Electronic Journal of Statistics

On the behaviour of Bayesian credible intervals in partially identified models

Paul Gustafson

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Abstract

Partially identified models typically involve set identification rather than point identification. That is, the distribution of observables is consistent with a set of values for the target parameter, rather than a single value. Interval estimation procedures therefore behave differently than for identified models. For instance, a Bayesian credible set arising from a proper prior distribution will tend to a non-degenerate set as the sample size goes to infinity. A natural question arising is for what parameter values does the limit of the Bayesian credible interval fail to cover its target? Intuition suggests this would arise for parameter values which are not very consistent with the prior distribution. The aim of this paper is to quantify this intuition.

Article information

Source
Electron. J. Statist. Volume 6 (2012), 2107-2124.

Dates
First available in Project Euclid: 2 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1351865119

Digital Object Identifier
doi:10.1214/12-EJS741

Mathematical Reviews number (MathSciNet)
MR3020258

Zentralblatt MATH identifier
1336.62102

Keywords
Bayesian inference credible interval partial identification

Citation

Gustafson, Paul. On the behaviour of Bayesian credible intervals in partially identified models. Electron. J. Statist. 6 (2012), 2107--2124. doi:10.1214/12-EJS741. https://projecteuclid.org/euclid.ejs/1351865119


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References

  • [1] Barankin, E. W. (1960). Sufficient parameters: Solution of the minimal dimensionality problem., Annals of the Institute of Mathematical Statistics 12 91-118.
  • [2] Bollinger, C. and van Hasselt, M. (2009). A Bayesian analysis of binary misclassification: inference in partially Identified models., Preprint.
  • [3] Chickering, D. M. and Pearl, J. (1996). A Clinician’s Tool for Analyzing Non-compliance. In, Proceedings of the Thirteenth National Conference on Artificial Intelligence (AAAI-96), Portland, OR 2 1269-1276.
  • [4] Dawid, A. P. (1979). Conditional independence in statistical theory., Journal of the Royal Statistical Society, Series B 41 1-31.
  • [5] Greenland, S. (2003). The impact of prior distributions for uncontrolled confounding and response bias: A case study of the relation of wire codes and magnetic fields to childhood leukemia., Journal of the American Statistical Association 98 47–55.
  • [6] Greenland, S. (2005). Multiple-bias modelling for analysis of observational data., Journal of the Royal Statistical Society, Series A 168 267-306.
  • [7] Gustafson, P. (2005). On model expansion, model contraction, identifiability, and prior information: two illustrative scenarios involving mismeasured variables (with discussion)., Statistical Science 20 111-140.
  • [8] Gustafson, P. (2010). Bayesian inference for partially identified models., International Journal of Biostatistics 6 issue 2 article 17.
  • [9] Gustafson, P. (2011a). Bayesian inference in partially identified models: Is the shape of the posterior density useful?, Technical Report #265, Department of Statistics, University of British Columbia.
  • [10] Gustafson, P. (2011b). Comment on ‘Transparent parameterizations of models for potential outcomes,’ by Richardson, Evans, and Robins. In, Bayesian Statistics 9: Proceedings of the Ninth Valencia International Meeting (J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West, eds.). Oxford University Press.
  • [11] Gustafson, P. and Greenland, S. (2009). Interval estimation for messy observational data., Statistical Science 24 328-342.
  • [12] Imbens, G. W. and Manski, C. F. (2004). Confidence intervals for partially identified parameters., Econometrica 72 1845-1857.
  • [13] Imbens, G. W. and Rubin, D. B. (1997). Bayesian inference for causal effects in randomized experiments with noncompliance., Annals of Statistics 25 305-327.
  • [14] Kadane, J. B. (1974). The role of identification in Bayesian theory. In, Studies in Bayesian Econometrics and Statistics, In Honor of Leonard J. Savage (S. E. Fienberg and A. Zellner, eds.) 175.
  • [15] Liao, Y. and Jiang, W. (2010). Bayesian analysis in moment inequality models., Annals of Statistics 38 275-316.
  • [16] Manski, C. F. (2003)., Partial Identification of Probability Distributions. Springer.
  • [17] Moon, H. R. and Schorfheide, F. (2012). Bayesian and frequentist inference in partially identified models., Econometrica 80 755-782.
  • [18] Pearl, J. (2000)., Causality: Models, Reasoning, and Inference. Cambridge University Press.
  • [19] Poirier, D. J. (1998). Revising beliefs in nonidentified models., Econometric Theory 14 483–509.
  • [20] Richardson, T. S., Evans, R. J. and Robins, J. M. (2011). Transparent parameterizations of models for potential outcomes. In, Bayesian Statistics 9: Proceedings of the Ninth Valencia International Meeting (J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West, eds.) 569-610. Oxford University Press.
  • [21] Romano, J. P. and Shaikh, A. M. (2008). Inference for identifiable parameters in partially identified econometric models., Journal of Statistical Planning and Inference 138 2786-2807.
  • [22] Vansteelandt, S., Goetghebeur, E., Kenward, M. G. and Molenberghs, G. (2006). Ignorance and uncertainty regions as inferential tools in a sensitivity analysis., Statistica Sinica 16 953-979.
  • [23] Zhang, Z. (2009). Likelihood-based confidence sets for partially identified parameters., Journal of Statistical Planning and Inference 139 696-710.