeGroot, D.V. Lindley, and A.V.M. Smith, eds.), 45–65. Oxford: Oxford University Press.

    Google Scholar 

  • Berliner, L. M. and Goel, P. (1990). Incorporating partial prior information: ranges of posterior probabilities. In Bayesian and Likelihood Methods in Statistics and Econometrics: Essays in Honor of George A. Barnard (S. Geisser, J. Hodges, F. J. Press and A. Zellner, eds.), 397–406. Amsterdam: North-Holland.

    Google Scholar 

  • Betrò, B. and Guglielmi, A. (1996). Numerical robust Bayesian analysis under generalized moment conditions. In Bayesian Robustness, IMS Lecture Notes — Monograph Series, (J.O. Berger, B. Betrò, E. Moreno, L.R. Pericchi, F. Ruggeri, G. Salinetti, and L. Wasserman, eds.), 3–20. Hayward: IMS.

    Google Scholar 

  • Betrò, B. and Guglielmi, A. (2000). Methods for global prior robustness under generalized moment conditions. In Robust Bayesian Analysis, (D. Ríos Insua and F. Ruggeri, eds.). New York: Springer-Verlag.

    Google Scholar 

  • Bose, S. (1994). Bayesian robustness with mixture classes of priors. Annals of Statistics, 22, 652–667.

    Article  MathSciNet  MATH  Google Scholar 

  • Bradley, S.P. and Frey, S.C. (1974). Fractional programming with homogeneous functions. Operations Research, 22, 350–357.

    Article  MathSciNet  MATH  Google Scholar 

  • Charnes, A. and Cooper, W.W. (1962). Programming with linear fractional functionals. Naval Research Logistic Quarterly, 9, 181–186.

    Article  MathSciNet  MATH  Google Scholar 

  • Cozman, F. (1999). Calculation of posterior bounds given convex sets of prior probability measures and likelihood functions. To appear in Journal of Computational and Graphical Statistics.

    Google Scholar 

  • Dall’aglio, M. (1995). Problema dei momenti e programmazione lineare semi infinita nella robustezza bayesiana. Ph.D. Thesis, Dipartimento di Statistica, Probabilità e Statistiche Applicate dell’Università di Roma “La Sapienza.”

    Google Scholar 

  • Derobertis, L. and Hartigan, J. (1981). Bayesian inference using intervals of measures. Annals of Statistics, 9, 235–244.

    Article  MathSciNet  Google Scholar 

  • Dinkelbach, W. (1967). On nonlinear fractional programming. Management Science, 13, 492–498.

    Article  MathSciNet  MATH  Google Scholar 

  • Ibaraki T. (1981). Solving mathematical programming problems with fractional objective functions. In Generalized Concavity in Optimization and Economics (S. Schaible and W.T. Ziemba, eds.), 441–472.

    Google Scholar 

  • Isbell, J.R. and Marlow, W.H. (1956). Attrition games. Naval Research Logistic Quarterly, 3, 71–93.

    Article  MathSciNet  Google Scholar 

  • Jagannathan, R. (1966). On some properties of programming problems in parametric form pertaining to fractional programming. Management Science, 12, 609–615.

    Article  MathSciNet  MATH  Google Scholar 

  • Kemperman, J.H.B. (1987). Geometry of the moment problem. In Moments in mathematics, Proceedings of Symposia in Applied Mathematics, 37, 16–53.

    MathSciNet  Google Scholar 

  • Lavine, M. (1991). Sensitivity in Bayesian statistics: the prior and the likelihood. Journal of the American Statistical Association, 86, 396–399.

    Article  MathSciNet  MATH  Google Scholar 

  • Lavine, M. (1994). An approach to evaluating sensitivity in Bayesian regression analyses. Journal of Statistical Planning and Inference, 40, 242–244.

    Article  MathSciNet  Google Scholar 

  • Lavine, M., Wasserman, L. and Wolpert, R.L. (1991). Bayesian inference with specified prior marginals. Journal of the American Statistical Association, 86, 964–971.

    Article  MathSciNet  MATH  Google Scholar 

  • Lavine, M., Wasserman, L. and Wolpert, R.L. (1993). Linearization of Bayesian robustness problems. Journal of Statistical Planning and Inference, 37, 307–316.

    Article  MathSciNet  MATH  Google Scholar 

  • Liseo, B., Moreno, E. and Salinetti, G. (1996). Bayesian robustness for classes of bidimensional priors with given marginals. In Bayesian Robustness, IMS Lecture Notes — Monograph Series, (J.O. Berger, B. Betrò, E. Moreno, L.R. Pericchi, F. Ruggeri, G. Salinetti, and L. Wasserman, eds.), 101–115. Hayward: IMS.

    Google Scholar 

  • Liseo, B., Petrella, L. and Salinetti, G. (1993). Block unimodality for multivariate Bayesian robustness. Journal of the Italian Statistical Society, 2, 55–71.

    Article  MATH  Google Scholar 

  • Moreno, E. (2000). Global Bayesian robustness for some classes of prior distributions. In Robust Bayesian Analysis (D. Ríos Insua and F. Ruggeri, eds.). New York: Springer-Verlag.

    Google Scholar 

  • O’Hagan, A. and Berger, J.O. (1988). Ranges of posterior probabilities for quasiunimodal priors with specified quantiles. Journal of the American Statistical Association, 83, 503–508.

    Article  MathSciNet  Google Scholar 

  • Perone Pacifico, M., Salinetti, G. and Tardella L. (1994). Fractional optimization in Bayesian robustness. Technical Report, A 23, Dipartimento di Statistica, Probabilit Statistiche Applicate, Universiti Roma “La Sapienza.”

    Google Scholar 

  • Perone Pacifico, M., Salinetti, G. and Tardella, L. (1996). Bayesian robustness on constrained density band classes. Test, 5, 395–409.

    Article  MathSciNet  Google Scholar 

  • Perone Pacifico, M., Salinetti, G. and Tardella, L. (1998). A note on the geometry of Bayesian global and local robustness. Journal of Statistical Planning and Inference, 69, 51–64.

    Article  MathSciNet  Google Scholar 

  • Salinetti, G. (1994). Discussion to “An Overview of Robust Bayesian Analysis” by J.O. Berger. Test, 3, 109–115.

    Google Scholar 

  • Schaible, S. (1976). Fractional programming I, Duality. Management Science, 22, 858–867.

    Article  MathSciNet  MATH  Google Scholar 

  • Schaible, S. (1981). A survey of fractional programming, in Generalized Concavity in Optimization and Economics (S. Schaible and W.T. Ziemba, eds.), 417–440.

    Google Scholar 

  • Walley, P. (1991). Statistical Reasoning with Imprecise Probabilities. London: Chapman and Hall.

    Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Science+Business Media New York

About this chapter

Cite this chapter

Lavine, M., Pacifico, M.P., Salinetti, G., Tardella, L. (2000). Linearization Techniques in Bayesian Robustness. In: Insua, D.R., Ruggeri, F. (eds) Robust Bayesian Analysis. Lecture Notes in Statistics, vol 152. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1306-2_14

Download citation

  • DOI: https://doi.org/10.1007/978-1-4612-1306-2_14

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-0-387-98866-5

  • Online ISBN: 978-1-4612-1306-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics