Polson, N., G. (1996). Convergence of Markov chain Monte Carlo algorithms. In Bayesian Statistics  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... 1997) There has been much work on developing rigorous, a priori, quantitative bounds on the convergence time; see for example Sinclair Jerrum (1989) Diaconis Stroock (1991) Frieze, Kannon Polson (1994) Ingrassia (1994) Meyn Tweedie (1994) Rosenthal (1995) Mengersen Tweedie (1996) Polson (1996) and Frigessi, Martinelli Stander (1997) Application of many of these results is difficult in practice except to the simplest of problems, and the constant of proportionality is not always available. In this paper, we show how coupling methodology can be used to give precise, a priori bounds on ....

Polson, N.G. (1996). Convergence of Markov chain Monte Carlo algorithms (with Discussion) . In Bayesian Statistics 5. Ed. J.M. Bernardo, A.P. Dawid, and A.F.M. Smith, pp. 297--321. Oxford: Oxford University Press.

.... number of iterations needed for convergence (see, amongst others, 4, 10, 9] for references) 2 Main results 3 Most of this work has involved using the total variation distance (the L 1 norm) from stationarity of positive recurrent Markov chains, although there has been some work in other norms [8]. The purpose of this paper is to show that geometric convergence in two of these norms, namely the L 1 and L 2 norms, are equivalent, at least in the reversible case which is relevant in most MCMC and many queueing contexts. We also point out that this result can be made quantitative, in ....

N.G. Polson. Convergence of Markov chain Monte Carlo algorithms. In J. Bernardo, J. Berger, A.P. Dawid, and A.F.M. Smith, editors, Bayesian Statistics 5, pages 297--321. Oxford University Press, 1995. References 9

.... (1997) There has been much work on developing rigorous, a priori, quantitative bounds on the convergence time (for example, Sinclair Jerrum (1989) Diaconis Stroock (1991) Frieze, Kannon Polson (1994) Ingrassia (1994) Meyn Tweedie (1994) Rosenthal (1995) Mengersen Tweedie (1996) Polson (1996), Frigessi, Martinelli Stander (1997) Many of these results are difficult to apply in practice except to the simplest of problems, and the constant of proportionality is not always available. In this paper, we develop precise, a priori bounds for a simplified problem in Bayesian image ....

Polson, N.G. (1996). Convergence of Markov Chain Monte Carlo Algorithms, in Bayesian Statistics 5, ed. Bernardo, J.M., Dawid, A.P. & Smith, A.F.M. Oxford University Press.

....use p 1 (x) 1 N P N i=1 p 1 (xj i;2 ; i;k ) In general the Metropolis algorithm does not seem to provide such straight forward density estimates. Various mathematical convergence results for Markov chain methods have been obtained; see Tierney (1991) Schervish and Carlin (1992) Polson (1993) and Baxter and Rosenthal (1994) Convergence can be very slow, however. For a simple example of this, involving Gibbs sampling and a function f with two modes, see Evans, Gilula and Guttman (1993) Typically when f is unimodal with roughly ellipsoidal contours, experience suggests that a ....

Polson, N. (1993). Convergence of Markov chain Monte Carlo algorithms. Working paper 93-148, Grad. Sch. Bus., U. of Chicago.

....that will insure convergence in total variation distance to within a specified tolerance of the true stationary distribution. Notice that this goes beyond merely proving that a certain algorithm will converge for a given problem, or even providing a rate for this convergence. For example, Polson (1994) develops polynomial time convergence bounds for a discrete jump Metropolis algorithm operating on a log concave target distribution in a discretized state space. Rosenthal (1993, 1995b, 1995c) instead uses Markov minorization conditions, providing bounds in continuous settings involving finite ....

Polson, N.G. (1994), "Convergence of Markov Chain Monte Carlo Algorithms," to appear in Bayesian Statistics 5, eds. J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith, Oxford: Oxford University Press.

.... Markov chain Monte Carlo (MCMC) methods have received considerable attention as tools for investigating complex probability distributions as those arising in Bayesian statistics, image analysis and spatial statistics, see e.g. Besag and Green, 1993) Besag et al. 1995) Gilks et al. 1996) (Polson, 1996), Smith and Roberts, 1993) and (Tierney, 1994) Especially the algorithms due to (Metropolis et al. 1953) and (Hastings, 1970) have received new interest as general and easy to implement methods for simulating from a probability density (x) known only up to scale. In this paper we nd ....

Polson, N. (1996). Convergence of Markov chain Monte Carlo algorithms (with discussion) . Bayesian Statistics 5, pages 297321.

....bound of 530 iterations required to achieve 1 accuracy in total variation distance. Previous rigorous quantitative bounds for MCMC samplers have generally been established only for very specific models (Meyn and Tweedie, 1994; Rosenthal, 1995) or have involved large undetermined constants (Polson, 1996). Indeed, we know of no comparable result which gives a reasonable uniform bound on the convergence rate of a realistic sampling algorithm, over such a broad class of distributions. Of course, it may not always be easy to implement a slice sampler for a particular problem. For example, the sets ....

Polson, N.G. (1996), Convergence of Markov chain Monte Carlo algorithms. In Bayesian Statistics V, 599-608, Clarendon press, Oxford.

....in Step 1 one needs to draw the i from the marginal distribution of . However, drawing i and i together is often easier than drawing i alone from its marginal distribution. In such case can be considered as an auxiliary variable or a latent variable. As Besag and Green (1993) and Polson (1996) pointed out, use of latent variables in Monte Carlo sampling will greatly ease implementation difficulty and dramatically accelerate convergence. Furthermore, if one uses the aforementioned grid numerical integration method to approximate c( the i can be used as part of grid points. For ....

Polson, N.G. (1996). Convergence of Markov chain Monte Carlo algorithms. In Bayesian Statistics 5, J.M. Bernado, J.O. Berger, A.P. Dawid and A.F.M. Smith (Eds.), Oxford University Press, 297-322.

....size, would result in a slowly mixing chain. For the given problem, the factor c = 0:25 turned out to be adequate in the sense that around 50 of the candidates end up being rejected. For a specific case Gelman, Roberts and Gilks (1996) show that an acceptance rate of around 25 is optimal. See Polson (1996) for a formal discussion of the underlying theoretical issues. For a practical implementation we recommend to pick a reasonable step size factor by trial and error . Also, the number of n = 4 iterations in this mini Metropolis is an arbitrary choice. Our rationale was that we did not want to ....

....Drawing candidates very close to the current point (i.e. small c) would lead to acceptance probabilities close to one. Neither is desirable. In the first case, the chain would get trapped in the current state, in the latter case the chain would take too long to move around the parameter space. Polson (1996) formalizes this argument in terms of the implied second eigenvalues of the Markov chain. A Simplified Mixture of Dirichlet Process (MDP) Model. The drawback of simulating the transition from p( t jD t Gamma1 ) to p( t jD t ) by the Metropolis scheme is that it requires pointwise evaluation of ....

Polson, N.G. (1996) Convergence of Markov chain Monte Carlo algorithms. In Bayesian Statistics 5, pp. 292-322, eds. J.M. Bernardo, J.O. Berger, A.P. Dawid, A.F.M. Smith, Oxford University Press.

No context found.

Polson, N., G. (1996). Convergence of Markov chain Monte Carlo algorithms. In Bayesian Statistics

No context found.

Polson, N.G. (1995). "Convergence of Markov chain Monte Carlo algorithms". In: Bayesian Statistics 5, (Eds. J. Bernardo, J. Berger, A.P. Dawid and A.F.M. Smith) Oxford University Press, Oxford (to appear).

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