Markov Chains for Exploring Posterior Distributions

@article{Tierney1994MarkovCF,
  title={Markov Chains for Exploring Posterior Distributions},
  author={Luke Tierney},
  journal={Annals of Statistics},
  year={1994},
  volume={22},
  pages={1701-1728}
}
  • L. Tierney
  • Published 1994
  • Mathematics
  • Annals of Statistics
Several Markov chain methods are available for sampling from a posterior distribution. Two important examples are the Gibbs sampler and the Metropolis algorithm. In addition, several strategies are available for constructing hybrid algorithms. This paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues. On the theoretical side, results from the theory of general state space Markov chains can be used to obtain convergence rates, laws… Expand
MARKOV CHAIN MONTE CARLO METHODS: COMPUTATION AND INFERENCE
TLDR
This chapter provides background on the relevant Markov chain theory and provides detailed information on the theory and practice of MarkovChain sampling based on the Metropolis-Hastings and Gibbs sampling algorithms. Expand
Markov Chain Monte Carlo
Markov chain Monte Carlo (MCMC) is a technique for estimating by simulation the expectation of a statistic in a complex model. Successive random selections form a Markov chain, the stationaryExpand
Markov Chain Monte Carlo Methods for Statistical Inference
These notes provide an introduction to Markov chain Monte Carlo methods and their applications to both Bayesian and frequentist statistical inference. Such methods have revolutionized what can beExpand
An Introduction to Markov Chain Monte Carlo
TLDR
The Bayesian approach to statistics is introduced, and the necessary continuous state space Markov chain theory is summarized, and two common algorithms for generating random draws from complex joint distribution are presented. Expand
Monte Carlo Methods and Bayesian Computation : MCMC
Markov chain Monte Carlo (MCMC) methods use computer simulation of Markov chains in the parameter space. The Markov chains are defined in such a way that the posterior distribution in the givenExpand
Studying Convergence of Markov Chain Monte Carlo Algorithms Using Coupled Sample Paths
Abstract I describe a simple procedure for investigating the convergence properties of Markov chain Monte Carlo sampling schemes. The procedure uses coupled chains from the same sampler, obtained byExpand
Over-relaxation methods and coupled Markov chains for Monte Carlo simulation
TLDR
A new algorithm for simulating from multivariate Gaussian densities is proposed, which combines ideas from coupled Markov chain methods and from an existing algorithm based only on over-relaxation. Expand
Metropolis Sampling
TLDR
This document describes in details all the elements involved in the MH algorithm and the most relevant variants, providing a quick but exhaustive overview of the current Metropolis-based sampling’s world. Expand
Hypothesis Tests of Convergence in Markov Chain Monte Carlo
Abstract Deciding when a Markov chain has reached its stationary distribution is a major problem in applications of Markov Chain Monte Carlo methods. Many methods have been proposed ranging fromExpand
Markov chain Monte Carlo for dynamic generalised linear models
SUMMARY This paper presents a new methodological approach for carrying out Bayesian inference about dynamic models for exponential family observations. The approach is simulationbased and involvesExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 78 REFERENCES
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, severalExpand
Optimum Monte-Carlo sampling using Markov chains
SUMMARY The sampling method proposed by Metropolis et al. (1953) requires the simulation of a Markov chain with a specified 7i as its stationary distribution. Hastings (1970) outlined a generalExpand
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 andExpand
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. WhenExpand
On the Convergence of Successive Substitution Sampling
Abstract The problem of finding marginal distributions of multidimensional random quantities has many applications in probability and statistics. Many of the solutions currently in use are veryExpand
Finite Markov chains
TLDR
This lecture reviews the theory of Markov chains and introduces some of the high quality routines for working with Markov Chains available in QuantEcon.jl. Expand
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 theExpand
On the Geometric Convergence of the Gibbs Sampler
SUMMARY The rate of convergence of the Gibbs sampler is discussed. The Gibbs sampler is a Monte Carlo simulation method with extensive application to computational issues in the Bayesian paradigm.Expand
Markov Chains and Stochastic Stability
TLDR
This second edition reflects the same discipline and style that marked out the original and helped it to become a classic: proofs are rigorous and concise, the range of applications is broad and knowledgeable, and key ideas are accessible to practitioners with limited mathematical background. Expand
Constrained Monte Carlo Maximum Likelihood for Dependent Data
Maximum likelihood estimates (MLEs) in autologistic models and other exponential family models for dependent data can be calculated with Markov chain Monte Carlo methods (the Metropolis algorithm orExpand
...
1
2
3
4
5
...