Inference from Iterative Simulation Using Multiple Sequences

@article{Gelman1992InferenceFI,
  title={Inference from Iterative Simulation Using Multiple Sequences},
  author={Andrew Gelman and Donald B. Rubin},
  journal={Statistical Science},
  year={1992},
  volume={7},
  pages={457-472}
}
The Gibbs sampler, the algorithm of Metropolis and similar iterative simulation methods are potentially very helpful for summarizing multivariate distributions. Used naively, however, iterative simulation can give misleading answers. Our methods are simple and generally applicable to the output of any iterative simulation; they are designed for researchers primarily interested in the science underlying the data and models they are analyzing, rather than for researchers interested in the… 

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References

SHOWING 1-10 OF 41 REFERENCES

How Many Iterations in the Gibbs Sampler

Abstract : When the Gibbs sampler is used to estimate posterior distributions (Gelfand and Smith, 1990) the question of how many iterations are required is central to its implementation. When

Monte Carlo Sampling Methods Using Markov Chains and Their Applications

SUMMARY A generalization of the sampling method introduced by Metropolis et al. (1953) is presented along with an exposition of the relevant theory, techniques of application and methods and

Spatial Statistics and Bayesian Computation

The early development of MCMC in Bayesian inference is traced, some recent computational progress in statistical physics is reviewed, based on the introduction of auxiliary variables, and its current and future relevance in Bayesesian applications are discussed.

Sampling-Based Approaches to Calculating Marginal Densities

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

Exploring Posterior Distributions Using Markov Chains

Abstract : Several Markov chain-based methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several

Metropolis Methods, Gaussian Proposals and Antithetic Variables

We investigate various aspects of a class of dynamic Monte Carlo methods, that generalises the Metropolis algorithm and includes the Gibbs sampler as a special case. These can be used to estimate

Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments

Methods for spectral analysis are used to evaluate numerical accuracy formally and construct diagnostics for convergence in the normal linear model with informative priors, and in the Tobit-censored regression model.

Accurate Approximations for Posterior Moments and Marginal Densities

These approximations to the posterior means and variances of positive functions of a real or vector-valued parameter, and to the marginal posterior densities of arbitrary parameters can also be used to compute approximate predictive densities.

Using EM to Obtain Asymptotic Variance-Covariance Matrices: The SEM Algorithm

This article defines and illustrates a procedure that obtains numerically stable asymptotic variance–covariance matrices using only the code for computing the complete-data variance-covarance matrix, the code of the expectation maximization algorithm, and code for standard matrix operations.

Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling

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