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Dynamic Display of Changing Posterior in Bayesian Survival Analysis

  • Hani Doss
  • B. Narasimhan
Part of the Lecture Notes in Statistics book series (LNS, volume 133)

Abstract

We consider the problem of estimating an unknown distribution function F in the presence of censoring under the conditions that a parametric model is believed to hold approximately. We use a Bayesian approach, in which the prior on F is a mixture of Dirichlet distributions. A hyperparameter of the prior determines the extent to which this prior concentrates its mass around the parametric family. A Gibbs sampling algorithm to estimate the posterior distributions of the parameters of interest is reviewed. An importance sampling scheme enables us to use the output of the Gibbs sampler to very quickly recalculate the posterior when we change the hyperparameters of the prior. The calculations can be done sufficiently fast to enable the dynamic display of the changing posterior as the prior hyperparameters are varied.

Keywords

Markov Chain Posterior Distribution Conditional Distribution Importance Sampling Borel Subset 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Antoniak, C. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2, 1152–1174.MathSciNetzbMATHCrossRefGoogle Scholar
  2. Diaconis, P. and Freedman, D.A. (1986a). On the consistency of Bayes estimates. Ann. Statist. 14, 1–26.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Diaconis, P. and Freedman, D.A. (1986b). On inconsistent Bayes estimates of location. Ann. Statist. 14, 68–87.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Doss, H. (1984). Bayesian estimation in the symmetric location problem. Z. Wahrsch. verw. Gebiete 68, 127–147.MathSciNetzbMATHCrossRefGoogle Scholar
  5. Doss, H. (1985a). Bayesian nonparametric estimation of the median; Part I: computation of the estimates. Ann. Statist. 13, 1432–44.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Doss, H. (1985b). Bayesian nonparametric estimation of the median; Part II: asymptotic properties of the estimates. Ann. Statist. 13, 1445–64.MathSciNetzbMATHCrossRefGoogle Scholar
  7. Doss, H. (1994). Bayesian nonparametric estimation for incomplete data via successive substitution sampling. Ann. Statist. 22, 1763–1786.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Doss, H. and Huffer, F. (1998). Monte Carlo methods for Bayesian analysis of survival data using mixtures of Dirichlet priors. Technical Report, Department of Statistics, Ohio State University.Google Scholar
  9. Doss, H. and Narasimhan, B. (1998). Dynamic display of changing posterior in Bayesian survival analysis: the software. Technical Report, Department of Statistics, Stanford University.Google Scholar
  10. Ferguson, T.S. (1973). A Bayesian analysis of some nonparametric problems. Ann. Statist. 1, 209–230.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Ferguson, T.S. (1974). Prior distributions on spaces of probability measures. Ann. Statist. 2, 615–629.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Hastings, W.K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109.zbMATHCrossRefGoogle Scholar
  13. Hjort, N.L. (1990). Goodness of fit tests in models for life history data based on cumulative hazard rates. Ann. Statist. 18, 1221–1258.MathSciNetzbMATHCrossRefGoogle Scholar
  14. Klein, J. and Moeschberger, M. (1997). Survival Analysis. Springer-Verlag, New York.zbMATHGoogle Scholar
  15. Knuth, D.E. (1992). Literate Programming Center for Study of Language and Information, Stanford University.Google Scholar
  16. Kong, A., Liu, J.S. and Wong, W.H. (1994). Sequential imputations and Bayesian missing data problems. J. Amer. Statist. Assoc. 89, 278–288.zbMATHCrossRefGoogle Scholar
  17. Pfanzagl, J. (1979). Conditional distributions as derivatives. Ann. Probab. 7, 1046–1050.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Tierney, L. (1990). Lisp-Stat: An Object-Oriented Environment for Sta- tistical Computing and Dynamic Graphics. Wiley, New York.CrossRefGoogle Scholar
  19. Turnbull, B.W. (1974). Nonparametric estimation of a survivorship function with doubly censored data. J. Amer. Statist. Assoc. 69, 169–173.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Turnbull, B.W. (1976). The empirical distribution function with arbitrarily grouped, censored, and truncated data.J. Roy. Statist. Society, Ser. B 38, 290–295.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Hani Doss
  • B. Narasimhan

There are no affiliations available

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