G.O. Roberts, Convergence diagnostics of the Gibbs sampler, in Bayesian Statistics 4 (ed. J.M. Bernardo et. al.), Oxford University Press, Oxford, 777-784.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[BYM] p. 6) This may be risky since Gibbs sampling sometimes converges very slowly; see for example [M] Now, it may be possible to use convergence diagnostics to check if the distribution after (say) 1000 steps is indeed close to the distribution to which the chain appears to converge; see [G] [Rob]. On the other hand, see [GR] for warnings about possible problems. In any case, it would be comforting to have theoretical results regarding how many iterations are required before the chain has in fact converged. There has been limited analysis of this question to date (though it can be ....

G.O. Roberts, Convergence diagnostics of the Gibbs sampler, in Bayesian Statistics 4 (ed. J.M. Bernardo et. al.), Oxford University Press, Oxford, 777-784.

....we can test a sampler. However, we can compare the representation produced by the sampler to methods that are significantly cheaper computationally. 5.1. Reality Checks: Has the Sampler Burnt in and is it Mixing There are convergence diagnostics for MCMC methods (e.g. see Besag et al. 1995; Roberts, 1992), but these can suggest convergence where none exists; it is easy to produce a chain that can pass these tests without having burnt in. Instead, we rely on general methods. Firstly, we check to ensure that the sampler can move to a nearmaximal value of the posterior from any start position within ....

....the first k samples must be discarded to allow the sampler to burn in . The rest represent the posterior; but what is k . The usual approach is to start different sequences at different points, and then confirm that they give comparable answers (e.g. Gelman and Rubin, 1993; Geweke, 1992; Roberts, 1992). Another approach is to prove that the proposal process has rapid mixing properties (which is extremely difficult, e.g. Jerrum and Sinclair, 1996) Rapid mixing is desirable, because the faster the sampler mixes, the lower the variance of expectations estimated using samples (Geyer, 1999) The ....

Roberts, G.O. 1992. Convergence diagnostics of the gibbs sampler. In Bayesian Statistics 4, J.M. Bernardo, J. Berger, A.P. Dawid, and A.F.M. Smith (Eds.), Oxford: Clarendon Press.

....are taken. The algorithm should be continued past burn in until some reasonable number of independent samples ought to have been obtained. Because a state retains memory of its predecessors, many intervening steps may be needed before samples become e#ectively independent. Progress has been made (Roberts, 1992), but unfortunately there are as yet no generally useful formal estimates of convergence times. Thus, in practice, MCMC algorithms are continued until the spread of samples appears to have converged, according to some ad hoc criterion. One such criterion is that the log likelihood log Pr(D X) fit ....

Roberts, G.O. (1992) Convergence diagnostics of the Gibbs sampler. In Bayesian Statistics (ed. J.M. Bernardo, J.O. Berger, A.P. Dawid & A.F.M.

....rat growth In this example, coupled MCMC sampling is used to investigate the convergence properties of the Gibbs sampling scheme proposed by Gelfand, Hills, Racine Poon, and Smith (1990) for analyzing growth rates of 30 rats. Similar convergence studies have been performed on the this data set by Roberts and Hills (1992) and Garren and Smith (1994) The data for this study were collected by weighing k = 30 rats every week for five weeks. A linear growth model of the form y ij N(ff i fi i x j ; oe 2 ) i = 1; k; j = 1; 5; was used to model the weight of the i th rat, y ij , at the j th ....

Roberts, G.O. (1992), "Convergence diagnostics of the Gibbs sampler," in Bayesian Statistics 4, eds. Bernardo, Berger, Dawid, and Smith, Oxford University Press, 777-784.

....from the difficulties of precise analysis of complicated models, and have largely tended to concentrate on relatively simple problems, and or to provide impractically large upper bounds. Consequently, most applied users of MCMC techniques have used convergence diagnostics (see for example, Roberts, 1992; Gelman and Rubin, 1992; Raftery and Lewis, 1992) to assess convergence. These diagnostics often work well in practice; however they are not completely understood and offer no guarantees. See Cowles and Carlin (1996) for a comprehensive review. In this paper, we present a way to make use of ....

Roberts, G.O. (1992), "Convergence Diagnostics of the Gibbs Sampler." In Bayesian Statistics 4 (J.M. Bernardo et al., eds.), 777-784. Oxford University Press.

....on the output of any Gibbs sampler. One drawback is the assumption of approximate normality of the variable of interest and the cases that are of more applied interest are those where the assumption of normality is not justified. The most mathematically sound convergence diagnostic was proposed by Roberts (1992). This method requires that we run a reversible Gibbs sampler which in one iteration cycles from the first component of X to the last and then from the last component back to the first again. Based on this sampler Roberts defines a distributional norm such that fi fi fi fi (n) Gamma fi ....

Roberts, G.O. (1992), "Convergence Diagnostics of the Gibbs Sampler" In Bayesian Statistics 4 (J.M. Bernardo, J.O.

....of Gelman and Rubin (1992) and of Raftery and Lewis (1992) currently are the most popular amongst the statistical community, at least in part because computer programs for their implementation are available from their creators. In addition to these two, we discuss the methods of Geweke (1992) Roberts (1992, 1994) Ritter and Tanner (1992) Zellner and Min (1995) Liu, Liu, and Rubin (1992) Garren and Smith (1993) Johnson (1994) Mykland, Tiermey, and Yu (1995) Yu (1994) and Yu and Mykland (1994) Furthermore, we mention some related ideas from the operations research literature, focussing on ....

....of Geweke s method include that it is sensitive to the specification of the spectral window. In addition, while his diagnostic is quantitative, Geweke does not specify a procedure for applying it but instead leaves that to the experience and subjective choice of the statistician. 2. 4 Roberts (1992, 1994) Roberts (1992) presents a one dimensional diagnostic intended to assess convergence of the entire joint distribution. His method is applicable when the distributions of the iterates of the Gibbs sampler have continuous densities. Roberts method requires a symmetrized Gibbs sampler ....

[Article contains additional citation context not shown here]

Roberts, G.O. (1992), "Convergence Diagnostics of the Gibbs Sampler." In Bayesian Statistics 4 (eds.

....[BYM] p. 6) This may be risky since Gibbs sampling sometimes converges very slowly; see for example [M] Now, it may be possible to use convergence diagnostics to check if the distribution after (say) 1000 steps is indeed close to the distribution to which the chain appears to converge; see [G] [Rob]. On the other hand, see [GR] for warnings about possible problems. In any case, it would be comforting to have theoretical results regarding how many iterations are required before the chain has in fact converged. There has been limited analysis of this question to date (though it can be expected ....

G.O. Roberts, Convergence diagnostics of the Gibbs sampler, in Bayesian Statistics 4 (ed. J.M. Bernardo et. al.), Oxford University Press, Oxford, 777-784.

No context found.

G.O. Roberts (1992), Convergence diagnostics of the Gibbs sampler. In Bayesian Statistics 4 (J.M. Bernardo et al., eds.), 777--784. Oxford University Press.

No context found.

G.O. Roberts, Convergence diagnostics of the Gibbs sampler, in Bayesian Statistics 4 (ed. J.M. Bernardo et. al.), Oxford University Press, Oxford, 777-784.

No context found.

Roberts, G. O. (1992). Convergence Diagnostics of the Gibbs Sampler. In Bayesian Statistics 4, ( J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith, Eds. ). Oxford University Press, Oxford, pp. 169--193.

No context found.

Roberts, G.O. (1992), "Convergence diagnostics of the Gibbs sampler," in Bayesian Statistics 4, eds. Bernardo, Berger, Dawid, and Smith, Oxford University Press, 777-784.

No context found.

G.O. Roberts (1992), Convergence diagnostics of the Gibbs sampler. In Bayesian Statistics 4 (J.M. Bernardo et al., eds.), 777-784. Oxford University Press.

No context found.

Roberts, G.O. (1992), "Convergence diagnostics of the Gibbs sampler," in Bayesian Statistics 4, eds. Bernardo, Berger, Dawid, and Smith, Oxford University Press, 777-784.

No context found.

G. O. Roberts (1992), Convergence Diagnostics of the Gibbs Sampler. Bayesian Statistics, 4, 775-782. 117

No context found.

Roberts, G. O. (1992), "Convergence Diagnostics of the Gibbs Sampler," In Bayesian Statistics, J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith (eds.), 4, 775-782.

No context found.

Roberts, G.O. (1992), "Convergence diagnostics of the Gibbs sampler," in Bayesian Statistics 4, eds. Bernardo, Berger, Dawid, and Smith, Oxford University Press, 777-784.

No context found.

Roberts, G. O. (1992). Convergence Diagnostics of the Gibbs Sampler. In Bayesian Statistics, J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith (Eds.) 4 775-782.

No context found.

G.O. Roberts (1992), Convergence diagnostics of the Gibbs sampler. In Bayesian Statistics 4 (J.M. Bernardo et al., eds.), 777-784. Oxford University Press.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC