A. F. M. Smith and G. O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. Royal Statistical Society, B55(1):3--23, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....might run at di erent speeds; they might have di erent user loads on them; one or more of them might be down; etc. Handling these issues correctly is crucial to the success of parallel Monte Carlo. In addition, Markov chain Monte Carlo algorithms are now very common (see for example [17] [51], 53] 22] 45] and parallelising them presents additional diculties such as determining appropriate burn in time. We note that similar issues have been considered in various contexts in the operations research literature. In particular, in an excellent series of papers ( 23] 24] 25] ....

....allowed to depend on the speed at which the simulation happens to run, then it follows that a second run would not produce identical results even if started with the same pseudo random number seed. 4. Parallel Markov chain Monte Carlo. Markov chain Monte Carlo (MCMC) algorithms (see e.g. 17] [51], 53] 22] 45] such as the Gibbs sampler and the Metropolis Hastings algorithm, have become extremely popular in statistics (especially Bayesian statistics) as a method of approximately computing dicult high dimensional integrals. They are also used in theoretical computer science for ....

A.F.M. Smith and G.O. Roberts (1993), Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Stat. Soc. Ser. B 55, 3-24.

....algorithm is quite different in its principle. The vectors Xt, Kt and rt are considered to be random variables with prior densities. Samples are then obtained iteratively from their joint posterior using a proper MCMC technique, namely the Gibbs Sampler. This method has been studied in [32] 33] [34] [35] or [36] for instance. It can be run sequentially at each time period. Gibbs Sampler is a special case of the Metropolis Hasting algorithm with the proposal densities being the conditional distributions, and the acceptance probability being consequently always equal to one. The interested ....

A. Smith and G. Roberts. Bayesian computation via the Gibbs Sampler and related Markov chain Monte Carlo methods. J. Royal Statist. Soc., Series B 55(1):3-24, 1993.

.... form of the density [86] In both cases little analytical progress can be made, and we must turn instead to simulation methods, whereby we try to build up a picture of the characteristics of a distribution, or a function of that distribution, from a number of samples drawn from that distribution [93]. Let p(x) be the joint distribution of the variables fx i : i = 1 : ng, and for simplicity let the x i take values from a discrete state space Omega Gamma We wish to generate samples x distributed as p(x) For some forms of p(x) it may be possible to generate samples directly, by some ....

....Consider again the joint distribution (x 1 : x n ) Begin with x and proceed as . n (x n jx n Gamma1 ) where indicates is drawn from the distribution of . This completes one cycle of the Gibbs sampler and is equivalent to a transition probability of [93] T (x We now show that this set of transition probabilities has (x) as its unique invariant distribution. We do this directly. x 1 jx n ) x n Gamma1 jx 1 x 2 : x n Gamma2 x 2 : x n ) x 1 jx n ) x n Gamma1 jx 1 ....

A.F.M Smith and G.O. Roberts. Bayesian computation via the Gibbs sampler and related Markov-chain Monte-Carlo methods. JRSS-B, 55(1):3--23, 1993.

....The Bayesian literature on nonparametric methods has grown rapidly since the theoretical background for the construction of priors on function spaces was developed, e.g. the work of Ferguson (1973, 1974) on the Dirichlet process. Markov chain Monte Carlo (MCMC) methods (Gelfand and Smith, 1990, Smith and Roberts, 1993, Tierney, 1994) made their practical use feasible. Walker, et al. 1999) provide a summary of some of the methods in Bayesian nonparametrics. Semiparametric regression modeling is especially attractive in this context. See, for instance, Brunner (1995) and Kuo and Mallick (1997) as well as ....

Smith, A.F.M., and Roberts, G.O. (1993), "Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods," Journal of the Royal Statistical Society, Ser. B, 55, 3-23.

....high dimensionality) These methods appeared in the statistical literature in the early 1980s and are used extensively in the O. Cappe et al. Signal Processing 73 (1999) 3 25 9 fields of image processing and computational statistics. MCMC techniques are well documented in the literature (see [4,25,28,57,61] and references therein) and only a brief account of these methods is given here. The idea is very simple. Suppose that we need to sample from a distribution f (x) where x#(x # , 2 ,x n )3X #n which is known (perhaps up to multiplicative constant) f will be referred to as the target ....

....mechanism, and techniques to control the convergence to the limit distribution. 4.1.1. Gibbs sampler The Gibbs sampler was first introduced for image restoration by Geman and Geman [26] and Besag [3] An extensive account of the Gibbs sampler may be found in the tutorials by Smith and Roberts [61], Gelfand and Smith [25] and Besag et al. 4] The Gibbs sampler proceeds by splitting the state vector into a number of components and updating each in turn by a series of Gibbs transitions. Suppose that the state vector is split into q)n components (x # , 2 , x q ) Having selected component ....

A.F.M. Smith, G.O. Roberts, Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, J. Royal Statist. Soc. Ser. B 55 (1) (1993) 3---23.

....multivariate distributions (generally of high dimensionality) These methods appeared in the statistical literature in the early 80 s and are very useful in the elds image processing and computational statistics. MCMC techniques are well documented in the literature (see [4] 25] 28] 57] [61] and references therein) and only a brief account of these methods is given here. The idea is very simple. Suppose that we need to sample from a distribution f (x) where x , x 1 ; x n ) 2 X R n which is known (perhaps up to multiplicative constant) f will be referred to as the ....

....transition mechanism, and techniques to control the convergence to the limit distribution. Gibbs Sampler The Gibbs sampler was rst introduced for image restoration by Geman and Geman [26] and Besag [3] An extensive account of the Gibbs sampler may be found in the tutorials by Smith and Roberts [61], Gelfand and Smith [25] and Besag et al. [4] The Gibbs sampler proceeds by splitting the state vector into a number of components and updating each in turn by a series of Gibbs transitions. Suppose that the state vector is split into q n components x 1 ; x q . Having selected ....

A. F. M. Smith and G. O. Roberts. Bayesian computation via the gibbs sampler and related Markov chain Monte Carlo methods. J. Royal Statist. Soc. Ser. B, 55(1):3-23, 1993.

....sampling is sometimes computationally ine#cient. We sometimes have the case where rejection sampling does not work well, depending on the underlying assumptions on the functional form or the error terms. Thus, in their paper, the specific state space models are taken. Gordon, Salmond and Smith (1993), Kitagawa (1996) and Kitagawa and Gersch (1996) proposed both filtering and smoothing using the resampling procedure, where random draws from the filtering density and the smoothing density are recursively generated at each time and the random draws from the smoothing are based on those from the ....

....procedure is quite robust to choice of the proposal density, but use of the transition equation might be recommended for safety because Proposal Density I has shown a good performance for all the simulation studies examined in this paper. 18 Appendices Appendix 1: Metropolis Hastings Algorithm Smith and Roberts (1993), Tierney (1994) Chib and Greenberg (1995, 1996) and Geweke (1996) discussed the Metropolis Hastings algorithm, which is the random number generation method such that we can generate random draws from any density function. Consider generating a random draw of z from P (z) which is called the ....

Smith, A.F.M. and Roberts, G.O., 1993, " Bayesian Computation via Gibbs Sampler and Related Markov Chain Monte Carlo Methods, " Journal of the Royal Statistical Society, Ser.B, Vol.55, No.1, pp.3 -- 23.

....at present time depends on that at the past time, accumulation of computational errors possibly become large as time goes by. Therefore, recently, some attempts are made to generate random draws directly from prediction, filtering and smoothing distribution functions. Gordon, Salmond and Smith (1993), Kitagawa (1996, 1998) and Kitagawa and Gersch (1996) utilized the resampling method to generate random draws from prediction, filtering and smoothing densities. For generation of filtering and smoothing random draws, one step ahead prediction random draws are chosen with the corresponding ....

....(1995) Tanizaki (1993a, 1996) and Tanizaki and Mariano (1994) applied Monte Carlo integration with importance sampling to evaluate each integration. Carlin, Polson and Sto#er (1992) Carter and Kohn (1994, 1996) and Chib and Greenberg (1996) utilized the Gibbs sampler. Gordon, Salmond and Smith (1993), Kitagawa (1996, 1998) and Kitagawa and Gersch (1996) proposed a nonlinear filter using a resampling procedure. Hurzeler and Kunsch (1998) Mariano and Tanizaki (2000) Tanizaki (1996, 1999a) and Tanizaki and Mariano (1998) introduced a nonlinear and nonnormal filter with rejection sampling. ....

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Smith, A.F.M. and Roberts, G.O., 1993, " Bayesian Computation via Gibbs Sampler and Related Markov Chain Monte Carlo Methods, " Journal of the Royal Statistical Society, Ser.B, Vol.55, No.1, pp.3 -- 23.

....Monte Carlo integration with importance sampling to derive nonlinear and non Gaussian state space models. Tanizaki (1996, 1999a) Mariano and Tanizaki (1999) and Tanizaki and Mariano (1998) utilized rejection sampling to generate random draws directly from filtering densities. Gordon, Salmond and Smith (1993) and Kitagawa (1996) Kitagawa and Gersch (1996) Tanizaki (1997) and Tanizaki and Mariano (1996) also obtained filtering means by random draws. As for nonlinear and non Gaussian filters and smoothers with Markov chain Monte Carlo, Carlin, Polson and Sto#er (1992) and Carter and Kohn (1994, ....

....state space model, it is known that convergence of the Gibbs sampler is very slow. However, blocking of the random vectors is very ad hoc in practical exercises. And sometimes it is not feasible since we have the case where p(x, y z) is not available. Appendix 1. 2: Metropolis Hastings Algorithm Smith and Roberts (1993), Tierney (1994) Chib and Greenberg (1995, 1996) and Geweke (1996) discussed the Metropolis Hastings algorithm, which is the random number generation method such that we can generate random draws from any density function. Consider generating a random draw of z from p(z) which is called the ....

Smith, A.F.M. and Roberts, G.O., 1993, " Bayesian Computation via Gibbs Sampler and Related Markov Chain Monte Carlo Methods, " Journal of the Royal Statistical Society, Ser.B, Vol.55, No.1, pp.3 -- 23.

....Rosenbluth Teller 1953, Hastings 1970) and the Gibbs sampler (Geman Geman 1984) have provided a good alternative as they are able to handle problems of very large size. For recent reviews, see (Gelfand Smith 1990, Thomas, Spiegelhalter Gilks 1992, Gelman Rubin 1992, Geyer 1992, Smith Roberts 1993). MCMC methods simulate realizations from probability distributions whose densities are known up to a normalizing factor. If h(x) is a probability distribution on the sample space, the Gibbs sampler and Metropolis Hastings algorithm simulate a Markov chain whose equilibrium distribution is ....

Smith, A. F. M. & Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, Journal of the Royal Statistical Society, Series B 55(1): 5--23.

....A Markov Chain fX(s)g is called irreducible if for all x; y 2 Omega there exist some integer s 0 for which P s x;y 0, i.e. if there is a positive probability that in a finite number of transitions state y can be reached starting from state x. We follow the definition in Smith and Roberts [34] for aperiodicity: Definition 2.2 A Markov Chain is called aperiodic if there does not exist a pairwise disjoint partition Omega = S r Gamma1 i=0 Omega i for some r 2, such that P (X(s) 2 Omega smod(r) jX(0) 2 Omega 0 ) 1 for all s, i.e. if the Markov Chain is not cycling through the ....

....x2 Omega (x) 1 (2.10) and (A) Z x2 Omega P (X(1) 2 AjX(0) x) x)d(x) 2.11) for all ( measurable) A. Furthermore X(s) d = s 1 X (x) 2.12) and for every bounded real valued ( measurable) f 1 s s X t=1 f(X(t) s 1 E [f(X) almost surely. 2. 13) See Smith and Roberts [34], Tierney [37] or Nummelin [26] for proof and further details. 2.2 Simulation from complex distributions The previous section discussed Markov Chains in general terms. In this section we discuss how we may apply this theory to simulate from the distribution (x) The basic idea for sampling ....

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A. F. M. Smith and G. O. Roberts. Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods. Journal of Royal Statistical society, series B, 55(1):3--23, 1993.

....4.3. Posterior Inferences The maximum posterior estimates and asymptotic uncertainties provide a good approximation to the posterior that can be used to address a variety of questions, including but not limited to localization. By using Markov Chain Monte Carlo (MCMC) simulation techiques [59, 54, 8, 27], we can refine the computed posterior and can account for more complex features in the model. The result of an MCMC simulation is a sample from the full posterior distribution of the parameters, including variation across sub models. Any functional of this distribution can be easily estimated ....

A. F. M. Smith and G. O. Roberts. Bayesian computation via the gibbs sampler and related markov chain monte carlo methods. J. Roy. Statist. Soc. B, 55(1), 1993.

....conditional posterior distribution of the model parameters. This produces a Markov chain, which converges under mild conditions. The in this way obtained draws can be seen as a sample from the posterior distribution. For an introduction and details about the Gibbs sampling algorithm we refer to Smith and Roberts (1993) and Tierney (1994) In Appendix B we derive the full conditional posterior distributions which are necessary in the Gibbs sampler. We focus on the most general Markov trend stationary model (20) The full conditional posterior distributions of the other models can be derived in a similar way. ....

Smith, A.F.M. and G.O. Roberts, 1993, Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods, Journal of the Royal Statistical Society B 55, 3--23.

....hyperpriors are chosen for all blocks with values a and b equal to 0.001. 2. 4 Computational Issues We used Markov chain Monte Carlo to sample from the posterior distribution implied by the above formulation, applying univariate Metropolis steps for each parameter, as described for example in Smith and Roberts (1993). The number of parameters in interaction models is extremely high, so tuning of the Metropolis steps was done in an automatic fashion. Specifically, the spread of each Metropolis proposal was fixed so that the corresponding acceptance rate of each parameter was between 30 and 50 . Hyperparameters ....

Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). Journal of the Royal Statistical Society B, 55, 3--23.

....manner. In many cases the appraisal of the results is based on only a subset of the ensemble, with the rest discarded. For example, this is the case if the objective is to locate a single optimal model (in some sense) and also with statistical methods, such as importance sampling (e.g. Smith Roberts 1993; Mosegaard Tarantola 1995) which use only a statistically independent subset of the ensemble. In principle, however, the entire ensemble may provide useful information from which to draw inferences. In some cases, models which fit the data poorly may tell us just as much as those which fit the ....

.... within a geophysical context the reader is referred to Tarantola (1987) Duijndam (1988a,b) Cary Chapman (1988) Mosegaard Tarantola (1995) Gouveia Scales (1998) Useful books on posterior simulation are Gelman, et al. 1995) and Tanner (1996) and summary papers are by Smith (1991) and Smith Roberts (1993). From the Bayesian viewpoint, the solution to the inverse problem is the posterior probability density function (PPD) This quantity is used to represent all information available on the model. Its calculation depends upon the data, any prior information, and the statistics of all noise present, ....

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Smith, A.F.M. & Roberts, G.O., 1993. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, J. R. statist. Soc., B., 55, 3--23.

....chain, so such algorithms are called Markov chain Monte Carlo (MCMC) methods (Hastings, 1970; Gelfand Smith, 1990) As a formal detail, any MCMC algorithm should be capable of reaching all states (or at least those within the support of #) all recommended algorithms can do this. Smith (1991) Smith Roberts (1993) and Besag Green (1993) are good sources. There is an analogy with statistical mechanics, where a physical system evolves under essentially random influences towards a thermal equilibrium described by #(X) # exp( #E(X) 17) where E is energy and # is reciprocal temperature, conventionally ....

Smith, A.F.M. & Roberts, G.O. (1993) Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods. J. Roy. Statist. Soc. B 55, 3--23.

....in a spirit of exploration, fuelled by the fact that complicated functionals of high dimensional posterior distributions can be calculated with comparative ease by Markov chain Monte Carlo (MCMC) methods. We shall 3 have rather little to say about MCMC itself and, for further details, refer to Smith and Roberts (1993), Besag and Green (1993) Tierney (1994) Besag et al. 1995) Green (1995) and the accompanying references and discussions. The analyses in this paper were all carried out using simple or block Gibbs samplers, except for the occasional inclusion of some Hastings steps. It would be of interest to ....

Smith, A. F. M. and Roberts, G. O. (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. R. Statist. Soc. B, 55, 3--23, 53--102.

.... non gaussian dynamic models like those in Carter and Kohn (1994) and FruhwirthSchnatter (1994) or applications like the one presented in Cargnoni et al. 1997) Methods used to update the distributions of the parameters sequentially, as observations become available, appear in Salmon and Smith (1993), Sheppard (1994) and Pitt and Sheppard (1997) Recent applications of DLMs to spatio temporal modelling are found in Tonellato (1997a) and Tonellato (1997b) In Sans o and Guenni (1997) a DLM is considered for the latent process w in (1) when there is only one station, so that no spatial ....

....degrees of freedom of the student density ff. The choice of all the hyperparameters will be discussed as part of the analysis of the data. 3. FITTING THE MODEL In order to fit the model presented in the previous section we use a Markov Chain Monte Carlo method (MCMC) as described for example in Smith and Roberts (1993), to obtain samples from the posterior distribution of oe 2 ; and = 1 ; T ) In analogy to the technique used to fit models for censored observations, we consider latent variables v t to model dry periods. We also use latent variables u t to account for the missing values. Note ....

Smith, A. F. M. and Roberts, G. O. (1993) Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods. Journal of the Royal Statistical Society, B, 55, 3--25; with discussion.

....proceed by generating samples from the posterior distribution via MCMC. The sample is produced by constructing a partial realization of a Markov chain whose stationary distribution is given by (3) Detailed descriptions of the methodology can be found in the Royal Statistical Society collection (Smith and Roberts, 1993; Besag and Green, 1993; Gilks, Clayton, Spiegelhalter, Best, McNeil, Sharples and Kirby, 1993) Geyer, 1992; Tierney, 1994; Besag, Green, Higdon, and Mengersen, 1995, including the accompanying discussions. In this application we initialize the chain at the point estimate obtained above. The ....

Smith, A. F. M. and Roberts, G. O. (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, Journal of the Royal Statistical Society (Series B) 55, 3--23.

....) the acceptance probability simplifies to a(i; i ) min(f(i) f( i ) 1) This describes the original Metropolis algorithm. If g(i; i ) g(i) we get a( i ; i) min(w(i) w( i ) 1) where w(i) f(i) g(i) which can be interpreted as importance weights. For details we refer to Smith and Roberts (1993) and Tierney (1994) If we opt for a M H algorithm, we can take the posterior of the parameters of the unrestricted error correction model (25) as candidate generating density function, since we have already shown how to sample from this distribution. However, in this case we also sample which ....

Smith, A.F.M. and G.O. Roberts, 1993, Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods, Journal of the Royal Statistical Society B 55, 3--23.

....; aN ) de 1 : deM dh 1 : dhM da 1 : daN : To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm; for background see, e.g. Smith and Roberts [23]; Tierney [24] Gilks et al. 11] Roberts and Rosenthal [21] The Metropolis Algorithm proceeds by starting all the 2M N parameter values at 1. It then attempts, for each parameter in turn, to add an independent N(0; random variable to the parameter. It then accepts this new value with ....

A.F.M. Smith and G.O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). Journal of the Royal Statistical Society, Series B, 55:3--24, 1993.

..... a n ) de 1 . de n dh 1 . dh n da 1 . da n . To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm (for background see, e.g. Smith and Roberts [48]; Tierney [49] Gilks et al. 22] Roberts and Rosenthal [45] We denote this algorithm as BAYESIAN. The Metropolis Algorithm proceeds by starting all the 3n parameter values at 1. It then attempts, for each parameter in turn, to add an independent N(0, # ) random variable to the parameter. ....

A.F.M. Smith and G.O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). Journal of the Royal Statistical Society, Series B, 55:3--24, 1993.

....of this strategy, where we write P : Q d Q d 1 Q 1 , where Q k is the Markov kernel that replaces the k th coordinate by a draw from (dx k jfx j g j 6=k ) leaving x j xed for j 6= k. The random scan Gibbs sampler, P : d 1 P d i=1 Q i is sometimes used instead (see Smith and Roberts [22], Tierney [23] When the full conditional distributions (dx i jfx j g j 6=i ) are dicult to sample, one can instead de ne new operators P i (e.g. one dimensional Metropolis algorithms) which are easily implemented, such that P i converges to Q i (in an appropriate sense) as n goes to in nity. ....

A.F.M. Smith and G.O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. R. Stat. Soc. Ser. B, 55(1):3-24, 1993.

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Smith, A.F.M. and Roberts, G.O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Stat. Soc. Ser. B 55, 3-24.

....are still more complicated if we consider the extended model proposed in section 7.2. However, we need to estimate all the components of the model, including the low level parameters, to disaggregate the time series of observed rainfall amounts to any arbitrary level. The MCMC methodology [2, 21, 23] offers much flexibility for this kind of problems: the use of almost noninformative priors (at the expense however of a slower convergence) the reconstruction of the posterior distributions of the parameters (which is far more than finding the posterior mean) and the possibility to deal easily ....

.... do not allow much variability between the Y k s and the Z k s and thus, when updating the X ij s, they are likely to generate high rejection rates within the MCMC algorithm. 3. 3 The MCMC algorithm In order to sample from the joint posterior distribution, we use the Gibbs sampling algorithm [2, 21, 23] which samples in turn from each of the full conditionals. Thus, at each iteration, we update in turn the two following boxes: ffl The higher level parameters: ff; OE; conditionally on everything else. ffl The storms and cells configuration: N; fM i g; fj i g; fO i g; fT i g; fO ij ....

Smith A.F., Roberts G.O. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. Royal Stat. Soc. B, 55(1993), 3--23.

....of this strategy, where we write P : Q d Q d 1 Q 1 , where Q k is the Markov kernel that replaces the k th coordinate by a draw from (dx k jfx j g j 6=k ) leaving x j xed for j 6= k. The random scan Gibbs sampler, P : d 1 P d i=1 Q i is sometimes used instead (see Smith and Roberts [21], Tierney [22] When the full conditional distributions (dx i jfx j g j 6=i ) are dicult to sample, one can instead de ne new operators P i (e.g. one dimensional Metropolis algorithms) which are easily implemented, such that P n i converges to Q i (in an appropriate sense) as n goes to in nity. ....

A.F.M. Smith and G.O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. R. Stat. Soc. Ser. B, 55(1):3-24, 1993.

....; aN ) de 1 : de M dh 1 : dhM da 1 : daN : To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm; for background see e.g. Smith and Roberts [14]; Tierney [15] Gilks et al. 6] Roberts and Rosenthal [12] 7 There is of course some arbitrariness in the specification of this Bayesian algorithm, e.g. in the form of the prior distributions and in the precise formula for the probability of a link from i to j. However, the model appears to ....

A.F.M. Smith and G.O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). Journal of the Royal Statistical Society, Series B, 55:3--24, 1993.

....: aN ) de 1 : deM dh 1 : dhM da 1 : daN : 8 To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm; for background see, e.g. Smith and Roberts [22]; Tierney [23] Gilks et al. 11] Roberts and Rosenthal [20] The Metropolis Algorithm proceeds by starting all the 2M N parameter values at 1. It then attempts, for each parameter in turn, to add an independent N(0; 2 ) random variable to the parameter. It then accepts this new value with ....

A.F.M. Smith and G.O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). Journal of the Royal Statistical Society, Series B, 55:3--24, 1993.

..... aN ) de1 . deM dh1 . dhM da1 . daN . To actually compute these conditional means is non trivial. To accomplish this, we used a Metropolis Algorithm. The Metropolis algorithm is an example of a Markov chain Monte Carlo Algorithm; for background see, e.g. Smith and Roberts [13]; Tierney [14] Gilks et al. 6] Roberts and Rosenthal [12] There is, of course, some arbitrariness in the specification of this Bayesian algorithm, e.g. in the form of the prior distributions and in the precise formula for the probability of a link from i to j. However, the model appears to ....

A.F.M. Smith and G.O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). Journal of the Royal Statistical Society, Series B, 55:3--24, 1993.

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, pages 95110. Smith, A. and Roberts, G. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. R. Statist. Soc. B 55, pages

....of Hastings and Metropolis algorithms in more than one dimension, and to use these results to provide central limit theorems for such algorithms. This work builds on that in [4] which concentrated on the one dimensional case. It is becoming increasingly well recognised and almost commonplace [1, 10, 9, 7, 12] that efficient simulation of a probability density (x) on IR d which is only known analytically up to a factor (that is, when only (x) y) is known, such as is the case for the posterior distribution in many Bayesian contexts) can be carried out using various forms of Markov chain Monte Carlo ....

....we denote the first entry time to A by A = min(n 0 : Phi n 2 A) and probabilities conditional on Phi 0 = x by P x . For chains with the structure (2) and (3) it is simple to check that the chain Phi is irreducible if (y) 0 ) q(x; y) 0; x 2 X: 10) Weaker conditions are possible (see [2, 10, 11], amongst others) The theory of such chains is described in Meyn and Tweedie [6] As discussed there, in order to develop criteria for rates of convergence, we need the concepts of small sets and aperiodicity. It is known [6, Chapter 5] that for a irreducible chain, any set A with (A) 0 ....

A.F.M. Smith and G.O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Statist. Soc. Ser. B, 55:3--24, 1993.

....the conditional probability density function that results when w i Theta max k w (k) i has conditional density proportional to Q i (w; m) The posterior for Delta is not amenable to an analytical solution. Numerical techniques must be used, and we use a Gibbs sampling (Geman and Geman, 1984; Smith and Roberts, 1993) strategy to obtain approximate samples from the desired posterior distribution. Gibbs techniques can be used to obtain a sample from a desired target distribution by simulating realizations from a Markov chain whose stationary distribution is equal to the target. The target distribution is the ....

....we can simulate approximate realizations from the joint posterior. The distribution of sampled points converges to the posterior distribution because the conditionals in equations (8) 12) have the entire parameter space as its support, thus producing an aperiodic irreducible Markov chain (Smith and Roberts, 1993). We use the quantile method to obtain posterior interval estimates. The details of the implementation, sampling, burn in, simulation length, and subsampling are similar to those in Givens et al. 1997) and we do not repeat them here. 3 Performance on Simulated Data Sets 3.1 Simulated Data Sets ....

Smith, A.F.M. and Roberts, G.O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). Journal of the Royal Statistical Society Series B, 55:3--23.

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Roberts, G.O. and Smith, A.F.M. (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, J. Roy. Stat. Soc. B 55, 1, 3-23.

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Smith A.F.M. and Roberts G.O (1993) Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, J. Roy. Stat. Soc. B 55, 1, 1993, 3-23.

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A. F. M. Smith and G. O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. Royal Statistical Society, B55(1):3--23, 1991.

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A. F. M. Smith and G. O. Roberts. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. Royal Statistical Society, B55(1):3--23, 1991.

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A.F.M. Smith and G.O. Roberts (1993), Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion). J. Roy. Stat. Soc. Ser. B 55, 3-24.

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A. Smith and G. Roberts, "Bayesian computation via the Gibbs Sampler and related Markov chain Monte Carlo methods," J. R. Statist. Soc., B, vol. 55, no. 1, pp. 3--24, 1993.

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Smith, A.F.M. and G.O. Roberts (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods. J. Royal Statistical Society B 55, 323.

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A.F.M. Smith and G.O. Roberts, Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion), J. Roy. Statist. Soc. Ser. B 55 (1993) 3-24.

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Smith, A. F. M. and Roberts, G. O. (1993). Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion) . Journal of the Royal Statistical Society, Series B 55 3-23.

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Smith, A. F. M., and Roberts, G. O. (1993). Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods. Journal of the Royal Statistical Society B, 55, 3--23.

No context found.

A.F.M. Smith and G.O. Roberts, Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion), J. Roy. Statist. Soc. Ser. B 55 (1993) 3-24.

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Smith, A.F.M. and Roberts, G.O., 1993. Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods, Journal of the Royal Statistical Society, B, 55(1):3-23.

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Smith, A.F.M. and Roberts, G.O. (1993), "Bayesian computation via the Gibbs sampler and related Markov Chain Monte Carlo Methods," Journal of the Royal Statistical Society, B 55, 3-23.

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Smith, A.F.M. and Roberts, G.O. (1993). "Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods". J. Roy. Statist. Soc. B, 55, 3-23.

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Smith, A. F. M. and Roberts, G. O. (1993), "Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods," Journal of the Royal Statistical Society B, 55, 3--23.

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Rev. Biochem, 61, 1053. Smith, A. F. M. and Roberts, G.O. (1993), "Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods", J. R. Statist. Soc. B, 55,3-23.

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Smith, A and Roberts, G. (1993), Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods (with discussion), Jour. Roy. Stat. Soc., B, 55, 3-24..

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, 499-524. Smith, A.F.M. and Roberts, G.O. (1993). `Bayesian computation via the Gibbs sampler and related Markov chain Monte Carlo methods (with discussion)', Journal of the Royal Statistical Society, Series B, 55,

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