A.E. Gelfand, S.E. Hills, A. Racine-Poon, and A.F.M. Smith (1990), Illustration of Bayesian inference in normal data models using Gibbs sampling, J. Amer. Stat. Soc. 85, 972-985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... Thus, the main thrust of the current paper is not that Data Augmentation should be used in these 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 the following Upper Bound Lemma , inspired by the discussion on page 151 of [A] It is closely ....

A.E. Gelfand, S.E. Hills, A. Racine-Poon, and A.F.M. Smith (1990), Illustration of Bayesian inference in normal data models using Gibbs sampling, J. Amer. Stat. Soc. 85, 972-985.

....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 models as described in [GS] Section 3.4, and de ned herein in Section 3. See also [BT] and [GHRS]. Brie y, this model involves an overall location parameter , and K di erent parameters 1 ; K which are normally distributed around . For each i there are J di erent observations Y i1 ; Y iJ , normally distributed around i . The point of view is that , the i , and the ....

....case of variance component models (Theorem 1) This will allow us to say how large k must be to make the variation distance less than a given 0. 3. Variance Component Models and Main Result. We will follow the de nition of variance component models given in [GS] See also [BT] and [GHRS]. We suppose that there is some overall parameter , and that the location parameters 1 ; K are independently normally distributed around : i N( 1 i K) We further suppose that for each i , there are J data points Y ij independently normally distributed ....

A.E. Gelfand, S.E. Hills, A. Racine-Poon, and A.F.M. Smith (1990), Illustration of Bayesian inference in normal data models using Gibbs sampling, J. Amer. Stat. Soc. 85, 972-985.

....time in covariates on the response (decision) with the correlated measurement and give us more insight in what criteria are important to decision makers and the weight of each. To handle the noise and the outlier data, we use Gibbs sampling under Bayesian framework with some prior information [5] [12], 13] Based on the estimations of AGEE with additive noise model, Adaptive Bayes (AB) classifiers are proposed which uses modified Naive Bayes algorithm [3] We believe that combining advanced statistical approaches with data mining techniques through using data from human detailers to build new ....

Gelfand, A. E., Hills, S. E., Racine-Poon, A., Smith, A. F. M.: Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association, 85(412):pp. 972--985,1990.

....General state space; Geometric ergodicity; Gibbs sampler; Hierarchical random e ects model; Metropolis algorithm; Minorization condition; Regeneration; Splitting; Uniform ergodicity. 2 1 Introduction 1. 1 The questions During the decade or so since the appearance of the seminal paper by Gelfand and Smith (1990), Markov chain Monte Carlo (MCMC) methods have revolutionized statistical computing. While the Bayesians have certainly made the most use of MCMC, applications have popped up in many di erent areas of statistics. For example, MCMC techniques can be used to calculate p values in exact conditional ....

....data and f denotes a generic density. The integrals required for inferences through this posterior are not available in closed form. Thus, we might resort to MCMC techniques like the Gibbs sampler. Indeed, variance component models have been advocated as an ideal application for the Gibbs sampler (Gelfand and Smith 1990, Gelfand, Hills, Racine Poon and Smith 1990) The data in Table 1 were simulated according to model (M) with K = 6, m = 8, a 1 = a 2 = b 1 = b 2 = 0 = 1 and 0 = 0. We now pretend that the origin of the data is unknown and that we desire the posterior expectations of and e under three ....

[Article contains additional citation context not shown here]

Gelfand, A. E., Hills, S. E., Racine-Poon, A. and Smith, A. F. M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling, Journal of the American Statistical Association 85: 972-985.

....a random draw of # from P (# A T , Y T , #) v) Repeat (ii) iv) N times to obtain N random draws of A T , # and #. In Steps (ii) v) the random draws of A T , # and # are updated, which sampling method is called the Gibbs sampler. See Geman and Geman (1984) Tanner and Wong (1987) Gelfand, Hills, Racine Poon and Smith (1990), Gelfand and Smith (1990) Carlin and 4 Typically, the smoothed estimates based on the extended Kalman filter are taken for # t , t = 0, 1, T . The extended Kalman filter is one of the traditional nonlinear filters, where the nonlinear measurement and transition equations given by equations ....

....#) v) Repeat (ii) iv) N times to obtain N random draws of A T , # and #. In Steps (ii) v) the random draws of A T , # and # are updated, which sampling method is called the Gibbs sampler. See Geman and Geman (1984) Tanner and Wong (1987) Gelfand, Hills, Racine Poon and Smith (1990) Gelfand and Smith (1990), Carlin and 4 Typically, the smoothed estimates based on the extended Kalman filter are taken for # t , t = 0, 1, T . The extended Kalman filter is one of the traditional nonlinear filters, where the nonlinear measurement and transition equations given by equations (1) and (2) are linearized ....

Gelfand, A.E., Hills, S.E., Racine-Poon, H.A. and Smith, A.F.M., 1990, " Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling, " Journal of the American Statistical Association, Vol.85, No.412, pp.972 -- 985.

....does not exist and (ii) it takes a long time when the acceptance probability #( is close to zero. See, for example, Knuth (1981) Boswell, Gore, Patil and Taillie (1993) O Hagan (1994) and Geweke (1996) for rejection sampling. Gibbs Sampling: Geman and Geman (1984) Tanner and Wong (1987) Gelfand and Smith (1990), Gelfand, Hills, Racine Poon and Smith (1990) and so on developed the Gibbs sampling theory, which is concisely described as follows (also see Geweke (1996, 1997) Consider two random variables x and y. Let P x y (x y) P y x (y x) and P xy (x, y) be the conditional density of x given y, the ....

....it takes a long time when the acceptance probability #( is close to zero. See, for example, Knuth (1981) Boswell, Gore, Patil and Taillie (1993) O Hagan (1994) and Geweke (1996) for rejection sampling. Gibbs Sampling: Geman and Geman (1984) Tanner and Wong (1987) Gelfand and Smith (1990) Gelfand, Hills, Racine Poon and Smith (1990) and so on developed the Gibbs sampling theory, which is concisely described as follows (also see Geweke (1996, 1997) Consider two random variables x and y. Let P x y (x y) P y x (y x) and P xy (x, y) be the conditional density of x given y, the conditional density of y given x and the joint ....

Gelfand, A.E., Hills, S.E., Racine-Poon, H.A., and Smith, A.F.M., 1990, " Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling, " Journal of the American Statistical Association, Vol.85, No.412, pp.972 -- 985.

....algorithm and Gibbs sampler. Furthermore, the candidates of the proposal density required to perform the Metropolis Hastings algorithm have been examined. APPENDICES APPENDIX 1: MARKOV CHAIN MONTE CARLO METHODS Appendix 1. 1: Gibbs Sampling Geman and Geman (1984) Tanner and Wong (1987) Gelfand, Hills, RacinePoon and Smith (1990), Gelfand and Smith (1990) Carlin and Polson (1991) Zeger and Karim (1991) and so on developed the Gibbs sampling theory. Carlin, Polson and Sto#er (1992) Carter and Kohn (1994, 1996) and De Jong and Shephard (1995) applied to the Gibbs sampler to the nonlinear and or non Gaussian state space ....

....the candidates of the proposal density required to perform the Metropolis Hastings algorithm have been examined. APPENDICES APPENDIX 1: MARKOV CHAIN MONTE CARLO METHODS Appendix 1. 1: Gibbs Sampling Geman and Geman (1984) Tanner and Wong (1987) Gelfand, Hills, RacinePoon and Smith (1990) Gelfand and Smith (1990), Carlin and Polson (1991) Zeger and Karim (1991) and so on developed the Gibbs sampling theory. Carlin, Polson and Sto#er (1992) Carter and Kohn (1994, 1996) and De Jong and Shephard (1995) applied to the Gibbs sampler to the nonlinear and or non Gaussian state space models. The Gibbs sampling ....

Gelfand, A.E., Hills, S.E., Racine-Poon, H.A. and Smith, A.F.M., 1990, " Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling, " Journal of the American Statistical Association, Vol.85, No.412, pp.972 -- 985.

....for the transition rates (f 1 ) the clinical states (ffi) and the measurement parameters (f 2 ) These distributions will be specified in the section 2.3. 2. 2 MCMC implementation The joint posterior distribution of all the parameters was simulated by Gibbs sampling (Gelfand and Smith (1990) Gelfand et al. (1990)) as is now current practice in many implementation of Bayesian hierarchical models (Gilks, Richardson and Spiegelhalter (1996) We list below the full conditional distributions (used in the Gibbs sampling algorithm) which can be derived from the model assumptions above. By full conditional ....

Gelfand, A.E., Hills, S.E., Racine-Poon, A. and Smith, A.F.M. (1990) Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association 85, 972--985.

.... with very large elements on the diagonals of the prior variance matrices M i for the link specific parameters i : Useful background to normal, linear hierarchical models and prior forms can be found in, for example, Lindley and Smith (1972) Smith (1973) and more recently and specifically in Gelfand et al. (1990), and in several of the contributed chapters in Gilks et al. (1996) Analysis of the nine days of observed data involves computing posterior distributions for all model parameters and the population hyperparameters. Analysis is via standard methods of Bayesian simulation using Markov chain Monte ....

Gelfand, A.E., Hills, S.E., Racine-Poon, A. and A.F.M. Smith, "Illustration of Bayesian inference in Normal data models using Gibbs sampling," Journal of the American Statistical Association 85, 972-985 (1990).

....analysis can be challenging. Little and Rubin [13] and Tanner [26] provide overviews of the extensive literature. Among the available methods, Bayesian approaches have proven successful in fully addressing uncertainty deriving from incomplete observations (Tanner and Wong [27] Gelfand et al. [8], Kong, Liu and Wong [12] Conjugate prior distributions (Raiffa and Schleifer [23] are commonly used in Bayesian parametric inference, because of their computational convenience and relative ease of elicitation and interpretation. While recent progress in simulation methods has opened several ....

....of the missing values and the parameters given the observed values. These algorithms generate samples of the missing values, and conditional on the missing values, generate samples from the posterior distribution of and Sigma. Following work of Tanner and Wong [27] and Gelfand et al. [8] these methods have become standard in Bayesian inference. See for example Thomas, Spiegelhalter and Gilks [28] More recent efforts have concentrated on improving efficiency of sampling, for example, by the sequential importance sampling method (Kong, Liu, and Wong, 12] Results of Liu, Wong ....

Gelfand, A.E., Hills, S.E., Racine-Poon, A., Smith, A.F.M. (1990) "Illustration of Bayesian inference in normal data models using Gibbs sampling" Journal of the American Statistical Association 85 972--985.

....inequality; Drift condition; Geometric ergodicity; Gibbs sampler; Hierarchical random e ects model; Metropolis algorithm; Minorization condition; Regeneration; Slice sampler; Splitting; Uniform ergodicity. 2 1 Introduction During the decade or so since the appearance of the seminal paper by Gelfand and Smith (1990), Markov chain Monte Carlo (MCMC) methods have revolutionized statistical computing. While the Bayesians have certainly made the most use of MCMC, applications have popped up in many di erent areas of statistics. For example, MCMC techniques can be used to calculate P values in exact conditional ....

....f denotes a generic density. The integrals required for inferences through this posterior are not available in closed form. Thus, we might resort to MCMC techniques like the Gibbs sampler. Indeed, variance component models have been advocated as an ideal application for the Gibbs sampler (see e.g. Gelfand and Smith, 1990 and Gelfand, Hills, Racine Poon and Smith, 1990) 6.2 The Block Gibbs Sampler The two obvious xed scan Gibbs samplers that can be employed to sample from the posterior are (i) the ordinary one at a time version that updates each component sequentially, and (ii) a block version in which all ....

[Article contains additional citation context not shown here]

Gelfand, A. E., Hills, S. E., Racine-Poon, A. and Smith, A. F. M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling, Journal of the American Statistical Association 85: 972-985.

....is investigated. In another example, the effects of multiple modes on the convergence of coupled paths are explored using mixtures of bivariate normal distributions. The technique is also used to evaluate the convergence properties of a Gibbs sampling scheme applied to a model for rat growth rates (Gelfand et al. 1990). Acknowledgements I would like to thank Steve MacEachern, Julian Besag, Donald Rubin, Alyson Wilson and Peter Muller for suggestions on presentation and content. Support was provided by PHS grant NCI CA56671 03 and NSF SCREMS award DMS 9305699. 1 Introduction By providing statisticians with ....

Gelfand, A.E., Hills, S.E., Racine-Poon, A., and Smith, A.F.M. (1990), "Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling," Journal of the American Statistical Association, 85, 972-985.

....is established, the Gibbs sampling algorithm can be applied for image synthesis or segmentation. The Gibbs sampler [7] provides a Bayesian inference mechanism for applications like image analysis and other areas of multivariate statistics that previously suffered from computational prohibition [6]. The Gibbs sampler updating scheme proceeds as follows. Given an arbitrary starting set of values S (0) s (0) 1 ; 1 1 1 ; s (0) p ) we update a single component s (0) 1 to s (1) 1 according to the distribution P (s 1 js (0) 2 ; 1 1 1 ; s (0) p ) in a stochastic manner (i.e. ....

A. Gelfand, S. E. Hills, A. Racine-Poon, A. F. M. Smith. Illustration of Bayesian inference in normal data models using Gibbs Sampling. Journal of the American Statistical Association, 85(412):972-985, 1990.

....and therefore are a valid measure of #surprise. 4 Examples We consider two examples in this paper, both of whichinvolve hierarchical models. The #rst example is a hierarchical linear model applied to a data set describing growth of 30 rats, with measurements at #ve time points for each rat #Gelfand, Hills, Racine Poon, and Smith 1990#. The second example is a complex hierarchical model for ordinal survey data with #NA responses; for each item, each subject either provided an ordinal rating or declined to respond #Bradlow 1994, Bradlow and Zaslavsky 1997b#. A summary of each model and of the methods used to calculate ....

Gelfand, A.E., Hills, S.E., Racine-Poon, A., and Smith, A.F.M. #1990#, Illustration of Bayesian inference in normal data models using Gibbs sampling, Journal of the American Statistical Association, 85, 972#985.

.... Sigma ff ) and the prior distribution by p( Omega 1 j Omega 2 ) Inferences for the model parameters Omega 1 and Omega 2 were derived by obtaining samples from the marginal posterior distributions p( Omega 1 jY#X) and p( Omega 2 jY#X) using a Markov chain Monte Carlo (MCMC) sampler (Gelfand et al. 1990, Rossi et al. 1996) For each of Model 1, Model 2, and Model 3, we report results obtained by running three independentchains for 3000 draws from overdispersed starting positions, discarding the initial 500 draws of each chain after determining convergence (Gelman and Rubin (1992) and ....

Gelfand, Alan E., Susan E. Hills, Amy Racine-Poon, and Adrian F.M. Smith (1990), "Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling", Journal of the American Statistical Association, Vol. 85, 972-985.

.... and Geman and Geman (1984) and to Bayesian statistics (Smith and Roberts, 1993) spatial statistics (Besag and Green, 1993, and Graham, 1994) expert systems (Pearl, 1987, Spiegelhalter, Dawid, Lauritzen and Cowell, 1993) incomplete data problems (Tanner and Wong, 1987) and hierarchical models (Gelfand, Hills, Racine Poon, A. and Smith, 1990). The algorithm of Metropolis, Metropolis, Rosenbluth, Teller and Teller (1953) and its generalization by Hastings (1970) construct MCMC samplers as follows. Let K(x; dy) be a candidate transition distribution on E. Write x (dy) for the one point probability measure with mass at x. Find a ....

....Rao Blackwell theorem in the context of stochastic simulation, and for Markov chain Monte Carlo (MCMC) in particular. See Casella and Robert (1996) and the references cited therein. Early references are Kalos and Whitlock (1986, Section 4.2) and Pearl (1987) See also Neal (1993, Section 6. 3) Gelfand and Smith (1990, 1991) consider i.i.d. runs of a Gibbs sampler. In the empirical estimator based on the nal value of each run, they replace f by a conditional expectation under . For long runs, the nal values are distributed approximately according to , so the classical Rao Blackwell theorem (4.2) implies ....

Gelfand, A. E., S. E. Hills, A. Racine-Poon and A. F. M. Smith (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling. J. Amer. Statist.

No context found.

Gelfand, A.E., Hills, S., Racine-Poon, A. and Smith, A.F.M. (1990). "Illustration of Bayesian inference in normal data models using Gibbs sampling". J. Amer. Statist. Assoc., 85, 972-985. 16

No context found.

A.E. Gelfand, S.E. Hills, A. Racine-Poon, and A.F.M. Smith (1990), Illustration of Bayesian inference in normal data models using Gibbs sampling, J. Amer. Stat. Soc. 85, 972-985.

No context found.

Gelfand, A. E., Hills, S. E., Rancine-Poon, A., and Smith, A. F. M. (1990). Illustration of Bayesian inference in Normal Data Models Using Gibbs Sampling. Journal of the American Statistical Association, 85, 972--985.

No context found.

Gelfand, A.E., Hills, S.E., Racin-poon, A., and Smith, A.F.M.,1990. Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association, 85:972-985.

No context found.

Gelfand, A.E., Hills, S.E., Racine-Poon, A., and Smith, A.F.M., 1990. Illustration of Bayesian inference in normal data models using Gibbs sampling, Journal of the American Statistical Association, 85(412):972-985.

No context found.

Gelfand, A.E., Hills, S.E., Racine-Poon, A., and Smith, A.F.M. (1990), "Illustration of Bayesian Inference in Normal Data Models Using Gibbs Sampling," Journal of the American Statistical Association, 85, 972-985. 30

No context found.

Gelfand, A.E., Hills, S.E., Racine-Poon, A., and Smith, A.F.M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association, 85:972-985. 17

No context found.

Gelfand, A.E., Hills, S.E., Racine-Poon, A., and Smith, A.F.M. (1990), "Illustration of Bayesian inference in normal data models using Gibbs sampling.", Journal Of the American Statistical Association, 85, 972985.

No context found.

Blood, 21, 699--716. Gelfand, A. E., Hills, S. E., Racine-Poon, A., and Smith, A. F. M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling. J Amer Statist Assoc, 85, 972--85.

First 50 documents  Next 50

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