## Rates of Convergence for Data Augmentation on Finite Sample Spaces (1993)

Venue: | Ann. Appl. Prob |

Citations: | 25 - 13 self |

### BibTeX

@ARTICLE{Rosenthal93ratesof,

author = {Jeffrey S. Rosenthal},

title = {Rates of Convergence for Data Augmentation on Finite Sample Spaces},

journal = {Ann. Appl. Prob},

year = {1993},

volume = {3},

pages = {819--839}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper, we examine this rate of convergence more carefully. We restrict our attention to the case where

### Citations

4020 |
Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...) posterior distribution of certain parameters given certain data. Their approach, which they call Data Augmentation, is closely related to the Gibbs Sampler algorithm as developed by Geman and Geman =-=[GG]-=-. It is used in the following situation. Suppose we observe data ~ Y , and wish to compute the posterior of a parameter ~ ` give ~ Y . Suppose further that there is some some other data ~ X which is n... |

877 |
Sampling-Based Approaches to Calculating Marginal Densities
- Gelfand, Smith
- 1990
(Show Context)
Citation Context ...of running this iterative process when given a large but finite amount of data. In [R], similar results are obtained for a more complicated model, namely the variance component models as discussed in =-=[GS]-=-. The plan of this paper is as follows. In Section 2 we review the definition of the Data Augmentation algorithm, and state the key lemma to be used in proving convergence results. In Section 3 we pro... |

626 |
The Calculation of Posterior Distributions By Data Augmentation
- Wong
- 1987
(Show Context)
Citation Context ...ata Augmentation, Gibbs Sampler, Harris Recurrence, Convergence Rate. AMS 1980 Subject Classifications. 60J10, 62F15. Running Title. Convergence of Data Augmentation. 1. Introduction. Tanner and Wong =-=[TW]-=- have defined an iterative process for obtaining closer and closer approximations to the (Bayes) posterior distribution of certain parameters given certain data. Their approach, which they call Data A... |

601 |
Applied Probability and Queues
- Asmussen
- 2003
(Show Context)
Citation Context ...ds used here may be applicable to many other Markov chain problems. The main tool used in proving the above result will be the following "Upper Bound Lemma", inspired by the discussion on pa=-=ge 151 of [A]-=-. It is closely related to the notions of Doeblin and Harris-recurrence (see [A], [AN], [AMN], [N], [Do]). In fact, a very similar (but less quantitative) result appears as Theorem 6.15 in [N]. But si... |

402 |
An introduction to chaotic dynamical systems
- Devaney
- 2003
(Show Context)
Citation Context ..., to get the desired result. Theorem 5 suggests that we further analyze the dynamical system given by (). This appears difficult in general. While there is a huge literature on dynamical systems (see =-=[De]-=-, [PdM], and references therein), including the promising theory of Liapounov functions (see [De] p. 176) for showing convergence to fixed points, we are unable to adapt this literature to our present... |

259 |
General irreducible Markov chains and nonnegative operators
- Nummelin
- 1984
(Show Context)
Citation Context ...the above result will be the following "Upper Bound Lemma", inspired by the discussion on page 151 of [A]. It is closely related to the notions of Doeblin and Harris-recurrence (see [A], [AN=-=], [AMN], [N]-=-, [Do]). In fact, a very similar (but less quantitative) result appears as Theorem 6.15 in [N]. But since this Lemma will be crucial to what follows, we include a complete proof. Lemma 2. Let P (x; \D... |

184 |
Illustration of Bayesian inference in normal data models using Gibbs sampling
- Gelfand, Hills, et al.
- 1990
(Show Context)
Citation Context ...cases, but rather that the convergence results obtained here may provide some insight into using Data Augmentation and Gibbs Sampler in more complicated examples, such as those considered in [GS] and =-=[GHRS]-=-. We intend to consider some of those examples elsewhere [R]. Also, the methods used here may be applicable to many other Markov chain problems. The main tool used in proving the above result will be ... |

112 |
A new approach to the limit theory of recurrent Markov chains
- Athreya, Ney
- 1978
(Show Context)
Citation Context ...d in proving the above result will be the following "Upper Bound Lemma", inspired by the discussion on page 151 of [A]. It is closely related to the notions of Doeblin and Harris-recurrence =-=(see [A], [AN]-=-, [AMN], [N], [Do]). In fact, a very similar (but less quantitative) result appears as Theorem 6.15 in [N]. But since this Lemma will be crucial to what follows, we include a complete proof. Lemma 2. ... |

100 |
Geometric Theory of Dynamical Systems
- Palis, Melo
- 1982
(Show Context)
Citation Context ...et the desired result. Theorem 5 suggests that we further analyze the dynamical system given by (). This appears difficult in general. While there is a huge literature on dynamical systems (see [De], =-=[PdM]-=-, and references therein), including the promising theory of Liapounov functions (see [De] p. 176) for showing convergence to fixed points, we are unable to adapt this literature to our present purpos... |

38 | Rates of Convergence for Gibbs Sampling for Variance Components Models
- Rosenthal
- 1995
(Show Context)
Citation Context ...pproach the true posterior does not grow too quickly with the amount of observed data. This suggests the feasibility of running this iterative process when given a large but finite amount of data. In =-=[R]-=-, similar results are obtained for a more complicated model, namely the variance component models as discussed in [GS]. The plan of this paper is as follows. In Section 2 we review the definition of t... |

20 |
On Coupling of Markov Chains
- Pitman
- 1976
(Show Context)
Citation Context ...ion, and brc is the greatest integer not exceeding r. (In particular, the stationary distribution is unique.) Proof. The proof shall be by a coupling argument. (For background on coupling, see, e.g., =-=[P] or chap-=-ter 4E of [Di].) We let fX k g be the Markov chain beginning in the distribution �� 0 , and let fY k g be the Markov chain beginning in the distribution ��. We realize each Markov chain as fol... |

19 |
Limit theorems for semi-Markov processes and renewal theory for Markov chains
- Athreya, McDonald, et al.
- 1978
(Show Context)
Citation Context ...roving the above result will be the following "Upper Bound Lemma", inspired by the discussion on page 151 of [A]. It is closely related to the notions of Doeblin and Harris-recurrence (see [=-=A], [AN], [AMN]-=-, [N], [Do]). In fact, a very similar (but less quantitative) result appears as Theorem 6.15 in [N]. But since this Lemma will be crucial to what follows, we include a complete proof. Lemma 2. Let P (... |