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  • Markov chain Monte Carlo  (2)
  • Springer  (2)
  • 1
    Electronic Resource
    Electronic Resource
    Analysis of the Gibbs sampler for a model related to James-Stein estimators (1996)
    Springer
    Statistics and computing 6 (1996), S. 269-275 
    Details
    ISSN: 1573-1375
    Keywords: Convergence rate ; James-Stein estimator ; Gibbs sampler ; Markov chain Monte Carlo
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract We analyse a hierarchical Bayes model which is related to the usual empirical Bayes formulation of James-Stein estimators. We consider running a Gibbs sampler on this model. Using previous results about convergence rates of Markov chains, we provide rigorous, numerical, reasonable bounds on the running time of the Gibbs sampler, for a suitable range of prior distributions. We apply these results to baseball data from Efron and Morris (1975). For a different range of prior distributions, we prove that the Gibbs sampler will fail to converge, and use this information to prove that in this case the associated posterior distribution is non-normalizable.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00140871
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  • 2
    Electronic Resource
    Electronic Resource
    A simulation approach to convergence rates for Markov chain Monte Carlo algorithms (1998)
    Springer
    Statistics and computing 8 (1998), S. 115-124 
    Details
    ISSN: 1573-1375
    Keywords: Drift condition ; Gibbs sampler ; Markov chain Monte Carlo ; Metropolis–Hastings algorithm ; minorization condition ; ordinal probit ; variance components
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract Markov chain Monte Carlo (MCMC) methods, including the Gibbs sampler and the Metropolis–Hastings algorithm, are very commonly used in Bayesian statistics for sampling from complicated, high-dimensional posterior distributions. A continuing source of uncertainty is how long such a sampler must be run in order to converge approximately to its target stationary distribution. A method has previously been developed to compute rigorous theoretical upper bounds on the number of iterations required to achieve a specified degree of convergence in total variation distance by verifying drift and minorization conditions. We propose the use of auxiliary simulations to estimate the numerical values needed in this theorem. Our simulation method makes it possible to compute quantitative convergence bounds for models for which the requisite analytical computations would be prohibitively difficult or impossible. On the other hand, although our method appears to perform well in our example problems, it cannot provide the guarantees offered by analytical proof.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1008982016666
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