BERNSTEIN-VON MISES THEOREM FOR LINEAR FUNCTIONALS OF THE DENSITY

By Vincent Rivoirard, Judith Rousseau. 2009

127 citations recorded by the source corpus.

In this paper, we study the asymptotic posterior distribution of linear functionals of the density by deriving general conditions to obtain a semi-parametric version of the Bernstein–von Mises theorem. The special case of the cumulative distributive function, evaluated at a specific point, is widely considered. In particular, we show that for infinite-dimensional exponential families, under quite general assumptions, the asymptotic posterior distribution of the functional can be either Gaussian or a mixture of Gaussian distributions with different centering points. This illustrates the positive, but also the negative, phenomena that can occur in the study of Bernstein–von Mises results.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5cea5dce544a7ee99350634f0d17974530aeeef4

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Citations in the corpus

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.”
Bernstein - von Mises Theorem for growing parameter dimension (2013, background): “the i.i.d. case; Ghosal (1999), Ghosal (2000) for 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 ve”
Critical dimension in the semiparametric Bernstein—von Mises theorem (2014, methodology): “In [24] the semiparametric BvM theorem is derived for linear functionals of density, and in [9] the aMoscow Institute of Physics and Technology (State University), Institutskii per.”
On Bayesian supremum norm contraction rates (2013, methodology): “5). In the logLipschitz case, the choice ϕ=ϕH,τ in (13) with any 0≤τ<1 combined with σl=2−lα satisﬁes (14)–(15). Both claims follow from minor adaptations of Theorem 4.5 in [38] and Theorem 2.1 in [27], respectively; see Lemma 8. In both Gaussian and log-Lipschitz cases, we in fact expect (15) to”
Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors (2010, methodology): “Rivoirard and Rousseau (2009) obtain the Bernstein–von Mises theorem for linear functionals of the density of the observations, in the context of a sieve approximation: sequences of spaces with an increasing dimension kn are used to approximate an infinite-dimensional space.”
The semiparametric Bernstein-von Mises theorem (2010, background): “Rivoirard and Rousseau [35] prove a version for linear functionals over the model, using a class of nonparametric priors based on infinite-dimensional exponential families.”
A Tutorial on Fisher Information (2017, methodology): “p, 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), in statistical decision t”
Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator (2015, background): “15) can be seen either as a log - linear model or as an infinite dimensional exponential family, see for instance [35], [33], [22], [23] and [1].”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, methodology): “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].”
On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures (2013, methodology): “The resulting posterior distributions (·|X(n)) attain minimax optimal contraction rates up to logarithmic terms in Hellinger and L2-distance [5, 32, 37] and L∞-distance [3].”
Adaptive Bayesian Procedures Using Random Series Priors (2014, methodology): “This coincides with the adaptation results for white noise models in Lian (2011) and for density estimation and regression models in Rivoirard and Rousseau (2012b). Example 6 (Multivariate B-splines).”
Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, methodology): “” 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.”
Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, result): “[47] Rivoirard, V.”
High dimensional Bernstein-von Mises: simple examples (2010, background): “Rivoirard and Rousseau [19] give conditions under which Bernstein-von Mises holds for linear functionals of a nonparametrically specified probability density function.”