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27
Assessment and Propagation of Model Uncertainty
, 1995
"... this paper I discuss a Bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible, by virtue of recent computational advances, and examine its implementation in examples that involve forecasting the price of oil and estimating the ..."
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Cited by 108 (0 self)
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this paper I discuss a Bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible, by virtue of recent computational advances, and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the U.S. Space Shuttle.
The Consistency of Posterior Distributions in Nonparametric Problems
 Ann. Statist
, 1996
"... We give conditions that guarantee that the posterior probability of every Hellinger... ..."
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Cited by 79 (4 self)
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We give conditions that guarantee that the posterior probability of every Hellinger...
Rates of Convergence of Posterior Distributions
, 1998
"... We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include infinite dimensional cases. The rates are driven by two quantities: the size of the space, as measure by metric entropy or bracketing entropy, and ..."
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Cited by 47 (0 self)
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We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include infinite dimensional cases. The rates are driven by two quantities: the size of the space, as measure by metric entropy or bracketing entropy, and the degree to which the prior concentrates in a small ball around the true parameter. We apply the results to several examples. In some cases, natural priors give suboptimal rates of convergence and better rates can be obtained by using sievebased priors such as those introduced by Zhao (1993, 1998). AMS 1990 classification: Primary, 62A15, Secondary: 62E20, 62G15. KEYWORDS: Bayesian inference, asymptotic inference, nonparametric models, sieves. 1 Introduction. Nonparametric Bayesian methods have become quite popular lately, largely because of advances in computing; see Dey, Mueller and Sinha (1998) for a recent account. Because of their growing popularity, it is important to understand ...
Convergence rates of posterior distributions
 Ann. Statist
, 2000
"... We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, D ..."
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Cited by 43 (11 self)
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We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, Dirichlet processes and interval censoring. 1. Introduction. Suppose
Consistency issues in Bayesian Nonparametrics
 IN ASYMPTOTICS, NONPARAMETRICS AND TIME SERIES: A TRIBUTE
, 1998
"... ..."
Consistent semiparametric Bayesian inference about a location parameter
, 1995
"... We consider the problem of Bayesian inference about the centre of symmetry of a symmetric density on the real line based on independent identically distributed observations. A result of Diaconis and Freedman shows that the posterior distribution of the location parameter may be inconsistent if (symm ..."
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Cited by 11 (5 self)
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We consider the problem of Bayesian inference about the centre of symmetry of a symmetric density on the real line based on independent identically distributed observations. A result of Diaconis and Freedman shows that the posterior distribution of the location parameter may be inconsistent if (symmetrized) Dirichlet process prior is used for the unknown distribution function. We choose a symmetrized Polya tree prior for the unknown density and independently choose ` according to a continuous and positive prior density on the real line. Suppose that the parameters of Polya tree depend only on the level m of the tree and the common values r m 's are such that P 1 m=1 r \Gamma1=2 m ! 1. Then it is shown that for a large class of true symmetric densities, including the trimodal distribution of Diaconis and Freedman, the marginal posterior of ` is consistent. AMS subject classification: Primary 62G20, 62F15. Key words: Consistency, KullbackLeibler number, location parameter, Polya ...
On optimality of Bayesian wavelet estimators
, 2004
"... We investigate the asymptotic optimality of several Bayesian wavelet estimators corresponding to dierent losses, namely, posterior mean, posterior median and Bayes Factor. The considered prior is a mixture of a mass function at zero and a Gaussian density. We show that in terms of the mean squared e ..."
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Cited by 11 (0 self)
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We investigate the asymptotic optimality of several Bayesian wavelet estimators corresponding to dierent losses, namely, posterior mean, posterior median and Bayes Factor. The considered prior is a mixture of a mass function at zero and a Gaussian density. We show that in terms of the mean squared error, for the properly chosen hyperparameters of the prior all the three resulting Bayesian wavelet estimators achieve optimal minimax rates within any prescribed Besov p;q for p 2. For 1 p < 2, the Bayes Factor is still optimal for (2s+2)=(2s+1) p < 2 and always outperforms the posterior mean and the posterior median that can achieve only the best possible rates for linear estimators in this case. Key words: Bayes Factor, Bayes model; Besov spaces; minimax estimation; nonlinear estimation; nonparametric regression; posterior mean; posterior median; wavelets. 1
On the uniform consistency of Bayes estimates for multinomial probabilities
, 1988
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 9 (1 self)
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Consistency of Bayes estimates for nonparametric regression: normal theory
 Bernoulli
, 1998
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 9 (1 self)
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Misspecification in infinitedimensional Bayesian statistics
 Annals of Statistics
, 2006
"... We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution P0, which may not be in the support of the prior, we show that the posterior concentrates its mass near the points in the support of the pri ..."
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Cited by 7 (0 self)
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We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution P0, which may not be in the support of the prior, we show that the posterior concentrates its mass near the points in the support of the prior that minimize the Kullback–Leibler divergence with respect to P0. An entropy condition and a priormass condition determine the rate of convergence. The method is applied to several examples, with special interest for infinitedimensional models. These include Gaussian mixtures, nonparametric regression and parametric models.