J.S. Rosenthal (1993), Rates of convergence for data augmentation on finite sample spaces, Ann. Appl. Prob., Vol. 3, No. 3, 819-839.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 sample spaces and certain hierarchical models. While approaches like these hold promise, they typically involve sophisticated mathematics, as well as laborious calculations that ....

Rosenthal, J.S. (1993) "Rates of Convergence for Data Augmentation on Finite Sample Spaces," Annals of Applied Probability, 3, 819-839.

....is a way of running a Markov chain which ensures that the terminal value of the implementation is an exact draw from the stationary distribution of the chain. The idea was introduced by Propp and Wilson in 1996 [20] Since then, the research area has become extremely active, see for example [3, 6, 7, 9, 10, 11, 12, 13, 16, 17, 18]. 2 Slice Sampler Suppose f (x) x 2 X g is an unnormalized integrable density with respect to the measure and let be the corresponding probability measure: A) R A (x) dx) R X (x) dx) for all measurable A. The slice sampler is an auxiliary variables simulation method that ....

....there is a maximal state, namely x max = 0, but there is no minimal state. We thus construct a lower bounding process following the theory given in Section 6.2. Let lb (x) e Gammaqx x 0: 18) For 0 q 1 condition (14) is verified. In the simulation we have taken ff = q = 1. Following [18] once we get an observation from by running the CFTP algorithm we use it as the starting value of a regular slice sampler which we run forward for m iterations. Figure 7 shows a histogram and PP plot obtained after running r = 5 independent replicates of the perfect slice sampler and using each ....

D. Murdoch and J. S. Rosenthal. Rates of convergence for data augmentation on finite sample spaces. Preprint at http://www.stats.bris.ac.uk/MCMC.

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J.S. Rosenthal (1993), Rates of convergence for data augmentation on finite sample spaces, Ann. Appl. Prob., Vol. 3, No. 3, 819-839.

....[BD] See [D] for background, examples, and references. For random walks on compact Lie groups, there has been some recent progress; see [R1] For more general Markov chains, the notion of Harris recurrence (see [A] AN] N] has proven useful in obtaining rates of convergence (see e.g. T] [R2], R3] Finite state space Markov chains remain the simplest case to study, because their convergence can be analyzed directly in terms of the finite spectrum of their transition kernel; see e.g. DS] In this paper, we identify a class of Markov chains, which we call pseudo finite , which are ....

....the convergence rate of general state space Markov chains, but by no means the complete picture. On the other hand, Proposition 3 below shows that a very large class of Markov chains are almost pseudo finite in a certain natural sense. Our interest in pseudo finiteness arose in an application [R2] to Bayesian statistics. The Markov chains considered there were pseudo finite, with the P j being various beta distributions, and the f j ( being related to distributions of sums of binomial distributions. The notion of pseudo finiteness helped the author s analysis of these chains (though it ....

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J.S. Rosenthal (1991), Rates of convergence for data augmentation on finite sample spaces, to appear in Annals of Applied Probability.

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J.S. Rosenthal (1993), Rates of Convergence for Data Augmentation on Finite Sample Spaces. Ann. Appl. Prob. 3, 819-839.

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J.S. Rosenthal (1993a), Rates of convergence for data augmentation on finite sample spaces.

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J.S. Rosenthal (1993a), Rates of convergence for data augmentation on finite sample spaces.

....[BD] See [D] for background, examples, and references. For random walks on compact Lie groups, there has been some recent progress; see [R1] For more general Markov chains, the notion of Harris recurrence (see [A] AN] N] has proven useful in obtaining rates of convergence (see e.g. T] [R2], R3] Finite state space Markov chains remain the simplest case to study, because their convergence can be analyzed directly in terms of the finite spectrum of their transition kernel; see e.g. DS] In this paper, we identify a class of Markov chains, which we call pseudo finite , which are ....

....the convergence rate of general state space Markov chains, but by no means the complete picture. On the other hand, Proposition 3 below shows that a very large class of Markov chains are almost pseudo finite in a certain natural sense. Our interest in pseudo finiteness arose in an application [R2] to Bayesian statistics. The Markov chains considered there were pseudo finite, with the P j being various beta distributions, and the f j ( being related to distributions of sums of binomial distributions. The notion of pseudo finiteness helped the author s analysis of these chains (though it ....

[Article contains additional citation context not shown here]

J.S. Rosenthal (1991), Rates of convergence for data augmentation on finite sample spaces, to appear in Annals of Applied Probability.

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J.S. Rosenthal (1993a), Rates of convergence for Data Augmentation on finite sample spaces. Ann. Appl.

....L(X n Gammak j X 0 0 )k ; i.e. that total variation distance cannot increase with time. Note that this example is not forward sufficient. Furthermore, it is not backward sufficient either unless k = 1. 3. Two variable data augmentation (Tanner and Wong, 1984; Gelfand and Smith, 1990; Rosenthal, 1993). Here there is some target distribution ( Delta) on a product space X Theta Y, and a Markov chain f(X n ; Y n )g on X Theta Y is defined by alternately choosing Y n 1 (dy j x = X n ) and X n 1 (dx j y = Y n 1 ) for n = 0; 1; 2; In this case we clearly have L (X n j X 0 ; ....

....this case we clearly have L (X n j X 0 ; X n Gamma1 ; Y n ) dx j y = Y n 1 ) so that again fY n g is sufficient (and backward sufficient) for fX n g and also for f(X n ; Y n )g. Hence, to study convergence of f(X n ; Y n )g it suffices to study convergence of fY n g, a fact used in Rosenthal (1993). In this example, it is also true that fX n g is forward sufficient for f(X n ; Y n )g. In fact, we can write L ( X n ; Y n ) j (X n Gamma1 ; Y n Gamma1 ) R(X n Gamma1 ; Delta) for appropriate choice of R(x; Delta) Hence, setting h ( X n ; Y n ) X n , we can apply Proposition 4. We ....

Rosenthal, J.S. (1993), Rates of convergence for data augmentation on finite sample spaces. Ann. Appl. Prob., Vol. 3, No. 3, 819--839.

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J.S. Rosenthal (1993), Rates of convergence for Data Augmentation on finite sample spaces.

....of how much less than 1 this ff will be, and how large A is, or equivalently to give a quantitative estimate of how large k should be to make the variation distance less than some ffl. In [SC] several simple models are analyzed exactly, facilitating convergence results for these cases. In [R1], quantitative convergence rates are obtained for Data Augmentation for a two step hierarchical model involving Bernoulli random variables. Also, see [AKP] for an interesting analysis of a related discretization algorithm. In this paper we analyze the convergence rate of the variance component ....

....of a Markov chain to its stationary distribution in terms of the amount of overlap of the transition probabilities starting from different places. The lemma is closely related to the notion of Harris recurrence; see [A] AN] AMN] and [N] A special case of this lemma was described in [R1]. We wish to emphasize that the lemma is valid for any Markov chain, and may be useful in situations quite different from Gibbs sampling. We need the following notation. If Q 1 ( Delta) and Q 2 ( Delta) are probability measures, and ffl 0, then we will write Q 1 ( Delta) ffl Q 2 ( Delta) to ....

[Article contains additional citation context not shown here]

J.S. Rosenthal (1993), Rates of convergence for data augmentation on finite sample spaces, Ann. Appl. Prob., Vol. 3, No. 3, 819-839.

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J.S. Rosenthal (1993), Rates of convergence for Data Augmentation on finite sample spaces.

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Rosenthal, J. (1993a), "Rates of Convergence for Data Augmentation on Finite Sample Spaces," The Annals of Applied Probability, 3, 819-839.

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