Nonparametric Bernstein–von Mises theorems in Gaussian white noise

By I. Castillo, Richard Nickl. 2012

155 citations recorded by the source corpus.

Bernstein-von Mises theorems for nonparametric Bayes priors in the Gaussian white noise model are proved. It is demonstrated how such results justify Bayes methods as efficient frequentist inference procedures in a variety of concrete nonparametric problems. Particularly Bayesian credible sets are constructed that have asymptotically exact $1-\alpha$ frequentist coverage level and whose $L^2$-diameter shrinks at the minimax rate of convergence (within logarithmic factors) over H\"{o}lder balls. Other applications include general classes of linear and nonlinear functionals and credible bands for auto-convolutions. The assumptions cover nonconjugate product priors defined on general orthonormal bases of $L^2$ satisfying weak conditions.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/3de7c6d946d6deda986b0968f189575927f55f35

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Citations in the corpus

Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions (2015, unknown): “[7, 8] for the diffusion model considered here.”
Bernstein - von Mises Theorem for growing parameter dimension (2013, unknown): “dels with inﬁnite or growing parameter dimension p exist for some special situations, see e.g. Freedman (1999) and Ghosal (1999, 2000) for linear models or Bontemps (2011) for Gaussian regression, or Castillo and Nickl (2013) for the functional Gaussian case. Though the main arguments behind BvM res”
On Bayesian supremum norm contraction rates (2013, methodology): “Abstract: Building on ideas from Castillo and Nickl [6], a method is provided to study nonparametric Bayesian posterior convergence rates when ‘strong’ measures of distances, such as the sup-norm, are considered.”
Large ball probabilities, Gaussian comparison and anti-concentration (2017, background): “ompare two probabilities IP a+ξ ≤ r vs IP ξ G ≤ r for all r≥ 0. Existing results cover only very special cases; see e.g. Johnstone (2010); Bontemps (2011); Panov and Spokoiny (2015); Castillo (2012); Castillo and Nickl (2013); Belitser (2017) and references therein. Most of the mentioned results are”
On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures (2013, methodology): “Using also Theorem 1 in the sampling model case, all arguments from the proof of Theorem 1 in [4] go through.”
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “In more complex settings with infinite-dimensional parameter spaces and inverse problems, such results are more recent and the present article contributes to the programme developed in [6, 7, 9, 53, 41, 39, 43, 22, 42, 8, 10].”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, methodology): “See Giné and Nickl [21] and also Bull [6] for further discussion about this condition.”
Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, background): “For the Gaussian white noise model, [10] showed that a Bernstein von Mises (BvM) theorem holds in weaker topologies for some natural priors, which yields the correct asymptotic coverage of credible sets based on the weaker topology; however their result is not applicable for understanding the…”
Critical dimension in the semiparametric Bernstein—von Mises theorem (2014, background): “Some results for particular semi- and nonparametric problems are available in [18, 17, 22, 8].”
On the Frequentist Properties of Bayesian Nonparametric Methods (2016, methodology): “In (Castillo and Nickl, 2013), conditions for deriving a weak nonparametric Bernstein von Mises theorem are derived in the white noise and density models, which leads to the construction of credible bands with correct asymptotic frequentist coverage.”
Bayesian inference in a class of partially identified models (2016, background): “Just to give one recent example, Castillo and Nickl (2013) prove a non-parametric version of the Bernstein-von Mises theorem that could replace assumption 3.”
Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, methodology): “4 Bernstein-von Mises theorems for b We now adopt the framework of nonparametric Bernstein-von Mises theorems from [11, 12], see also the recent contributions [10, 41, 32, 33, 31].”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, background): “Finally, although there is no immediate analogue of the BvM property for infinite dimensional parameters, as pointed out by Cox [16] and Freedman [19], some recent contributions have introduced possible notions of nonparametric BvM; see Castillo and Nickl [13] and also Leahu [29].”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, background): “Castillo and Nickl [3] showed that for the Gaussian white noise model a BvM theorem can hold in weaker topologies for some natural priors, and the resulting credible sets appropriately modified will have asymptotically the correct coverage and optimal size.”