Gelman, A. B., Carlin, J. S., Stern, H. S., & Rubin, D. B. (1995). Bayesian data analysis. London: Chapman and Hall.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....maximum likelihood to obtain a point estimate, we use a Gibbs sampling procedure to obtain the posterior distribution of the parameters (see Appendix A; also Albert and Chib [1993] Gelfland and Smith [1990] and Tanner and Wong [1987] 9 4. 2 The Hierarchical Model The hierarchical model (see Gelman, Carlin, Stern, and Rubin [1996]) is a generalization of the regression model that allows each site to have its own value for the coefficients: 2 2 , s b s b itj j j t i itj itj x N x Y , 2) where x ijt is defined as in the previous section, except that we index the site as well, j=1, J. Let ) 1 jM ....

Gelman, Andrew, John Carlin, Hal Stern, and Donald Rubin (1996). Bayesian Data Analysis. London: Chapman and Hall.

....parameters and, potentially, families of models. Inference consists of computing conditional probabilities for unknown quantities given the known quantities typically data using the probability calculus. Accessible introduction to Bayesian inference concepts and methods are Gelman et al. [13] and Berry [14] Bayesian inference in medical guidelines development is discussed by Parmigiani et al. 15] In the context of this paper, Bayesian methods can provide a practical framework for assessing the consequences of uncertainty in the inputs on the output of a decision model. In the ....

Gelman A, Carlin J, Stern H, Rubin DB. Bayesian data analysis. London: Chapman and Hall, 1995.

....by attempting to restrict the simulation analysis to such a region by imposing constraints in the prior. This follows what is now essentially standard practice in other kinds of mixture models (e.g. West 1997) Finally, in order to explore predictive questions, for both model evaluation (as in Gelman et al. 1996) and for use in predicting potency of new compounds, we simulate predictive distributions based on the posterior parameters and latent variable samples. 3 A Simulated Dataset We report an analysis of a test dataset simulated to explore and validate the model and our implementation, which is ....

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (1996) Bayesian Data Analysis, London: Chapman and Hall.

.... #1 x 0 i #X 0 #,i# X #,i# # ,1 x i # 1=2 st n,k x 0 i # #,i# , where t n,k has the t distribution with n , k degrees of freedom, s and # #,i# are the usual estimators of # and # based on the n observations in Y #,i# , and X #,i# is the regressor matrix with row i deleted #Gelman et al. 1995, p. 239#. A straightforward calculation shows that the corre8 DRAFT sponding outlier p value is identical to that obtained by referring the studentized residual #y i , x i ##=s p 1 , h ii , where H = X#X 0 X# ,1 X 0 , to the t distribution with n , k degrees of freedom, if s is ....

Gelman, A., Carlin, J.B., Stern, H. and Rubin, D.B. #1995#, Bayesian Data Analysis, London: Chapman and Hall.

....to such a data as objects (vectors being only a special case) The basic problem with clustering objects is that there is no standard notion of distance among them. The simple Euclidean distance is not any more directly applicable. Some previous treatments of hierarchical models can be found in [Gelman et al. 1995]. The most natural way to introduce hierarchical models is to describe the appropriate generative model (Figure (4) ffl An individual is randomly drawn from the overall population (universe) and is indexed with letter i (the index represents the individual count ) q 3 Observed data on N ....

Gelman, A., Carlin, B., Stern, H. S., and Rubin, D., Bayesian Data Analysis, London: Chapman and Hall, 1995.

....a Markov chain whose equilibrium distribution is the desired distribution P ( jD) the integral in equation 33 is then approximated using samples from the Markov chain. Two standard methods for constructing MCMC methods are the Gibbs sampler and MetropolisHastings algorithms (see, e.g. Gelman et al., 1995). However, the conditional parameter distributions are not amenable to Gibbs sampling if the covariance function has the form given by equation 30, and the Metropolis Hastings algorithm does not utilize the derivative information that is available, which means that it tends to have an inefficient ....

Gelman, A., J. B. Carlin, H. S. Stern, and D. B. Rubin (1995). Bayesian Data Analysis. London: Chapman and Hall.

....i gjff; fi; z) By use of the Metropolis algorithm (Chib and Greenberg, 1995) a large number of values of (ff; fi) are simulated from the marginal distribution p(ff; fijz) Then for each simulated value of (ff; fi) values of the rates f i g are simulated from independent Gamma distributions. (Gelman, et al. (1995) suggest a similar simulation algorithm for simulating from a Poisson Gamma hierarchical model. The marginal posterior distributions of the rates f i g are summarized by the sets of simulated values. The median, 5th and 95th percentiles of the posterior distribution of i are computed by sorting ....

....analysis with respect to the model perturbation. Although there is general agreement about the need to check one s modeling assumptions, there is disagreement about the best way to perform this model criticism. Christiansen and Morris (1995) use classical methods to check the Poisson Gamma model, Gelman et al. (1995) advocate the use of the posterior predictive distribution, and other authors consider cross validation appoaches, such as the pseudo Bayes factor developed in Geisser and Eddy (1979) Bayes factors have not generally been used in model criticism of a given regression model (an exception is ....

Gelman, A., Carlin, J. B, Stern, H. S., and Rubin, D. B. (1995), Bayesian Data Analysis, London: Chapman and Hall.

....more restrictive conditions. It follows that BIC can also be a good approximation for the comparison of non nested models. 5 A brief introduction to Bayesian estimation is given in Raftery (1995, Section 3.1) several good references for further study are recommended there. To these I would add Gelman et al. 1995). in his Figure 1, with a normal prior and a single parameter, the Bayes factor for the null hypothesis (e.g. the independence hypothesis in Weakliem s 2 Theta 2 table examples) is roughly proportional to the prior standard deviation when the latter is large 6 . As a result, when the prior ....

Gelman, Andrew, John B. Carlin, Hal S. Stern and Donald B. Rubin 1995. Bayesian Data Analysis. London: Chapman and Hall.

....Bayesian Model Averaging 2.1. 1 General Principles Bayesian model averaging (BMA) is the Bayesian solution to the problem of inference in the presence of multiple competing models ( 10] 11] 12] 13] 14] 15] 16] 17] For general introductions to Bayesian inference, see [18] 19] and [20]. BMA starts by acknowledging that in the situation of equation (1) there are up to K = 2 q possible models (assuming that no interaction exists between the risk factors) defined by allowing each of X 1 ; X q to be either in or out of the model. We denote these models by M 1 ; ....

A. Gelman, J.B. Carlin, H.S. Stern, , and D.B. Rubin. Bayesian Data Analysis. London: Chapman and Hall, 1995.

....models. Finally, we must specify the predictors X and the model for the vectors fi and ff. The simplest task is to model ff, for which we use independent normal random effects: ff j iid N(0; 2 ) where is assigned a uniform prior distribution (as in the hierarchical models of chapter 5 of Gelman et al. 1995). There is no loss of generality in assuming a zero mean for the ff j s if a constant term is included in the set of predictors X . We choose the predictors X and model the coefficients fi using an informal Bayesian approach, by which we mean that we assign independent noninformative uniform ....

....by 39 zeroes, X = X W ff 0 I 39 , Sigma y = Diag(oe 2 V i ) Sigma = Sigma y 0 0 2 I 39 . The linear regression computation gives an estimate fl and variance matrix V fl , and the conditional posterior distribution, fljoe; X; y N(fl; V fl ) see, e.g. Gelman et al. 1995, chapter 8) The variance parameters oe and are not known, however, so we compute their posterior density numerically on a two dimensional grid. The marginal posterior density of the variance parameters can be computed using the formula, p(oe; jX; y) p(fl; oe; jX; y) p(fljoe; X; y) ....

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Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

....upon the deterministic point estimates, which we use as starting points. 3.4 Model checking The algorithm above yields a set of simulation draws (z l ; l ; y rep l ) l = 1; L. We can use these draws to check the model fit using posterior predictive checks (see Rubin, 1984, Gelman, Carlin, Stern, and Rubin, 1995, and Gelman, Meng, and Stern, 1996) as follows. First, define a test summary T (y; z) to summarize some aspect of interest of the data or its discrepancy from the model. Second, compute the realized values of the test summary, T (y; z l ) and the replicated values T (y rep l ; z l ) under ....

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

.... inferences, as long as (a) sample sizes are large enough that sampling distributions of estimands of interest are approximately normal, b) the inferences take into account design features such as stratification and clustering, and (c) the model uses noninformative prior distributions (see, e.g. Gelman et al. 1995, chapters 4 and 7) This similarity allows one to use model based calculations to get reasonable repeated sampling inference or, conversely, to use design based standard errors to make probability statements about unknown population quantities. With complex weighting schemes, the design based ....

.... 5 for an example) Standard errors for model based inferences such as in (4) come directly from the posterior distribution of the corresponding quantities of interest, such as in (2) which would be computed from posterior simulations of the parameter vector in a Bayesian analysis (e.g. Gelman et al. 1995). 3.3 Three simple examples illustrating the distinction between inverseprobability weights and poststratification weights In classical sampling theory, unit weights can be defined in two ways. Inverse probability weights (from Horvitz and Thompson, 1952) are defined as w i 1= j(i) and ....

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Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

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Gelman, A., Carlin, J. B., Stern, H. S. and Rubin, D. B. (1995). Bayesian Data Analysis, London: Chapman and Hall.

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Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

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Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

....this paper, we perform calculations of the probabilities of Type S errors for classical confidence statements and Bayesian posterior intervals under the proposed hierarchical model. The 100(1 Gamma ff) Bayesian posterior interval we consider is a so called central posterior interval (see, e.g. Gelman et al. 1995) bounded by the posterior ff=2 and 1 Gamma ff=2 quantiles. Our calculations are both frequentist and Bayesian: frequentist because they evaluate long term error rates (i.e. under repeated sampling) and Bayesian (or empirical Bayesian, in the sense of Morris, 1983) because the replications ....

....oe and the hyperparameters and are assumed to be known. Equivalently, we could say that there are a large number of studies and that the sample size within each study j are also large, such that , and oe can be accurately estimated from the data in the empirical Bayes sense; see, e.g. Gelman et al. 1995) or Carlin and Louis (1996) Given the data y and the parameters ; oe; the j s are independent with the following distributions: j jy; oe; N( j ; V j ) 6 where j = 1 oe 2 y j 1 2 1 oe 2 1 2 and V j = 1 1 oe 2 1 2 : The inverse of the posterior ....

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Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

....1994) and is very likely to give reliable answers Gelman: Hierarchical linear regression models and GLMs work pretty well. Of course, nonhierarchical models are even easier, but if the data make it possible to fit hierarchical models, I ll almost always do so (see Carlin and Louis 1996, and Gelman, Carlin, Stern, and Rubin 1995, for much more elaboration on this point) Mixture models (including Student t s) are tougher because the posterior distributions typically have multiple modes. Carlin: Certainly the simpler the model is, the better. I like the approach taken by the BUGS people to o#er not just a manual ....

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (1995), Bayesian Data Analysis, London: Chapman and Hall.

....As with the EM algorithm, data augmentation is most conveniently computed using properties of mathematical tractability of the augmented posterior distribution. Apart from solving the estimation problem, simulation of draws from the posterior distribution has three important advantages (see Gelman et al. 1995): 1) the computation of posterior intervals of any estimands of interest is straightforward, 2) the posterior sample can be used to investigate problems of local maxima and to trace trade off relations between parameters in the model and (3) the posterior sample can be used to check the fit of ....

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

....Mushrush, 1997, and Pinel et al. 1995) In our notation, y and are the logarithms of the measurement and the true ALAA radon level, respectively. The posterior distribution for is jM; y N( V ) 3) where = M S 2 y oe 2 1 S 2 1 oe 2 V = 1 1 S 2 1 oe 2 (4) see, e.g. Gelman et al. 1995). We base our decision analysis of when to measure and when to remediate on the distributions (2) and (3) Before moving to the decision analysis, we briefly discuss the relevance of the hierarchical aspect of our radon model. In a classical regression model, the estimated distributions of home ....

....review, and sometimes data analysis (in which case it is identified as a posterior rather than a prior distribution) However, it is not yet common for decision analyses to use the sorts of hierarchical models that are becoming standard in Bayesian statistics (see, e. g, Carlin and Louis, 1996, and Gelman et al. 1995), as we have done in the present paper. Indeed, we are unaware of any other case in which spatially varying recommendations have been made based on the output of a hierarchical model, with correct incorporation of spatially varying statistical uncertainties. A non hierarchical model that has ....

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995) Bayesian Data Analysis. London: Chapman and Hall.

....V is the average value of m j c var(l j ) We use V=m j in the hierarchical analysis because c var(l j ) is extremely variable for small states. Given the V j s, we estimate the parameters of the hierarchical model (14) Bayesianly, averaging over the hyperparameters and (see Rubin, 1981, and Gelman et al. 1995, chap. 5) We obtain the posterior medians of the parameters L j using simulation and use the inverse logits of these values as the fixed values of R W j in the subsequent analysis. The raw values R w j and the smoothed estimates b R W j for the pre election polls appear as the last two ....

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

....population quantities by summing N j j s. We view this approach as an approximation to the full Bayesian analysis, which averages over the parameters l . The two approaches will differ the most when components l are imprecisely estimated or are indistinguishable from 0 (see for example, Gelman et al. 1995, Section 5.5) In the example we consider here, this is not a problem because the various components are clearly estimated to be different from 0. If this were not the case, it would probably be worth putting in the additional programming effort for a full Bayes analysis. The focus of this paper, ....

....each l = 1; L. We solve the likelihood equations d dfi L(fijy; 0 using iteratively weighted least squares, involving a normal approximation to the likelihood p(yjfi) Q i p(y i jfi) based on locally approximating the logistic regression model by a linear regression model (see Gelman et al. 1995, p. 391) Let j i = Zfi) i be the linear predictor for the ith observation. Starting with the current guess of fi, let j = Z fi. Then a Taylor series expansion to L(y i jj i ) gives z i N(j i ; oe 2 i ) where z i = j i (1 exp(j i ) 2 exp(j i ) y i Gamma exp(j i ) 1 exp(j ....

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

....interesting and important contribution of viewing the pseudo data expression of the hierarchical linear model as a unifying tool in model checking. The analytical and computational use of the formulation for Bayesian inference is well known; see, e.g. Dempster, Rubin, and Tsutakawa, 1991, and Gelman et al. 1995, chapter 13. More generally, the author notes that hierarchical regression models are more difficult to understand than we might imagine, especially if predictors appear at more than one level in the hierarchy. We would like to echo that point with a statistical anecdote of our own (see Price, ....

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

....1994) and is very likely to give reliable answers Gelman: Hierarchical linear regression models and GLM s work pretty well. Of course, nonhierarchical models are even easier, but if the data make it possible to fit hierarchical models, I ll almost always do so (see Carlin and Louis, 1996, and Gelman et al. 1995, for much more elaboration on this point) Mixture models (including Student t s) are tougher because the posterior distributions typically have multiple modes. Carlin: Certainly the simpler the model is, the better. I like the approach taken by the BUGS people to offer not just a manual ....

Gelman, A., Carlin, J.B., Stern, H.S., and Rubin, D.B. (1995), Bayesian Data Analysis, London: Chapman and Hall.

.... are normally distributed conditional on the parameters, an assumption that is supported by Stern (1991) Rerunning the model with t distributions in place of normal distributions is straightforward since t distributions can be expressed as scale mixtures of normal distributions (see, e.g. Gelman et al. 1995) and Smith (1983) Rather than redo the entire analysis, we checked the sensitivity of our inferences to the normal assumption by reweighting the posterior draws from the Gibbs sampler by ratios of importance weights (relating the normal model to a variety of t models) The reweighting is easily ....

Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B.(1995), Bayesian Data Analysis, London: Chapman and Hall.

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Gelman, A. B., Carlin, J. S., Stern, H. S., & Rubin, D. B. (1995). Bayesian data analysis. London: Chapman and Hall.

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Gelman A. Bayesian data analysis. London: Chapman and Hall; 1995.

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Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995), Bayesian Data Analysis, London: Chapman and Hall.

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Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B. (1995). Bayesian data analysis. London: Chapman and Hall.

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Gelman, Andrew, John B. Carlin, Hal S. Stern, and Donald B. Rubin. 1995. Bayesian Data Analysis. London: Chapman and Hall.

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Gelman, A., Carlin, J. B., Stern, H. S., and Rubin, D. B., (1995), Bayesian Data Analysis, London: Chapman and Hall.

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Gelman, A., Carlin, B., Stern, H. S., and Rubin, D., Bayesian Data Analysis, London: Chapman and Hall, 1995.

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Gelman, A. and Carlin, J. B. and Stern, H. S. and Rubin, D. B. (1995). Bayesian Data Analysis. London: Chapman and Hall.

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