Local robustness of Bayesian parametric inference and observed likelihoods
Smith, J. Q., 1953- (2007) Local robustness of Bayesian parametric inference and observed likelihoods. Working Paper. Coventry: University of Warwick. Centre for Research in Statistical Methodology. (Working papers).
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Here a new class of local separation measures over prior densities is studied and their usefulness for examining prior to posterior robustness under a sequence of observed likelihoods, possibly erroneous, illustrated. It is shown that provided an approximation to a prior distribution satisfies certain mild smoothness and tail conditions then prior to posterior inference for large samples is robust, irrespective of whether the priors are grossly misspecified with respect to variation distance and irrespective of the form or the validity of the observed likelihood. Furthermore it is usually possible to specify error bounds explicitly in terms of statistics associated with the posterior associated with the approximating prior and asumed prior error bounds. These results apply in a general multivariate setting and are especially easy to interpret when prior densities are approximated using standard families or multivariate prior densities factorise.
|Item Type:||Working or Discussion Paper (Working Paper)|
|Subjects:||Q Science > QA Mathematics|
|Divisions:||Faculty of Science > Statistics|
|Library of Congress Subject Headings (LCSH):||Distribution (Probability theory), Bayesian statistical decision theory|
|Series Name:||Working papers|
|Publisher:||University of Warwick. Centre for Research in Statistical Methodology|
|Place of Publication:||Coventry|
|Number of Pages:||39|
|Status:||Not Peer Reviewed|
|Access rights to Published version:||Open Access|
|References:|| Andrade, J. A. A. and O'Hagan, A. (2006). Bayesian robustness modelling using regularly varying distributions. Bayesian Analysis 1, 169-188.  Berger, J. (1992) in discussion of Wasserman, L.(1992b) "Recent methodological advances in robust Bayesian inference (with discussion)" In Bayesian Statistics 4 J.M. Bernado et al (eds) 495 - 496 Oxford University Press  Bernardo, J.M. and Smith, A.F.M.(1996) "Bayesian Theory" Wiley Chichester  Blackwell, D. and Dubins, L.(1962) "Merging of opinions with increasing information" Annals of Mathematical Statistics, 33, 882 -886  Daneseshkhah,A (2004) "Estimation in Causal Graphical Models" PhD Thesis University of Warwick.  Dawid, A.P. (1973) "Posterior expectations for large observations" Bio- metrika 60 664 -667  DeRobertis, L. (1978) "The use of partial prior knowledge in Bayesian inference" Ph.D. idssertation, Yale Univ.  Devroye, L. and Gyor, L.(1985) "Non-parametric density estimation - the L1 view" Wiley New York  Gustafson, P. and Wasserman, L. (1995) "Local sensitivity diagnostics for Bayesiain inference" Annals Statist ,23 , 23, 2153 - 2167  Ghosal,S, Lember,J. and van der Vaart, A.W. (2002) "On Bayesian badap- tion" Proceedings 8th Vilnius Conference Probability and Statistics, B Grigelionis et al eds.  Ghosh, J.K. and Ramamoorthi, R.V.(2003) "Bayesian Nonparametrics" Springer  Ghosh, J.K. Ghosal, S. aad Samanta, T. (1994) "Stability and Convergence of posteriors in non-regular problems," In Statistical Decision theory and related topics 5 Springer 183 - 199.  West, M. and Harrison, P.J.(1997) "Bayesian Forecasting and Dynamic Models" Springer.  French, S. and Rios Insua, D.(2000) "Statistical Decision Theory" Kendall's Library of Statistics 9 Arnold  Kadane, J.B. and Ghuang, D.T. (1978) "Stable decision problems" Ann. Statist.6, 1095 -111  Lauritzen, S.L.(1996) "Graphicla Models" Oxford University Press.  Lenk, P.J.(1988) "The logistic normal distribution for Bayesian non- parametric predctie densities" J.Amer. Statist. Assoc. 83(402) 509 - 516.  Kruijer, W. and van der Vaart, A.W. (2005) "Posterior Convergence Rates for Dirichlet Mixtures of Beta Densities" Res.Rep. Dept. Mathematics, Univ. Amsterdam.  Marshall, A.W. and Olkin, (1979) "Inequalities: Theory of Majorisation and its Applications" Academic Press  Monhor, D. (2007) "A Chebyshev Inequality for Multivariate Normal Dis- tribution" Probability in the Engineering And Unformational Sciences Vol 21, 2,,289 - 300  Oakley, J. E. and O'Hagan, A. (2007). Uncertainty in prior elicitations: a nonparametric approach.Biometrika.(to appear).  O'Hagan, A.(1979) On outlier rejection phenomena in Bayesian inference J.R. Statist. Soc. B 41, 358 - 367  O'Hagan, A and Forster, J (2004) "Bayesian Inference" Kendall's Advanced Theory of Statistics, Arnold  Schervish, M.J. (1995) "The Theory of Statistics" Springer Verlag New York  Shafer, G., Gillett,P. R. and Scherl,R. (2003) A new understanding of subjective probability and its generalization to lower and upper prevision International Journal of Approximate Reasoning. 33 1-49.  Smith, J.Q.(1979) "A generalisation of the Bayesian steady forecasting model" J.R.Statist. Soc . B 41, 375 -87  Smith, J.Q.(1981) "The multiparameter steady model" J.R.Statist. Soc . B 43,2, 255-260  Smith,J.Q. and Croft, J. (2003) "Bayesian networks for discrete multivariae data: an algebraic approach to inference" J of Multivariate Analysis 84(2), 387 -402  Tong, Y.L.(1980) "Probability Inequalities in Multivariate Distributions" Academic Press New York  Wasserman, L.(1992a) "Invariance properties of density ratio priors" Ann Statist, 20, 2177- 2182  Wasserman, L.(1992b) "Recent methodological advances in robust Bayesian inference (with discussion)" In Bayesian Statistics 4 J.M. Bernado et al (eds) 483 - 502 Oxford University Press|
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