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363
Bayesian density estimation and inference using mixtures.
 J. Amer. Statist. Assoc.
, 1995
"... JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about J ..."
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Cited by 653 (18 self)
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JSTOR is a notforprofit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. We describe and illustrate Bayesian inference in models for density estimation using mixtures of Dirichlet processes. These models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions. Efficient simulation methods are used to approximate various prior, posterior, and predictive distributions. This allows for direct inference on a variety of practical issues, including problems of local versus global smoothing, uncertainty about density estimates, assessment of modality, and the inference on the numbers of components. Also, convergence results are established for a general class of normal mixture models. American Statistical Association
An Adaptive Metropolis algorithm
 Bernoulli
, 1998
"... A proper choice of a proposal distribution for MCMC methods, e.g. for the MetropolisHastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis Algorithm (AM), where the Gaussian proposal distribution is updated al ..."
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Cited by 216 (8 self)
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A proper choice of a proposal distribution for MCMC methods, e.g. for the MetropolisHastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis Algorithm (AM), where the Gaussian proposal distribution is updated along the process using the full information cumulated so far. Due to the adaptive nature of the process, the AM algorithm is nonMarkovian, but we establish here that it has the correct ergodic properties. We also include the results of our numerical tests, which indicate that the AM algorithm competes well with traditional MetropolisHastings algorithms, and demonstrate that AM provides an easy to use algorithm for practical computation. 1991 Mathematics Subject Classification: 65C05, 65U05. Keywords: adaptive MCMC, comparison, convergence, ergodicity, Markov Chain Monte Carlo, MetropolisHastings algorithm 1 Introduction It is generally acknowledged that the choice of an effective proposal...
Rates of convergence of the Hastings and Metropolis algorithms
 ANNALS OF STATISTICS
, 1996
"... We apply recent results in Markov chain theory to Hastings and Metropolis algorithms with either independent or symmetric candidate distributions, and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution ß. In the independence ca ..."
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Cited by 202 (17 self)
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We apply recent results in Markov chain theory to Hastings and Metropolis algorithms with either independent or symmetric candidate distributions, and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution ß. In the independence case (in IR k ) these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of ß; in the symmetric case (in IR only) we show geometric convergence essentially occurs if and only if ß has geometric tails. We also evaluate recently developed computable bounds on the rates of convergence in this context: examples show that these theoretical bounds can be inherently extremely conservative, although when the chain is stochastically monotone the bounds may well be effective.
General state space Markov chains and MCMC algorithm
 PROBABILITY SURVEYS
, 2004
"... This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform e ..."
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Cited by 177 (35 self)
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This paper surveys various results about Markov chains on general (noncountable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equation or direct regeneration constructions. Finally, optimal scaling and weak convergence results for MetropolisHastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.
An exact likelihood analysis of the multinomial probit model
, 1994
"... We develop new methods for conducting a finite sample, likelihoodbased analysis of the multinomial probit model. Using a variant of the Gibbs sampler, an algorithm is developed to draw from the exact posterior of the multinomial probit model with correlated errors. This approach avoids direct evalu ..."
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Cited by 164 (6 self)
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We develop new methods for conducting a finite sample, likelihoodbased analysis of the multinomial probit model. Using a variant of the Gibbs sampler, an algorithm is developed to draw from the exact posterior of the multinomial probit model with correlated errors. This approach avoids direct evaluation of the likelihood and, thus, avoids the problems associated with calculating choice probabilities which affect both the standard likelihood and method of simulated moments approaches. Both simulated and actual consumer panel data are used to fit sixdimensional choice models. We also develop methods for analyzing random coefficient and multiperiod probit models.
On choosing and bounding probability metrics
 INTERNAT. STATIST. REV.
, 2002
"... When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a mea ..."
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Cited by 153 (2 self)
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When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric.
Markov Chain Monte Carlo Simulation Methods in Econometrics
, 1993
"... We present several Markov chain Monte Carlo simulation methods that have been widely used in recent years in econometrics and statistics. Among these is the Gibbs sampler, which has been of particular interest to econometricians. Although the paper summarizes some of the relevant theoretical literat ..."
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Cited by 152 (9 self)
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We present several Markov chain Monte Carlo simulation methods that have been widely used in recent years in econometrics and statistics. Among these is the Gibbs sampler, which has been of particular interest to econometricians. Although the paper summarizes some of the relevant theoretical literature, its emphasis is on the presentation and explanation of applications to important models that are studied in econometrics. We include a discussion of some implementation issues, the use of the methods in connection with the EM algorithm, and how the methods can be helpful in model specification questions. Many of the applications of these methods are of particular interest to Bayesians, but we also point out ways in which frequentist statisticians may find the techniques useful.
Stability of Markovian processes III: FosterLyapunov criteria for continuoustime processes
 Adv. Appl. Prob
, 1993
"... In Part I we developed stability concepts for discrete chains, together with FosterLyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuoustime processes. In this paper we develop criteria for these forms of stability for continuousparameter ..."
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Cited by 143 (24 self)
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In Part I we developed stability concepts for discrete chains, together with FosterLyapunov criteria for them to hold. Part II was devoted to developing related stability concepts for continuoustime processes. In this paper we develop criteria for these forms of stability for continuousparameter Markovian processes on general state spaces, based on FosterLyapunov inequalities for the extended generator. Such test function criteria are found for nonexplosivity, nonevanescence, Harris recurrence, and positive Harris recurrence. These results are proved by systematic application of Dynkin's formula. We also strengthen known ergodic theorems, and especially exponential ergodic results, for continuoustime processes. In particular we are able to show that the test function approach provides a criterion for fnorm convergence, and bounding constants for such convergence in the exponential ergodic case. We apply the criteria to several specific processes, including linear stochastic systems under nonlinear feedback, workmodulated queues, general release storage processes and risk processes.
Bayesian Treatment of the Independent Studentt Linear Model
 JOURNAL OF APPLIED ECONOMETRICS
, 1993
"... This article takes up methods for Bayesian inference in a linear model in which the disturbances are independent and have identical Studentt distributions. It exploits the equivalence of the Studentt distribution and an appropriate scale mixture of normals, and uses a Gibbs sampler to perform the ..."
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Cited by 128 (2 self)
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This article takes up methods for Bayesian inference in a linear model in which the disturbances are independent and have identical Studentt distributions. It exploits the equivalence of the Studentt distribution and an appropriate scale mixture of normals, and uses a Gibbs sampler to perform the computations. The new method is applied to some wellknown macroeconomic time series. It is found that posterior odds ratios favor the independent Studentt linear model over the normal linear model, and that the posterior odds ratio in favor of difference stationarity over trend stationarity is often substantially less in the favored Studentt models.