# Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling

@article{Gelfand1990IllustrationOB, title={Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling}, author={Alan E. Gelfand and Susan E. Hills and Amy Racine‐Poon and Adrian F. M. Smith}, journal={Journal of the American Statistical Association}, year={1990}, volume={85}, pages={972-985} }

Abstract The use of the Gibbs sampler as a method for calculating Bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models, including variance components, unordered and ordered means, hierarchical growth curves, and missing data in a crossover trial. In all cases the approach is straightforward to specify distributionally and to implement computationally, with output readily adapted for required inference summaries.

## 1,075 Citations

### On Bayesian calculations for mixture likelihoods and priors.

- EconomicsStatistics in medicine
- 1999

The methodology can give substantially greater accuracy than the traditional approach and is illustrated using the model selection prior of George and McCulloch applied to logistic regression and to a mixture model for observations in a hierarchical random effects model.

### Bayesian Computations in Survival Models via the Gibbs Sampler

- Mathematics, Computer Science
- 1992

This paper illustrates how these apparent difficulties can be overcome, in both parametric and nonparametric settings, by the Gibbs sampler approach to Bayesian computation.

### MCMC methods to approximate conditional predictive distributions

- Computer ScienceComput. Stat. Data Anal.
- 2006

### Bayesian Inference for Generalized Linear and Proportional Hazards Models Via Gibbs Sampling

- Computer Science
- 1993

It is shown that Gibbs sampling, making systematic use of an adaptive rejection algorithm proposed by Gilks and Wild, provides a straightforward computational procedure for Bayesian inferences in a…

### Inference for nonconjugate Bayesian Models using the Gibbs sampler

- Computer Science
- 1991

The Gibbs sampler technique is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework and the result is a general strategy for obtaining marginal posterior densities under changing specification of the model error densities and related prior densities.

### Bayesian Analysis of Constrained Parameter and Truncated Data Problems

- Computer Science, Mathematics
- 1991

This paper illustrates how the Gibbs sampler approach to Bayesian calculation avoids these difficulties and leads to straightforwardly implemented procedures, even for apparently very complicated model forms.

### Hierarchical Bayesian Analysis of Changepoint Problems

- Computer Science
- 1992

A general approach to hierarchical Bayes changepoint models is presented, including an application to changing regressions, changing Poisson processes and changing Markov chains, which avoids sophisticated analytic and numerical high dimensional integration procedures.

### A bayesian analysis for estimating the common mean of independent normal populations using the gibbs sampler

- Mathematics
- 1997

Combining information from several independent normal populations to estimate a common parameter has applications in meta-analysis and is an important statistical problem. For this application a…

### Bayesian Analysis of Stochastically Ordered Distributions of Categorical Variables

- Mathematics, Computer Science
- 1997

A Bayesian approach to assess the strength of evidence produced by sampled data for a hypothesis of a specified stochastic ordering among the underlying distributions and to estimate these distributions subject to the ordering is presented.

## References

SHOWING 1-10 OF 33 REFERENCES

### Sampling-Based Approaches to Calculating Marginal Densities

- Computer Science
- 1990

Abstract Stochastic substitution, the Gibbs sampler, and the sampling-importance-resampling algorithm can be viewed as three alternative sampling- (or Monte Carlo-) based approaches to the…

### Parametric Empirical Bayes Inference: Theory and Applications

- Mathematics
- 1983

Abstract This article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical Bayes viewpoint, particularly in the case of parametric prior distributions. Some…

### Bayesian inference in statistical analysis

- Computer Science, Mathematics
- 1973

This chapter discusses Bayesian Assessment of Assumptions, which investigates the effect of non-Normality on Inferences about a Population Mean with Generalizations in the context of a Bayesian inference model.

### The implementation of the bayesian paradigm

- Computer Science
- 1985

A numerical integration strategy based on Gaussian quadrature, and an associated strategy for the reconstruction and display of distributions based on spline techniques are described.

### Econometric illustrations of novel numerical integration strategies for Bayesian inference

- Mathematics
- 1988

### Order restricted statistical inference

- Mathematics
- 1988

Isotonic Regression. Tests of Ordered Hypotheses: The Normal Means Case. Tests of Ordered Hypotheses: Generalizations of the Likelihood Ratio Tests and Other Procedures. Inferences about a Set of…

### The calculation of posterior distributions by data augmentation

- Computer Science
- 1987

If data augmentation can be used in the calculation of the maximum likelihood estimate, then in the same cases one ought to be able to use it in the computation of the posterior distribution of parameters of interest.

### INFERENCE AND MISSING DATA

- Geology
- 1975

Two results are presented concerning inference when data may be missing. First, ignoring the process that causes missing data when making sampling distribution inferences about the parameter of the…

### A Bayesian approach to nonlinear random effects models.

- Computer ScienceBiometrics
- 1985

Nonlinear random effects models are considered from the Bayesian point of view and the numerical method is related to the EM algorithm.