Rates of convergence of posterior distributions

By Xiaotong Shen, Larry A. Wasserman. 2001

338 citations recorded by the source corpus.

Subfields: Artificial Intelligence, Numerical Analysis, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/207038a4f52c8113c4ecdf7fdc20342ea9bac0a5

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Citations in the corpus

Bayesian inverse problems with non-conjugate priors (2012, background): “[11, 13, 16, 29, 31] for some examples), but the situation of inverse problems has only recently been considered and then only in the conjugate setting [1, 20, 21], where explicit posterior expressions are available.”
Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, background): “Although remarkable progress has been made in the understanding of the frequentist properties of nonparametric Bayes procedures (we refer here only to some of the key papers such as [18, 19, 34, 36, 38] and references therein), a fully satisfactory answer to the BvM-question seems not to have been f”
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”
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).”
Dynamics of Bayesian Updating with Dependent Data and Misspecified Models (2009, background): “These contributions also make (d) certain dynamical assumptions about the data-generating process, most often that it is IID [4, 26, 65] (in this setting, [27] and [59] in particular consider convergence rates), independent non-identically distributed [13, 29], or, in some cases, Markovian [28, 29];”
Learning Causal Bayesian Network Structures From Experimental Data (2008, background): “It is known that likelihood and Bayesian methods do not perform well if the parameter space has very large cardinality, unless sieve or penalty functions are used to reduce the effective cardinality (Birge and Massart 1993; Wong and Shen 1995; Shen and Wasserman 2001).”
Bayesian inference with rescaled Gaussian process priors (2007, background): “It is well known that in the frequentist setup in which the data are sampled from a ﬁxed true distribution, a prior on an inﬁnite-dimensional model has to be carefully chosen in order to obtain optimal rates of contraction of the posterior (cf. e.g. [3], [17], [19], [4], [25], [23]).”
Bayesian Goodness of Fit Testing with Mixtures of Triangular Distributions (2009, methodology): “From A3 we may apply lemma 1 of Shen & Wasserman (2001) to give that, when is small enough, Pn∗ {∫ n∏ i =1 p ( yi) p∗( yi) 0(d ) < n−d/2 } ≤ C (− log ) .”
Adaptive Bayesian Procedures Using Random Series Priors (2014, methodology): “06 25 v1 [ m at h. ST ] 3 M ar 2 01 We consider a prior for nonparametric Bayesian estimation which uses finite random series with a random number of terms.”
Dirichlet Process Mixtures of Generalized Linear Models (2009, methodology): “By decreasing the radius of the balls used to show consistency, consistency results are used to give asymptotic rates of posterior convergence in the models where L1 or Hellinger consistency has been shown (Ghosal et al., 2000; Shen and Wasserman, 2001; Walker et al., 2007; Ghosal and van der Vaart,”
Lower bounds for posterior rates with Gaussian process priors (2008, background): “[1]) and posterior rates of convergence (see for instance [5; 13]) are now available.”
The semiparametric Bernstein-von Mises theorem (2010, background): “However, general conditions for Bayesian consistency were formulated by Schwartz as early as 1965 [37]; positive results on posterior rates of convergence in the same spirit were obtained in Ghosal, Ghosh and van der Vaart [16] (see also, Shen and Wasserman [40]).”
Posterior concentration rates for infinite dimensional exponential families (2012, methodology): “The prior model proposed by Shen and Wasserman is a special case of the prior described in Section 2.”
Thomas Bayes’ walk on manifolds (2012, neutral): “Ghosh [12], [27].”
Remarks on consistency of posterior distributions (2008, methodology): “Regarding either posterior consistency or rate of convergence of posterior distributions, this model has been investigated by Shen and Wasserman [26], Walker [33] and Choi and Schervish [8].”