The consistency of posterior distributions in nonparametric problems

By Andrew R. Barron, Mark J. Schervish, Larry A. Wasserman. 1999

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Citations in the corpus

Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems (2013, background): “ncreases. Already consistency of fully acknowledged. panov, m.and spokoiny, v. 3 the posterior distribution in nonparametric and semiparametric models is a nontrivial problem; cf. Schwartz (1965) and Barron et al. (1996). Asymptotic normality of the posterior measure for these classes of models is e”
Posterior concentration rates for infinite dimensional exponential families (2012, methodology): “In this paper we derive adaptive non-parametric rates of concentration of the posterior distributions for the density model on the class of Sobolev and Besov spaces.”
Large sample asymptotics for the two-parameter Poisson–Dirichlet process (2007, background): “For some more results on the modern treatment of Bayesian consistency in nonparametric settings one may note, for instance, the works of Ghosal, Ghosh and Ramamoorthi(10), Barron, Schervish and Wasserman(2) and Ghosal, Ghosh and Van der Vaart (11); and the book of Ghosh and Ramamoorthi(12).”
Nonlinear and Nonparametric Regression and Instrumental Variables (2004, result): “If the number of knots increases to infinity, then we conjecture that P-splines are consistent by results of Barron, Schervish, and Wasserman (1999) , but we have been unable to verify that their assumptions hold.”
On the Frequentist Properties of Bayesian Nonparametric Methods (2016, background): “Consistency has been established in a variety of models, as in density estimation for i.i.d data, see (Schwartz, 1965, Barron et al., 1999, Lijoi et al., 2005) for generic results, regression function estimation (Choi and Schervish, 2007, Ghosal and Roy, 2006), or in models with non independent…”
New approaches to Bayesian consistency (2004, methodology): “Previous studies of Hellinger consistency [see, e.g., Barron, Schervish and Wasserman (1999) and Ghosal, Ghosh and Ramamoorthi (1999)] deal with the numerator and denominator in the expression for Π n ( A ) separately.”
On Bayesian supremum norm contraction rates (2013, background): “Two natural families of nonparametric priors are considered for density estimation: priors on log-densities, see for convergence properties e.g. Ghosal, Ghosh and van der Vaart [12], Scricciolo [28], Tokdar and Ghosh [34], van der Vaart and van Zanten [37], [3], Rivoirard and Rousseau [26], and rand”
Bayesian Nonparametric Causal Inference: Information Rates and Learning Algorithms (2017, methodology): “in [22], which studied the frequentist consistency of posterior distributions in the Hellinger metric, and the seminal work in [29], which proposed information-theoretic approaches for deriving the posterior contraction rates of Bayesian methods for nonparametric distribution estimation and regressi”
Convergence rates of posterior distributions for non-i.i.d. observations (2007, background): “The complementary assertion P (n) θ0 n( \ n|X(n)) → 0 may be handled either by a direct argument or by the following analog of Lemma 5 of [2].”
Asymptotically minimax empirical Bayes estimation of a sparse normal mean vector (2013, background): “Our ﬁrst result gives a bound on the denominator of Q n , like that which obtains from the familiar Kullback–Leibler property (e.g., Barron et al. 1999; Ghosal et al. 1999, 2000; Schwartz 1965; Shen and Wasserman 2001).”
Dirichlet Process Mixtures of Generalized Linear Models (2009, background): “Most current literature centers around consistency of the posterior density for DP Gaussian mixture models (Barron et al. 1999, Ghosal et al. 1999, Ghosh and Ramamoorthi 2003, Walker 2004, Tokdar 2006) and semi-parametric linear regression models (Amewou-Atisso et al. 2003, Tokdar 2006).”
Remarks on consistency of posterior distributions (2008, background): “8, the next (BSW) is due to [3] and the third (GGR) appears in [15].”
Rates of contraction for posterior distributions in Lr-metrics, 1 ≤ r ≤ ∞ (2011, methodology): “Consistency in stronger metrics was considered by Barron, Schervish and Wasserman [1] and by Ghosal, Ghosh and Ramamoorthi [9], and, shortly after, Ghosal, Ghosh and van der Vaart [10] and Shen and Wasserman [30] developed techniques that allow us to prove frequentist rates of contraction of the pos”
From ɛ-entropy to KL-entropy: Analysis of minimum information complexity density estimation (2006, methodology): “The consistency theorem given in (Barron et al., 1999) also relies on the upper covering number N ub (Γ , (cid:15) ).”
CONSISTENCY OF A RECURSIVE ESTIMATE OF MIXING DISTRIBUTIONS (2009, background): “With n = 50,000, L1-consistency of pNPB [2, 11] and p̄n has kicked in, and that of NPB and PARE follows by Remark 6.”