The semiparametric Bernstein-von Mises theorem

By Peter J. Bickel, B. Kleijn. 2010

132 citations recorded by the source corpus.

In a smooth semiparametric estimation problem, the marginal posterior for the parameter of interest is expected to be asymptotically normal and satisfy frequentist criteria of optimality if the model is endowed with a suitable prior. It is shown that, under certain straightforward and interpretable conditions, the assertion of Le Cam’s acclaimed, but strictly parametric, Bernstein-von Mises theorem [Univ. California Publ. Statist. 1 (1953) 277-329] holds in the semiparametric situation as well. As a consequence, Bayesian point-estimators achieve efficiency, for example, in the sense of Hajek’s convolution theorem [Z. Wahrsch. Verw. Gebiete 14 (1970) 323-330]. The model is required to satisfy differentiability and metric entropy conditions, while the nuisance prior must assign nonzero mass to certain Kullback-Leibler neighborhoods [Ghosal, Ghosh and van der Vaart Ann. Statist. 28 (2000) 500-531]. In addition, the marginal posterior is required to converge at parametric rate, which appears to be the most stringent condition in examples. The results are applied to estimation of the linear coefficient in partial linear regression, with a Gaussian prior on a smoothness class for the nuisance.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems (2013, methodology): “These values can be upper bounded under usual smoothness conditions on f , e.g. if f belongs to a Sobolev ball with a certain regularity; cf. Bontemps (2011); Bickel and Kleijn (2012); Castillo (2012).”
On the Frequentist Properties of Bayesian Nonparametric Methods (2016, background): “In (Castillo, 2010) and (Bickel and Kleijn, 2012) sufficient conditions are proposed to derive BvM in separated models in the form θ = (ψ, η).”
Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors (2010, methodology): “Castillo (2010) obtains Bernstein–von Mises theorems for Gaussian process priors, in the semiparametric framework where the unknown quantity is (θ, f ), with θ the parameter of interest and f an infinite-dimensional nuisance parameter.”
Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, result): “For the related question of obtaining semiparametric BvM-results, general sufficient conditions are given in Castillo [8] and Bickel and Kleijn [2], as well as in Rivoirard and Rousseau [32] for linear functionals of probability density functions.”
A Tutorial on Fisher Information (2017, methodology): “and Grasman, 2005; Waldorp, 2009; Waldorp, Christoﬀels and van de Ven, 2011; White, 1982), in the reconciliation of frequentist and Bayesian estimation methods through the Bernstein-vonMises theorem (Bickel and Kleijn, 2012; Rivoirard and Rousseau, 2012; van der Vaart, 1998; Yang and Le Cam, 2000), ”
Critical dimension in the semiparametric Bernstein—von Mises theorem (2014, background): “The recent paper [2] extends the BvM statement from the classical parametric case to a rather general i.”
Bayesian inference in a class of partially identified models (2016, background): “, Van der Vaart (1998), Shen (2002), or Bickel and Kleijn (2012)), taking μn(X) to be the maximum likelihood estimator and Σ0 to be the inverse Fisher information matrix.9 This assumption can also hold, for example, for the Bayesian bootstrap for nonparametric estimation of moments of an unknown dis”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, background): “ce and techniques of proofs arise. Results on semiparametric BvM applicable to general models and/or general priors include Shen [31], Castillo [10], Rivoirard and Rousseau [30] and Bickel and Kleijn [3]. The variety of possible interactions between prior and model and the subtleties of prior choice”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, background): “with BvM results in the nonparametric setting. Leahu [34] has studied the impact of prior smoothness on the existence of BvM theorems in the conjugate Gaussian sequence space model. Bickel and Kleijn [3], Castillo [8], Rivoirard and Rousseau [45] and Castillo and Rousseau [13] provide sucient condit”