Smith, A.F.M and A.E. Gelfand (1992). Bayesian statistics without tears: A sampling-resampling perspective. The American Statistican 46(2), 84--88.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....importance sampling for recursive Bayesian estimation. However, the paper of [8] with sequential Monte Carlo simulation techniques for solving the optimal estimation problem started the research in particle filtering techniques. It was the introduction of the crucial resampling step introduced in [14, 8] that solved previous divergence problems. In this section the presentation of the sequential Monte Carlo method, or particle filter theory is according to theory and notations described thoroughly in [8, 4, 13, 6] The particle filter method provides an approximative Bayesian solution to ....

A. Smith and A. Gelfand. Bayesian statistics without tears: A sampling-resampling perspective. The American Statistican, 46(2):84--88, 1992.

....an additional component in our measurement vector. Unfortunately, we will show in Section 2 that this is not practical with a Kalman filter (or an extended Kalman filter, for that matter) Instead, we resort to the samplebased nonparametric density estimation technique known as particle filtering [1, 2]. In addition to permitting the use of RCS as a component of our measurement vector, we will show that particle filtering also presents a useful framework for a joint approach to tracking and classification. A particle filtering approach could also be used for joint tracking and classification in ....

....approaches to density estimation. This motivation will lead us to the weighted bootstrap and its use for recursive Bayesian inference. Recursive Bayesian Inference via Weighted Bootstrap Sampling based approaches to density estimation are discussed extensively in the statistics literature [1, 2, 29, 30]. We can motivate these approaches by considering the posterior density. In the non recursive case, Bayes rule yields ( 2 # , 52) Thus, evaluation of the posterior requires knowledge of both the prior and the likelihood function , in addition to an integration to find ....

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A. F. M. Smith and A. E. Gelfand, "Bayesian statistics without tears: A sampling-resampling perspective," The American Statistician, vol. 46, pp. 84--88, May 1992.

....not come without a penalty. Namely, the lack of a closed form expression for RCS prevents us from using an extended Kalman filter or any of its variants for state estimation. Instead, we rely upon a sequential Monte Carlo based approach known as particle filtering to perform the needed inference [9 15]. In addition to allowing us to include RCS as a component of our measurement vector, we will also show that particle filtering simplifies the implementation of our nonlinear non Gaussian flight model. In summary, although others have suggested that tracking and classification should be performed ....

A. F. M. Smith and A. E. Gelfand, "Bayesian statistics without tears: A samplingresampling perspective," The American Statistician, vol. 46, pp. 84--88, May 1992.

....sequential Monte Carlo methods, or particle filters, could be used. The simulation based ideas have been discussed in [4] where the conditional mean and covariance were calculated using importance sampling for recursive Bayesian estimation. However, the crucial resampling step introduced in [5, 6] solved the divergence problems. In this section the presentation of the particle filter theory is according to [1, 3, 6, 7] Consider the following non linear discrete time system x t 1 = f(x t , v t ) 1a) y t = s t d t = h(x t , e t ) 1b) denotes the state of the system and where y ....

A.F.M Smith and A.E. Gelfand, "Bayesian Statistics Without Tears: A Sampling-Resampling Perspective," The American Statistican, vol. 46, no. 2, pp. 84--88, 1992.

....using range image sequences. Characteristic for these applications are the problems of tracking multiple, newly appearing or occluded objects which can not all be handled by the basic method. The rst step towards the Condensation algorithm was the development of several resampling techniques [28, 9, 29]. A sample set from one distribution is resampled to form samples from another distribution. The idea of recursive Bayesian ltering based on sample sets was then independently discovered by several research groups. Our work has evolved from the Condensation algorithm [13, 14] when it became ....

A. F. M. Smith and A. E. Gelfand, Bayesian Statistics Without Tears: A SamplingResampling Perspective, The American Statistician 46(2) May (1992), pp. 84-88.

.... Each such particle x is distributed according to the belief distribution Bel(x t ) Obviously, the pair is distributed according to the product distribution : 12) In accordance with the literature on the SIR algorithm (short for: Sampling importance resampling) [62,67,69], we will refer to this distribution q t as the proposal distribution. Its role is to propose samples of the desired posterior distribution, which is given in Eq. 9) however, it is not equivalent to the desired posterior. 3) Finally, correct for the mismatch between the proposal distribution ....

A.F.M. Smith, A.E. Gelfand, Bayesian statistics without tears: A sampling-resampling perspective, American Statistician 46 (2) (1992) 84--88.

....of this method depends critically on the choice of q(x) it must both approximate p(x) well and be easy to sample from. Often, especially in high dimensional problems, these two desiderata will be in conflict. This often leads to inefficient sample generation, as many proposals are rejected [92]. At the cost of producing a sequence of correlated samples, rather than a set of indepen dent ones, the theory of Markov chains provides a more efficient method of sampling from a distribution [43] In outline, we wish to construct a Markov chain which has as its invariant distribution the ....

A.F.M. Smith and A.E. Gelfand. Bayesian statistics without tears: a samplingresampling approach. The American Statistician, 46(2):84--88, May 1992.

....iteratively repeat the computational procedure to update the distribution of # given new observations of S, so that the posterior distribution of # for the current iteration will become the prior distribution of # for the next iteration. Therefore we choose the SIR algorithm (see Smith and Gelfand [20] and Pitt and Shepherd [17] a simulation procedure that requires only the ability to evaluate the likelihood function, and to sample from the prior density of #, rather than the Metropolis Hastings algorithm which requires the ability to evaluate the prior density (see Gilks et al. 9] The ....

A.F.M. Smith, A.E. Gelfand, Bayesian statistics without tears: a sampling-resampling perspective, American Statistician 46 1992 pp84-88

....system) and its heading direction . The belief over the state space is updated whenever the robot moves and senses. Monte Carlo Localization (in short: MCL [7] relies on sample based representations for the robot s belief and the sampling importance resampling algorithm for belief propagation [8, 9]. The sampling importance resampling algorithm has been introduced for Bayesian filtering of nonlinear, non Gaussian dynamic models. It is known alternatively as the bootstrap filter [10] the Monte Carlo filter [11] the Condensation algorithm [12] or the survival of the fittest algorithm [13] ....

A.F.M Smith and A.E. Gelfand. Bayesian statistics without tears: A sampling-resampling perspective. American Statistician, 46(2):84--88, 1992.

....In the rest of this section we write the prior as f(#) and the likelihood as f(y #) abstracting from subscripts and conditioning arguments, in order to briefly review these methods in this context. 2.3. 1 Sampling importance resampling (SIR) This method (due to Rubin (1987) Rubin (1988) and Smith and Gelfand (1992)) can be used to simulate from a posterior density f(# y) given an ability to: 1. simulate from the prior f(#) 2. evaluate (up to proportionality) the conditional likelihood f(y #) which is assumed to vary smoothly with #. The idea is to draw proposals # 1 , # R from f(#) and then ....

....out than the likelihood and so the second of these problems should not be typically important. However, the sensitivity to aberrant observations will be important. 2.3.2 Rejection sampling The SIR method has some similarities with rejection sampling (see, for example, Ripley (1987, pp. 60 62) and Smith and Gelfand (1992)) which is based on simulating from f(#) and accepting with probability #(#) f(y #) f(y # max ) where # max = arg max # f(y #) Again the rejection becomes worse if the var f #(#) is high. A fundamental di#erence is that rejection sampling produces a random sample regardless of the size of ....

Smith, A. F. M. and A. E. Gelfand (1992). Bayesian statistics without tears: a samplingresampling perspective. American Statistican 46, 84--88.

....value, then we sample from the conditional distribution of T i given (Y i ; Delta i ; and augment it to the observed data to form a complete data likelihood. 3 Model Fitting using MCMC Customary iterative approaches, using, e.g. rejection method or weighted bootstrap, as presented in Smith and Gelfand (1992) are difficult to implement for our models. However, the form in (5) is nicely suited for Markov Chain Monte Carlo (MCMC) simulation using Gibbs sampler of Gelfand and Smith (1990) Using the independent prior process formulation we express the sampling distribution as usual mixture model. In ....

Smith, A.F.M. and Gelfand, A.E. (1992). Bayesian statistics without tears: A sampling-resampling perspective. Amer. Statist., 46, 84--88.

.... Kitagawa (1996, 1998) and Kitagawa and Gersch (1996) proposed nonlinear filter and smoother using a resampling procedure (see, for example, 6 Note that we need N N order of computation for each integration in equations (13) 14) and (16) and N order of computation for equation (15) 18 Smith and Gelfand (1992) for the resampling procedure) Let us define # i,r s as the i th random draw of # r generated from the conditional density P (# r Y s ) We consider how to generate random draws # i,r s , i = 1, 2, N . Prediction: The prediction estimate is very simple and easy (see, for example, ....

Smith, A.F.M. and Gelfand, A.E., 1992, " Bayesian Statistics without Tears: A Sampling-Resampling Perspective, " The American Statistician, Vol.46, No.2, pp.84 -- 88.

....all the probability weights #(x # j ) j = 1, 2, N , have to be computed for IR. In Step (ii) practically we need to generate a uniform random draw between zero and one, denoted by u, and set x i = x # j when # j 1 # u # j , where # j # # j 1 #(x # j ) and # 0 # 0. For example, see Smith and Gelfand (1992) for the resampling procedure. To obtain N random draws from p(x) IR requires just N random draws from p # (x) but RS needs N(1 NR ) random draws from p # (x) For IR, when we have N di#erent random draws from the sampling density, we pick up one of them with the corresponding probability ....

Smith, A.F.M. and Gelfand, A.E. (1992). Bayesian Statistics without Tears: A Sampling-Resampling Perspective, The American Statistician, 46, 84--88.

....Monte Carlo #MCMC#. In the rest of this section we write the prior as f### and the likelihood as f#yj##, abstracting from subscripts and conditioning arguments, in order to brie#y describe these methods in this context. 2 2.2. 1 Sampling#importance resampling #SIR# This method #Rubin #1987# and Smith and Gelfand #1992## draws # 1 ; # R from f### and then associates with each of these draws the weights # j where w j = f#yj# j #; # j = w j P R i=1 w i ; j =1; R: The weighted sample will converge, as R 1, to a non random sample from the desired posterior f##jy# as R ,1 P R i=1 w i p ....

Smith, A. F. M. and A. E. Gelfand #1992#. Bayesian statistics without tears: a sampling-resampling perspective. American Statistican 46, 84#88.

....Importance Sampling 3.2.1 The Importance Sampling Algorithm Importance sampling appeared in the literature as early as the 60 s ( 41] 78] proposed using a sampling importance resampling algorithm for drawing imputations from the posterior distribution in Bayesian missing data problem. [84] discussed rejection and weighted bootstrapping methods to obtain posterior samples via random draws from the prior distribution. This section o ers an introduction to Bayesian importance sampling; detailed illustration of the application of importance sampling methods in Bayesian inference can be ....

A F M Smith and A E Gelfand. Bayesian statistics without tears: A samplingresampling perspective. The American Statistician, 46(2):84-88, 1992.

....posterior in closed form. The standard approach for posterior estimation in such situations is Monte Carlo Markov Chain (MCMC) Doucet et al. 2000; Gilks et al. 1996; Neal, 1993) In particular, our approach uses the popular Metropolis Hastings algorithm (Hastings, 1970; Metropolis et al. 1953; Smith Gelfand, 1992), for approximating the desired posterior summaries. However, Metropolis Hastings can be extremely inefficient in large spaces (Gilks et al. 1996) To remedy this problem, we propose a new proposal distribution based on chain flipping, which is specifically suited for data association problems. ....

....assignments. Experimental results show that this approach is orders of magnitude more efficient than commonly used approaches that ml.tex; 27 12 2000; 9:23; p. 3 4 Dellaert, Seitz, Thorpe Thrun consider only local changes in the MCMC sampling process (e.g. Gibbs sampling (Geman Geman, 1984; Smith Gelfand, 1992)) The M step calculates the location of the features in the scene, along with the camera positions. As pointed out, the SFM literature has developed a number of excellent algorithms for solving this problem under the assumption that the data association problem is solved. However, the E step ....

Smith, A., & Gelfand, A. (1992). Bayesian statistics without tears: a samplingresampling perspective. American Statistician, 46, 84--88.

....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 ....

....that will lead to this. Prior to considering these and other results we develop some background and auxiliary results relating the Hastings and Metropolis algorithms to more general Markov chain theory. 3 Irreducible Markov chains and Hastings Metropolis algorithms As has been observed often (cf [9, 12, 2]) the Hastings algorithms (and other algorithms of the Markov chain Monte Carlo type, such as the Gibbs sampler) can be 3 Irreducible Markov chains and Hastings Metropolis algorithms 4 analysed using the theory of irreducible Markov chains: that is, chains for which there exists a measure ....

A. F. M. Smith and A. E. Gelfand. Bayesian statistics without tears: A samplingresampling perspective. Amer. Statist., 46:84--88, 1992.

....posterior in closed form. The standard approach for posterior estimation in such situations is Monte Carlo Markov Chain (MCMC) Doucet et al. 2001; Gilks et al. 1996; Neal, 1993) In particular, our approach uses the popular Metropolis Hastings algorithm (Hastings, 1970; Metropolis et al. 1953; Smith Gelfand, 1992), for approximating the desired posterior summaries. However, the design of efficient Metropolis Hastings algorithms can be very difficult in high dimensional spaces (Gilks et al. 1996) In this paper, we propose a novel, efficient proposal distribution based on chain flipping, which is ....

....is a method that can quickly jump across globally different assignments. Experimental results show that this approach is orders of magnitude more efficient than commonly used approaches that consider only local changes in the MCMC sampling process (e.g. Gibbs sampling (Geman Geman, 1984; Smith Gelfand, 1992)) The M step calculates the location of the features in the scene, along with the camera positions. As pointed out, the SFM literature has developed a number of excellent algorithms for solving this problem under the assumption that the data association problem is solved. However, the E step ....

Smith, A., & Gelfand, A. (1992). Bayesian statistics without tears: A samplingresampling perspective. American Statistician, 46(2), 84--88.

....the variability of rainfall under given hypothetic scenarios, it is desirable to have a fast and cheap procedure to update the posterior distribution. A simple sampling procedure to update the distribution of the parameters of a likelihood for which some observations are available is presented in Smith and Gelfand (1992), this is known as sampling importance resampling or SIR and consists of resampling a sample of the parameters using the likelihood as weighing function. A modified SIR is developed in Pitt and Sheppard (1997) in the context of particle filters; these are algorithms developed to update a discrete ....

Smith, A. F. M. and Gelfand, A. E. (1992) Bayesian statistics without tears: a sampling-resampling perspective. The American Statistician, 46, 84--88.

....to the fact that the hyperparameters are defined as constants through the 1 These models were estimated as static models that are numerically equal to dynamic models without stochastic components in the transition equation. 3 sampling period 2 . According to the SIR techniques ( Rubin, 1988; Smith and Gelfand, 1992)) samples from the posterior distribution of , p( j D T ) can be obtained by sampling from p( and then computing for each sampled value i its likelihood marginal, L( i ; D T ) where D T = fy 1 ; y T ; D 0 g is the observed data up to time T , and D 0 is the set of prior information. ....

....be used with the importance function equal to the prior. More details about MCIS can be found in van Dijk and Kloek (1980, 1985) Efron (1982) and Gamerman (1998) The SIR technique and the MCIS, just mentioned, are comparable asymptotically. In the last section we have seen that Rubin (1988) and Smith and Gelfand (1992) present a result which guarantees that P j =M converges to E( j D T ) as n and M goes to infinity. On the other hand, the MCIS method guarantees, for example, that = P n i=1 i L( i ; D T ) P n i=1 L( i ; D T ) n Gamma 1 Gamma R p(y j ; D 0 )p( d R p(y j ; D 0 )p( d = E( j D T ....

Smith, A. F. M. and Gelfand, A. E. (1992) Bayesian statistics without tears: A samplingresampling perspective (corr: 93v47 p158). The American Statistician, 46, 84--88.

....random sequences at each timestep according to the observation density. Resampling in the context of static probability distributions is described by Rubin (1988) where he calls it the Sampling Importance Resampling (SIR) algorithm, and performed in a Bayesian context as a weighted bootstrap by Smith and Gelfand (1992). The extension of the resampling idea to the recursive filtering of time series data was recently independently discovered by several researchers (Gordon et al. 1993; Kitagawa, 1996; Isard and Blake, 1996) The algorithm to do this in the context of computer vision, which we denote Condensation ....

.... Jepson, 1998) communities (chapters 3 7 contain material originally published as (Isard and Blake, 1998a; Isard and Blake, 1998c; Isard and Blake, 1998d; Isard and Blake, 1998b) Gordon (1993; 1995) describes the algorithm, which he calls the bootstrap filter after the weighted bootstrap of (Smith and Gelfand, 1992), and applies it to the traditional filtering problem of bearings only tracking. He describes some ad hoc techniques to improve the efficiency of the sampling scheme, a discussion of which is deferred until section 8.3 on page 130 by which time the relevance to the work in this thesis will be ....

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Smith, A. and Gelfand, A. (1992). Bayesian statistics without tears: a sampling-resampling perspective. The American Statistician, 46, 84--88.

....remaining the same . ffl Changepoint in linear models : Bacon and Watts (1971) Ferreira (1975) Holbert and Broemeling (1977) Choy and Broemeling (1980) Moen, Salazar and Broemeling (1985) Smith and Cook (1990) ffl Hierarchical Bayes formulation of changepoint model : Carlin, Gelfand and Smith (1992). Most of these references provide methodology for off line problems. We are interested in process control applications where on line algorithms are needed. The next subsection describes the simplest possible on line scenerio. 2.2 Single Jump Before and After Change Parameter Values Known In ....

....close attention to numerical methods and substantial amounts of computer time. Also since our problem is an on line algorithm, finding the range of numerical integration at a later stage is not an easy task. This has led us to use the simulation based technique of rejection sampling, described by Smith and Gelfand(1992), to evaluate such integrals. Needed calculations are easy to program and can be carried out quite efficiently by using any standard mathematical numerical software package. The method of rejection sampling used here is described in detail in the appendix. We initially generate a large number of ....

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Smith, A. F. M. and Gelfand, A. E. (1992), Bayesian Statistics Without Tears: A Sampling-Resampling Perspective. The American Statistician, 46, 84-88.

....without modification. In sampling based methods one represents the density p(x k jZ k ) by a set of N random samples or particles S k = fs i k ; i = 1: Ng drawn from it. We are able to do this because of the essential duality between the samples and the density from which they are generated [33]. From the samples we can always approximately reconstruct the density, e.g. using a histogram or a kernel based density estimation technique. The goal is then to recursively compute at each timestep k the set of samples S k that is drawn from p(x k jZ k ) A particularly elegant algorithm to ....

.... of the mechanism underlying the CONDENSATION algorithm are given in [6, 29] The entire procedure of sampling, reweighting and subsequently resampling to sample from the posterior is called Sampling Importance Resampling (SIR) 30] and an accessible introduction to it can be found in [33]. 6 Experimental Results As a test bed for our approach we used the robotic tourguide Minerva, a prototype RWI B18 robot shown in Fig. 4. In the summer of 1998, Minerva functioned for two weeks as an interactive robotic tour guide in the Smithsonian s National Museum of American History. During ....

A.F.M. Smith and A.E. Gelfand. Bayesian statistics without tears: A sampling-resampling perspective. American Statistician, 46(2):84-- 88, 1992.

....i ffi Gamma D i Gamma i ffi D Gamma x 0 i ff) and ( Deltaj i ; oe 2 ) is a normal density with mean i ( i = in Model 3, and i = u 0 i fi in Model 4) and variance oe 2 . Obtaining a direct draw from this density can be difficult. Instead, we apply the weighted bootstrap (Smith and Gelfand, 1992), which is closely related to the SIR algorithm (Rubin, 1988) This can be applied as follows. The product of the second and third factors in (19) correspond to an unnormalized truncated normal density. We simulate eight values at random from this truncated normal density. Denote these eight ....

Smith, A.F.M. and Gelfand, A.E.(1992) "Bayesian statistics without tears: a sampling-resampling perspective," American Statistician, 46, 84--88.

....samples for ff t 1 (x) are obtained by sampling from the sample set representing ff t (x) see also Section 3.1) and the tree representing . The p x values of ff t 1 (x) are determined using the tree representing . This recursive resampling technique, known as sampling importance resampling [44, 49] applied to time invariant Markov chains, converges at the rate 1= p N (if T is finite, c.f. Section 3.2) Sampling importance resampling, which will further be discussed in Section 8, has been successfully applied in domains such as computer vision and robotics [11, 18] 6 Error Analysis and ....

.... of the ffs) is essentially equivalent to the condensation algorithm proposed by Isard and Blake [18] and the Markov localization algorithm proposed by Dellaert and colleagues [11, 10] both of which are basically versions of the well known sampling importance resampling (SIR) algorithm [44, 49]. Similar approaches are known as particle filters [41] bootstrap [14] and survival of the fittest [22] All these approaches are concerned with state estimation in an HMM like or Kalman filter like fashion. Thus, they rely on the a priori availability of and , which are learned from data by ....

A.F.M. Smith and A.E. Gelfand. Bayesian statistics without tears: a sampling-resampling perspective. American Statistician, 46:84--88, 1992.

....model is not amenable to analytical treatment. Inference about the model parameters, as well as functions of them, will be based on Monte Carlo exploration of the posterior distribution, using the practical duality between a density function and a (large) sample drawn from this density function (Smith and Gelfand, 1992). To this end, note that the posterior distribution of the model parameters can be factored as p( jz obs ) p( j ; #; z obs )p( j ; #; z obs )p( #; jz obs ) suggesting an algorithm to generate a sample from the posterior distribution. For each i = 1; m, generate: i ; # ....

Smith, A.F.M. and Gelfand, A.E. (1992). Bayesian Statistics Without Tears: A SamplingResampling Perspective. The American Statistician 46, 84-88.

....used to eliminate samples with low importance ratios and multiply samples with high importance ratios. It is possible to see an analogy to the steps in genetic algorithms. Many of the ideas on resampling have stemmed from the work of Efron (Efron 1982) Rubin (Rubin 1988) and Smith and Gelfand (Smith and Gelfand 1992). Various authors have described efficient algorithms for accomplishing this task in O(N) operations (Doucet 1998, Pitt and Shephard 1997) Resampling involves mapping the measure fw (i) k ; q (i) k g into an equally weighted random measure fw (j) k ; N Gamma1 g. This is accomplished ....

Smith, A. F. M. and Gelfand, A. E. (1992). Bayesian statistics without tears: a samplingresampling perspective, American Statistician 46: 84--88.

....by the model structure. 3. Changing the pre model distribution: Suppose one has carried out this procedure for one pre model distribution p 1 ( OE) and one wants results for another, p 2 ( OE) One may, of course, rerun the procedure from scratch, replacing p 1 by p 2 . As suggested by Smith and Gelfand (1992), one may also resample from the final sample of size obtained the first time with p 1 , using weights p 2 ( Phi( p 1 ( Phi( This is more convenient, because it does not require any new runs of the model. It will work well if p 2 is less diffuse than p 1 and is covered by it. ....

....important. It would be useful at least to assess the sensitivity of our results to such heterogeneity. 6.3 Other Approaches As an alternative to the SIR algorithm described in Section 2. 2, a rejection sampling method might be considered for generating the post model sample; this was suggested by Smith and Gelfand (1992) in the context of standard Bayesian inference (but not of simulation models) However, for the present problem the maximum likelihood problem is intractable so that their suggestion of using the prior scaled up by the likelihood evaluated at the MLE as the initial sampling envelope would not be ....

Smith, A.F.M. and Gelfand, A.E. (1992), "Bayesian statistics without tears: a sampling-- resampling perspective," The American Statistician, 46, pp. 84--88.

....of sampling n observations from the prior distribution, attaching weights to the sampled points according to their likelihood, and then sampling with replacement from this weighted discrete distribution. As n 1, the resulting set of values then approximates a sample from the required posterior (Smith and Gelfand, 1992). In the dynamic version, proposed by Gordon et al. 1993) the SIR algorithm is applied repeatedly as new data are acquired. One can think of the sample points in a SIR filter as a set of particles which move according to the state model and multiply or die depending on their fitness as ....

Smith, A. F. M. and Gelfand, A. E. (1992) Bayesian Statistics Without Tears: A SamplingResampling Perspective. The American Statistician, 46, 84--88.

.... simulation; in fact, we will use noniterative Monte Carlo with a suitable importance sampling density (ISD) This ISD enables both Monte Carlo integration for expectations regarding the posterior of (as in Geweke, 1989) as well as Monte Carlo sampling for other distributional features ( as in Smith and Gelfand, 1992). Though the ISD is of very high dimension and would require an astronomical amount of sampling to learn about the joint posterior of and W , since interest lies only in f( jY ) a much smaller number of draws may be sufficient. This latter number will grow large with n, resulting in long run ....

Smith, A.F.M. and Gelfand, A.E. (1992), " Bayesian Statistics without tears: a sampling-resampling perspective," The American Statistician, 46, 84--88.

....form weights i = f( i ) g( i ) Let = P V i=1 i and q i = i ; i = 1; V . Monte Carlo integration for any posterior expectation say E(b( jy) takes the form P q i b( i ) while resampling the i using the probabilities q i provides an approximate sample from the posterior (Smith and Gelfand, 1992). West s adaptive mixture method (1993) is used to iteratively construct a mixture distribution to use as a g( Since West s procedure employs mixtures of multivariate normal distributions, all parameters should be transformed to the real line to improve the resulting g( Starting from some g ....

Smith, A.F.M. and Gelfand, A.E. (1992). Bayesian statistics without tears: A samplingresampling perspective. American Statistician. 46(2), pp 84-88.

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Smith, A.F.M. and Gelfand, A.E. (1992), "Bayesian Statistics Without Tears: a Sampling-Resampling Perspective, " The American Statistician, 46, 2, 84-88.

.... i ) g( i ; i ) After calculating q i = v i P v i ; i = 1; V , Monte Carlo integration for any posterior expectation, E(b( jY ) takes the form P q i b( i ; i ) while resampling the ( i ; i ) using the probabilities q i provides an approximate sample from the posterior (Smith and Gelfand, 1992). g is obtained using West s (1993) adaptive mixture method after transforming all parameters to R 1 . 5 Analysis of the Scallop Data Since 1982, the NMFS has been annually sampling scallops o the Northeastern United States coastline from the Delmarva Peninsula to the Georges Bank. We analyze ....

Smith, A.F.M. and Gelfand, A.E. (1992). Bayesian statistics without tears: A samplingresampling perspective. American Statistician. 46(2), pp 84-88.

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Smith, A.F.M and A.E. Gelfand (1992). Bayesian statistics without tears: A sampling-resampling perspective. The American Statistican 46(2), 84--88.

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Smith A.F.M. and Gelfand A.E. (1992) Bayesian Statistics without Tears: a SamplingResampling Perspective. The American Statistician, 46, 84-88. 30

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A.F.M. Smith and A.E. Gelfand. Bayesian Statistics Without Tears: A Sampling-Resampling Perspective. The American Statistician, 46:84--88, 1992.

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Smith, A. F. M. and Gelfand, A. E. (1992). Bayesian statistics without tears: a samplingresampling perspective, American Statistician 46(2): 84-88.

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A. F. M. Smith and A. E. Gelfand. Bayesian statistics without tears:a sampling-resampling perspective. The American Statistician, 46:84--88, 1992.

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A.F.M. Smith and A.E. Gelfand, \Bayesian statistics without tears: A samplingresampling perspective," American statistican, vol. 46, no. 2, pp. 84-88, 1992.

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A.F.M. Smith and A.E. Gelfand. Bayesian statistics without tears: a samplingresampling perspective. American Statistician, 46(2):84--88, 1992.

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A.F.M. Smith and A.E. Gelfand, \Bayesian statistics without tears: A samplingresampling perspective," American statistican, vol. 46, no. 2, pp. 84-88, 1992.

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SMITH, A. F. M. AND GELFAND, A. E. `Bayesian Statistics Without Tears: A SamplingResampling Perspective', The American Statistician, 1992, 46, pp. 84--88.

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A.F.M. Smith and A.E. Gelfand. Bayesian statistics without tears: a sampling-resampling perspective. Am. Stat., 46(2):84--8, May 1992.

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A. F. M Smith and A. E. Gelfand. Bayesian statistics without tears: A samplingresampling perspective. American Statistician, 46(2):84--88, 1992.

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A. F. M. Smith and A. E. Gelfand. Bayesian statistics without tears: a sampling-resampling perspective. American Statistician, 46(2):84-88, 1992. 48

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A.F.M. Smith and A.E. Gelfand, \Bayesian statistics without tears: A samplingresampling perspective," American statistican, vol. 46, no. 2, pp. 84-88, 1992.

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Smith, A., Gelfand, A.: Bayesian statistics without tears: A sampling-resampling perspective. American Statistician 46 (1992) 84-88

No context found.

A.F.M. Smith andA. E. Gelfand, "Bayesian statistics without tears: A sampling- resampling perspective," The American Statistician, vol. 46, pp. 84-88, May 1992.

No context found.

Smith A.F.M. and Gelfand A.E. (1992) Bayesian Statistics without Tears: a SamplingResampling Perspective. The American Statistician, 46, 84-88. 30

No context found.

A. F. M. Smith and A. E. Gelfand. Bayesian statistics without tears: A sampling--resampling perspective. American Statistician, 46:84--88, 1992.

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