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58
Honest Exploration of Intractable Probability Distributions Via Markov Chain Monte Carlo
 STATISTICAL SCIENCE
, 2001
"... Two important questions that must be answered whenever a Markov chain Monte Carlo (MCMC) algorithm is used are (Q1) What is an appropriate burnin? and (Q2) How long should the sampling continue after burnin? Developing rigorous answers to these questions presently requires a detailed study of the ..."
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Cited by 74 (19 self)
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Two important questions that must be answered whenever a Markov chain Monte Carlo (MCMC) algorithm is used are (Q1) What is an appropriate burnin? and (Q2) How long should the sampling continue after burnin? Developing rigorous answers to these questions presently requires a detailed study of the convergence properties of the underlying Markov chain. Consequently, in most practical applications of MCMC, exact answers to (Q1) and (Q2) are not sought. The goal of this paper is to demystify the analysis that leads to honest answers to (Q1) and (Q2). The authors hope that this article will serve as a bridge between those developing Markov chain theory and practitioners using MCMC to solve practical problems. The ability to formally address (Q1) and (Q2) comes from establishing a drift condition and an associated minorization condition, which together imply that the underlying Markov chain is geometrically ergodic. In this paper, we explain exactly what drift and minorization are as well as how and why these conditions can be used to form rigorous answers to (Q1) and (Q2). The basic ideas are as follows. The results of Rosenthal (1995) and Roberts and Tweedie (1999) allow one to use drift and minorization conditions to construct a formula giving an analytic upper bound on the distance to stationarity. A rigorous answer to (Q1) can be calculated using this formula. The desired characteristics of the target distribution are typically estimated using ergodic averages. Geometric ergodicity of the underlying Markov chain implies that there are central limit theorems available for ergodic averages (Chan and Geyer 1994). The regenerative simulation technique (Mykland, Tierney and Yu 1995, Robert 1995) can be used to get a consistent estimate of the variance of the asymptotic nor...
Inference in Generalized Additive Mixed Models Using Smoothing Splines
, 1999
"... this paper, we propose generalized additive mixed models (GAMMs), which are an additive extension of generalized linear mixed models in the spirit of Hastie and Tibshirani (1990). This new class of models uses additive nonparametric functions to model covariate effects while accounting for overdispe ..."
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Cited by 45 (4 self)
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this paper, we propose generalized additive mixed models (GAMMs), which are an additive extension of generalized linear mixed models in the spirit of Hastie and Tibshirani (1990). This new class of models uses additive nonparametric functions to model covariate effects while accounting for overdispersion and correlation by adding random effects to the additive predictor. GAMMs encompass nested and crossed designs and are applicable to clustered, hierarchical and spatial data. We estimate the nonparametric functions using smoothing splines, and jointly estimate the smoothing parameters and the variance components using marginal quasilikelihood. This marginal quasilikelihood approach is an extension of the restricted maximum likelihood approach used by Wahba (1985) and Kohn, et al. (1991) in the classical nonparametric regression model (Kohn, et al. 1991, eq 2.1), and by Zhang, et al. (1998) in Gaussian nonparametric mixed models, where they treated the smoothing parameter as an extra variance component. In view of numerical integration often required by maximizing the objective functions, double penalized quasilikelihood (DPQL) is proposed to make approximate inference. Frequentist and Bayesian inferences are compared. A key feature of the proposed method is that it allows us to make systematic inference on all model components of GAMMs within a unified parametric mixed model framework. Specifically, our estimation of the nonparametric functions, the smoothing parameters and the variance components in GAMMs can proceed by fitting a working GLMM using existing statistical software, which iteratively fits a linear mixed model to a modified dependent variable. When the data are sparse (e.g., binary), the DPQL estimators of the variance components are found to be subject t...
Geoadditive Models
, 2000
"... this paper is a recent article on modelbased geostatistics by Diggle, Tawn and Moyeed (1998) where pure kriging (i.e. no covariates) is the focus. Our paper inherits some of its aspects: modelbased and with mixed model connections. In particular the comment by Bowman (1998) in the ensuing discussi ..."
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Cited by 33 (1 self)
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this paper is a recent article on modelbased geostatistics by Diggle, Tawn and Moyeed (1998) where pure kriging (i.e. no covariates) is the focus. Our paper inherits some of its aspects: modelbased and with mixed model connections. In particular the comment by Bowman (1998) in the ensuing discussion suggested that additive modelling would be a worthwhile extension. This paper essentially follows this suggestion. However, this paper is not the first to combine the notions of geostatistics and additive modelling. References known to us are Kelsall and Diggle (1998), Durban Reguera (1998) and Durban, Hackett, Currie and Newton (2000). Nevertheless, we believe that our approach has a number of attractive features (see (1)(4) above), not all shared by these references. Section 2 describes the motivating application and data in detail. Section 3 shows how one can express additive models as a mixed model, while Section 4 does the same for kriging and merges the two into the geoadditive model. Issues concerning the amount of smoothing are discussed in Section 5 and inferential aspects are treated in Section 6. Our analysis of the Upper Cape Cod reproductive data is presented in Section 7. Section 8 discusses extension to the generalised context.We close the paper with some disussion in Section 9. 2 Description of the application and data
Bayesian inference for generalized linear mixed models of portfolio credit risk
 Journal of Empirical Finance
, 2007
"... The aims of this paper are threefold. First we highlight the usefulness of generalized linear mixed models (GLMMs) in the modelling of portfolio credit default risk. The GLMMsetting allows for a flexible specification of the systematic portfolio risk in terms of observed fixed effects and unobserve ..."
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Cited by 26 (2 self)
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The aims of this paper are threefold. First we highlight the usefulness of generalized linear mixed models (GLMMs) in the modelling of portfolio credit default risk. The GLMMsetting allows for a flexible specification of the systematic portfolio risk in terms of observed fixed effects and unobserved random effects, in order to explain the phenomena of default dependence and timeinhomogeneity in empirical default data. Second we show that computational Bayesian techniques such as the Gibbs sampler can be successfully applied to fit models with serially correlated random effects, which are special instances of state space models. Third we provide an empirical study using Standard & Poor’s data on US firms. A model incorporating rating category and sector effects and a macroeconomic proxy variable for stateoftheeconomy suggests the presence of a residual, cyclical, latent component in the systematic risk.
p_2: A Random Effects Model with Covariates for Directed Graphs
, 2004
"... A random effects model is proposed for the analysis of binary dyadic data that represent a social network or directed graph, using nodal and/or dyadic attributes as covariates. The network structure is reflected by modeling the dependence between the relations to and from the same actor or node. Par ..."
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Cited by 15 (2 self)
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A random effects model is proposed for the analysis of binary dyadic data that represent a social network or directed graph, using nodal and/or dyadic attributes as covariates. The network structure is reflected by modeling the dependence between the relations to and from the same actor or node. Parameter estimates are proposed that are based on an iterated generalized least squares procedure. An application is presented to a data set on friendship relations between American lawyers.
Trends in Job Instability and Wages for Young Adult Men
"... Data and measurement problems have complicated the debate over trends in job instability in the United States. In this paper we compare two cohorts of young white men from the National Longitudinal Surveys (NLS), construct a rigorous measure of job change, and confirm earlier findings of a significa ..."
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Cited by 12 (1 self)
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Data and measurement problems have complicated the debate over trends in job instability in the United States. In this paper we compare two cohorts of young white men from the National Longitudinal Surveys (NLS), construct a rigorous measure of job change, and confirm earlier findings of a significant increase in job instability in recent years. Further validation of this increase is found when we benchmark the NLS against the other main datasets in the field and conduct a thorough attrition analysis. Extending the analysis to wages, we find that the wage returns to job changing have both declined and become more unequal for young men, mirroring trends in their longterm wage growth. 1. Introduction * While the perception of increased job instability is widespread, empirical documentation of this `fact' remains elusive. Data and measurement problems have led to a trail of conflicting findings, and the absence of clear evidence of rising instability has led some to question whether t...
A survey of Monte Carlo algorithms for maximizing the likelihood of a twostage hierarchical model
, 2001
"... Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternati ..."
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Cited by 10 (4 self)
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Likelihood inference with hierarchical models is often complicated by the fact that the likelihood function involves intractable integrals. Numerical integration (e.g. quadrature) is an option if the dimension of the integral is low but quickly becomes unreliable as the dimension grows. An alternative approach is to approximate the intractable integrals using Monte Carlo averages. Several dierent algorithms based on this idea have been proposed. In this paper we discuss the relative merits of simulated maximum likelihood, Monte Carlo EM, Monte Carlo NewtonRaphson and stochastic approximation. Key words and phrases : Eciency, Monte Carlo EM, Monte Carlo NewtonRaphson, Rate of convergence, Simulated maximum likelihood, Stochastic approximation All three authors partially supported by NSF Grant DMS0072827. 1 1
Geometric ergodicity of MetropolisHastings algorithms for conditional simulation in generalised linear mixed models
, 2001
"... Conditional simulation is useful in connection with inference and prediction for a generalised linear mixed model. We consider random walk Metropolis and LangevinHastings algorithms for simulating the random eects given the observed data, when the joint distribution of the unobserved random eects is ..."
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Cited by 9 (0 self)
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Conditional simulation is useful in connection with inference and prediction for a generalised linear mixed model. We consider random walk Metropolis and LangevinHastings algorithms for simulating the random eects given the observed data, when the joint distribution of the unobserved random eects is multivariate Gaussian. In particular we study the desirable property of geometric ergodicity, which ensures the validity of central limit theorems for Monte Carlo estimates. Keywords: conditional simulation; generalised linear mixed model; geometric ergodicity; LangevinHastings algorithm; Markov chain Monte Carlo; random walk Metropolis algorithm. 1
Analysis of spatial data using generalized linear mixed models and Langevintype Markov chain Monte Carlo
, 2000
"... Markov chain Monte Carlo methods are useful in connection with inference and prediction for spatial generalized linear mixed models, where the unobserved random effects constitute a spatially correlated Gaussian random field. We point out that socalled Langevintype updates are useful for Metropoli ..."
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Cited by 8 (3 self)
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Markov chain Monte Carlo methods are useful in connection with inference and prediction for spatial generalized linear mixed models, where the unobserved random effects constitute a spatially correlated Gaussian random field. We point out that socalled Langevintype updates are useful for MetropolisHastings simulation of the posterior distribution of the random eects given the data. Furthermore, we discuss the use of improper priors in Bayesian analysis of spatial generalized linear mixed models with particular emphasis on the socalled Poissonlog normal model. For this and certain other models nonparametric estimation of the covariance function of the Gaussian field is also studied. The methods are applied to various data sets including counts of weed plants on a field.
Bayesian covariance selection in generalized linear mixed models
 Biometrics
, 2006
"... SUMMARY. The generalized linear mixed model (GLMM), which extends the generalized linear model (GLM) to incorporate random effects characterizing heterogeneity among subjects, is widely used in analyzing correlated and longitudinal data. Although there is often interest in identifying the subset of ..."
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Cited by 8 (3 self)
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SUMMARY. The generalized linear mixed model (GLMM), which extends the generalized linear model (GLM) to incorporate random effects characterizing heterogeneity among subjects, is widely used in analyzing correlated and longitudinal data. Although there is often interest in identifying the subset of predictors that have random effects, random effects selection can be challenging, particularly when outcome distributions are nonnormal. This article proposes a fully Bayesian approach to the problem of simultaneous selection of fixed and random effects in GLMMs. Integrating out the random effects induces a covariance structure on the multivariate outcome data, and an important problem which we also consider is that of covariance selection. Our approach relies on variable selectiontype mixture priors for the components in a special LDU decomposition of the random effects covariance. A stochastic search MCMC algorithm is developed, which relies on Gibbs sampling, with Taylor series expansions used to approximate intractable integrals. Simulated data examples are presented for different exponential family distributions, and the approach is applied to discrete survival data from a timetopregnancy study.