A Bernstein–von Mises theorem for smooth functionals in semiparametric models

By I. Castillo, Judith Rousseau. 2013

123 citations recorded by the source corpus.

A Bernstein–von Mises theorem is derived for general semiparametric functionals. The result is applied to a variety of semiparametric problems in i.i.d. and non-i.i.d. situations. In particular, new tools are developed to handle semiparametric bias, in particular for nonlinear functionals and in cases where regularity is possibly low. Examples include the squared L2-norm in Gaussian white noise, nonlinear functionals in density estimation, as well as functionals in autoregressive models. For density estimation, a systematic study of BvM results for two important classes of priors is provided, namely random histograms and Gaussian process priors.

Subfields: Artificial Intelligence, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Kolyan Ray
2. Richard Nickl
3. Botond Szab'o
4. Maxim Panov
5. Vladimir Spokoiny
6. Judith Rousseau
7. F. Monard
8. Gabriel P. Paternain

In Statistics and Probability

1. Kolyan Ray
2. Richard Nickl
3. Judith Rousseau
4. Botond Szab'o
5. Maxim Panov
6. Vladimir Spokoiny
7. Gabriel P. Paternain
8. F. Monard

Citations in the corpus

• Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems (2013, background): “In Rivoirard and Rousseau (2012) semiparametric BvM theorem is derived for linear functionals of density and in forthcoming work Castillo and Rousseau (2013) the result is generalized to a broad class of models and functionals.”
• Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator (2015, methodology): “In this section, we parameterize F using piecewise constant functions, as in [6] for instance.”
• Bernstein - von Mises Theorem for growing parameter dimension (2013, background): “r high dimensional linear models; Boucheron and Gassiat (2009), Kim (2006) for some special non-Gaussian models; Shen (2002), Bickel and Kleijn (2012), Rivoirard and Rousseau (2012), Castillo (2012), Castillo and Rousseau (2013) for a semiparametric version of the BvM result for different models; Kle”
• Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, methodology): “We follow ideas from [7, 9, 41, 43] and prove a Bernstein-von Mises theorem by proving convergence of the moment generating functions (Laplace transforms) of √ N ( 〈θ,ψ〉L2(Z)|DN − Ψ̂N )”
• Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, methodology): “j ∑ λ≤κ,k aλ,k,jΦλ,k,j, κ fixed, and thus by Proposition 29 in [32] (or by the results in the supplement of [13]) we deduce joint weak convergence of the finite-dimensional distributions, in particular, for every fixed κ ∈ N, βV ⊗d κ (Π̄ D̄T κ ,Nb0,κ) →b0 0, as T → ∞.”
• Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, background): “Bickel and Kleijn [3], Castillo [8], Rivoirard and Rousseau [41] and Castillo and Rousseau [13] provide sufficient conditions for BvMs for semiparametric functionals.”