Bayesian Computations in Survival Models via the Gibbs Sampler

Survival models used in biomedical and reliability contexts typically involve data censoring, and may also involve constraints in the form of ordered parameters. In addition, inferential interest often focuses on non-linear functions of natural model parameters. From a Bayesian statistical analysis perspective, these features combine to create difficult computational problems by seeming to require (multi-dimensional) numerical integrals over awkwardly defined regions. This paper illustrates how these apparent difficulties can be overcome, in both parametric and nonparametric settings, by the Gibbs sampler approach to Bayesian computation.

[1]  E. Kaplan,et al.  Nonparametric Estimation from Incomplete Observations , 1958 .

[2]  T. Ferguson.  A Bayesian Analysis of Some Nonparametric Problems , 1973 .

[3]  C. Antoniak.  Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems , 1974 .

[4]  B. Turnbull.  Nonparametric Estimation of a Survivorship Function with Doubly Censored Data , 1974 .

[5]  J. V. Ryzin,et al.  Nonparametric Bayesian Estimation of Survival Curves from Incomplete Observations , 1976 .

[6]  A.F.M. Smith.  A Bayesian Note on Reliability Growth During a Development Testing Program , 1977, IEEE Transactions on Reliability.

[7]  G. J. Hahn,et al.  A Simple Method for Regression Analysis With Censored Data , 1979 .

[8]  T. Ferguson,et al.  Bayesian Nonparametric Estimation Based on Censored Data , 1979 .

[9]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[10]  M. Woodroofe.  Estimating a Distribution Function with Truncated Data , 1985 .

[11]  L. Kuo.  Computations of mixtures of dirichlet processes , 1986 .

[12]  L. Tierney,et al.  Accurate Approximations for Posterior Moments and Marginal Densities , 1986 .

[13]  A. F. M. Smith,et al.  Progress with numerical and graphical methods for practical Bayesian statistics , 1987 .

[14]  Duane L. Dietrich,et al.  A Bayes Reliability Growth Model for A Development Testing Program , 1987, IEEE Transactions on Reliability.

[15]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[16]  S. E. Hills,et al.  Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling , 1990 .

[17]  Adrian F. M. Smith,et al.  Bayesian Analysis of Constrained Parameter and Truncated Data Problems , 1991 .

[18]  Adrian F. M. Smith,et al.  Gibbs Sampling for Marginal Posterior Expectations , 1991 .

[19]  W. Gilks,et al.  Adaptive Rejection Sampling for Gibbs Sampling , 1992 .

[20]  Adrian F. M. Smith,et al.  Bayesian Inference for Generalized Linear and Proportional Hazards Models Via Gibbs Sampling , 1993 .