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28
Informationtheoretic asymptotics of Bayes methods
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1990
"... In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and sh ..."
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Cited by 140 (12 self)
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In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. We examine the relative entropy distance D,, between the true density and the Bayesian density and show that the asymptotic distance is (d/2Xlogn)+ c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D,,/n converges to zero at rate (logn)/n. The constant c, which we explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stockmarket portfolio selection.
An MCMC approach to classical estimation
, 2003
"... This paper studies computationally and theoretically attractive estimators called here Laplace type estimators (LTEs), which include means and quantiles of quasiposterior distributions dened as transformations of general (nonlikelihoodbased) statistical criterion functions, such as those in GMM, n ..."
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Cited by 97 (13 self)
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This paper studies computationally and theoretically attractive estimators called here Laplace type estimators (LTEs), which include means and quantiles of quasiposterior distributions dened as transformations of general (nonlikelihoodbased) statistical criterion functions, such as those in GMM, nonlinear IV, empirical likelihood, and minimum distance methods. The approach generates an alternative to classical extremum estimation and also falls outside the parametric Bayesian approach. For example, it o ers a new attractive estimation method for such important semiparametric problems as censored and instrumental quantile regression, nonlinear GMM and valueatrisk models. The LTEs are computed using Markov Chain Monte Carlo methods, which help circumvent the computational curse of dimensionality. Alarge sample theory is obtained and illustrated for regular cases.
Asymptotic normality of posterior distributions for exponential families when the number . . .
, 2000
"... We study consistency and asymptotic normality of posterior distributions of the natural parameter for an exponential family when the dimension of the parameter grows with the sample size. Under certain growth restrictions on the dimension, we show that the posterior distributions concentrate in neig ..."
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Cited by 44 (7 self)
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We study consistency and asymptotic normality of posterior distributions of the natural parameter for an exponential family when the dimension of the parameter grows with the sample size. Under certain growth restrictions on the dimension, we show that the posterior distributions concentrate in neighbourhoods of the true parameter and can be approximated by an appropriate normal distribution.
ON THE COMPUTATIONAL COMPLEXITY OF MCMCBASED ESTIMATORS IN LARGE SAMPLES
"... In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasiBayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the LaplaceBernsteinVon Mises central limit theorem, which states that ..."
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Cited by 14 (3 self)
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In this paper we examine the implications of the statistical large sample theory for the computational complexity of Bayesian and quasiBayesian estimation carried out using Metropolis random walks. Our analysis is motivated by the LaplaceBernsteinVon Mises central limit theorem, which states that in large samples the posterior or quasiposterior approaches a normal density. Using this observation, we establish polynomial bounds on the computational complexity of general Metropolis random walks methods in large samples. Our analysis covers cases, where the underlying loglikelihood or extremum criterion function is possibly nonconcave, discontinuous, and of increasing dimension. However, the central limit theorem restricts the deviations from continuity and logconcavity of the loglikelihood or extremum criterion function in a very specific manner. Under minimal assumptions for the central limit theorem framework to hold, we show that the Metropolis algorithm is theoretically
Asymptotic Normality of the Posterior in Relative Entropy
 IEEE Trans. Inform. Theory
, 1999
"... We show that the relative entropy between a posterior density formed from a smooth likelihood and prior and a limiting normal form tends to zero in the independent and identically distributed case. The mode of convergence is in probability and in mean. Applications to codelengths in stochastic compl ..."
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Cited by 8 (0 self)
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We show that the relative entropy between a posterior density formed from a smooth likelihood and prior and a limiting normal form tends to zero in the independent and identically distributed case. The mode of convergence is in probability and in mean. Applications to codelengths in stochastic complexity and to sample size selection are briey discussed. Index Terms: Posterior density, asymptotic normality, relative entropy. Revision submitted to Trans. Inform Theory , 22 May 1998. This research was partially supported by NSERC Operating Grant 554891. The author is with the Department of Statistics, University of British Columbia, Room 333, 6356 Agricultural Road, Vancouver, BC, Canada V6T 1Z2. 1 I.
Asymptotic Behaviour Of Bayes Estimates Under Possibly Incorrect Models
 Annals Statistics
, 1994
"... Introduction The frequentist asymptotic properties of Bayes estimators and of posterior distributions are wellknown and have been investigated in different directions, see e.g. Bickel and Yahav (1969), Ibragimov and Has'minskii (1981), Strasser (1981) or Lehmann (1983). The interesting genera ..."
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Cited by 4 (0 self)
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Introduction The frequentist asymptotic properties of Bayes estimators and of posterior distributions are wellknown and have been investigated in different directions, see e.g. Bickel and Yahav (1969), Ibragimov and Has'minskii (1981), Strasser (1981) or Lehmann (1983). The interesting generalization to a possibly incorrect model has been treated by Berk (1966), who proved under regularity conditions, that a.s. the posterior distribution converges weakly toward the Dirac measure at the pseudotrue parameter, assuming its uniqueness. This is the parameter value corresponding to the distribution in the model, which is nearest to the true distribution in the sense of the information distance. The result is proven in the general case of possibly nonunique pseudotrue parameters for a corresponding generalization of the above mentioned weak convergence. Unfortunately, many standard models with unbounded parameter space are not covered by his theorems (see Remark 1 in our Section 2
Asymptotic normality of the posterior given a statistic
 THE CANADIAN JOURNAL OF STATISTICS
, 2004
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A Bayesian Approach to Financial Model Calibration, Uncertainty Measures and Optimal Hedging
"... Michaelmas 2009This thesis is dedicated to the late ..."
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Cited by 3 (1 self)
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Michaelmas 2009This thesis is dedicated to the late
Posterior inference in curved exponential families under increasing dimensions
, 20079
"... The goal of this work is to study the large sample properties of the posteriorbased inference in the curved exponential family under increasing dimension. The curved structure arises from the imposition of various restrictions, such as moment restrictions, on the model, and plays a fundamental role ..."
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Cited by 3 (2 self)
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The goal of this work is to study the large sample properties of the posteriorbased inference in the curved exponential family under increasing dimension. The curved structure arises from the imposition of various restrictions, such as moment restrictions, on the model, and plays a fundamental role in various branches of data analysis. We establish conditions under which the posterior distribution is approximately normal, which in turn implies various good properties of estimation and inference procedures based on the posterior. In the process we revisit and improve upon previous results for the exponential family under increasing dimension by making use of concentration of measure. We also discuss a variety of applications including the multinomial model with moment restrictions, seemingly unrelated regression equations, and single structural equation models. In our analysis, both the parameter dimension and the number of moments are increasing with the sample size.
Convergence properties of sequential Bayesian Doptimal designs
 JOURNAL OF STATISTICAL PLANNING AND INFERENCE
, 2009
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