On the Bernstein-von Mises Theorem with Infinite Dimensional Parameters

By David Freedman. 1999

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Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness (2020, background): “It is worth noting that the Laplace approximation to the RFF posterior is asymptotically exact by the virtue of the Bernstein-von Mises (BvM) theorem and the fact that (8) is a finite-rank model [16, 23, 46, 57].”
Sharp asymptotics of the functional quantization problem for Gaussian processes (2004, background): “Ritter (2000, page 79) [see also Freedman (1999) for m = 1 and Gao, Hanning and Torcaso (2003)] has derived the asymptotic behavior of the eigenvalues of the covariance operator: as k → ∞ .”
A Bernstein–von Mises theorem in the nonparametric right-censoring model (2004, methodology): “Previous research on the Bernstein–von Mises theorem for nonparametric models includes Lo (1983, 1986, 1993), Brunner and Lo (1996), Diaconis and Freedman (1998), Conti (1999) and Freedman (1999).”
Critical dimension in the semiparametric Bernstein—von Mises theorem (2014, background): “See [12–14, 16] and references therein for some special phenomena arising in the Bayesian analysis when the parameter dimension increases.”
Bayesian inference in a class of partially identified models (2016, unknown): “See also Freedman (1999).”
Bayesian and likelihood-based methods in multilevel modeling 1 A comparison of Bayesian and likelihood-based methods for fitting multilevel models (2006, background): “† There is an expectation, expressed formally in the Bernstein-von Mises theorem (e.g., Freedman (1999) ; also see Samaniego and Reneau (1994) and Severini (1994)), that likelihood and difiuse-prior Bayesian results will be close in large samples, and this will typically occur when parametric model”
Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors (2010, background): “[6] gives negative results, and we recall some positive ones below.”
INFERENCE FOR SEMIPARAMETRIC MODELS: SOME QUESTIONS AND AN ANSWER (2001, background): “There are negative results—see Freedman (1963, 1965, 1999) on consistency, and Cox(1993) on some higher order phenomena.”
Convergence rates for Bayesian density estimation of infinite-dimensional exponential families (2006, background): “Several papers—including [1, 2, 4, 11, 14, 15, 16, 17, 18, 20, 21, 24, 25, 26]—have been devoted to the investigation of asymptotic properties of posterior distributions on infinite-dimensional parameter spaces.”
A Bayesian approach to non‐parametric monotone function estimation (2009, background): “However, the statistical literature (e.g. Freedman (1999) and Cox (1993)) has provided numerous surprises and anomalies when the number of parameters is infinite.”
The Bernstein–von Mises theorem for the proportional hazard model (2006, unknown): “These examples cast doubt on the Bernstein– von Mises theorem in nonparametric models, which states that the posterior distribution centered at the maximum likelihood estimator is asymptotically equivalent to the sampling distribution of the maximum likelihood estimator. See also [ 7 ].”
Integrated Brownian motions and exact L2-small balls (2003, background): “f(t0) = f ′(t1) = f ′′(t2) = · · · = f (t2m+1) = 0 (5)”
High dimensional Bernstein-von Mises: simple examples (2010, methodology): “In this section, we pay homage to a remarkable paper by Freedman [6], itself stimulated by Cox [4], which sets out the failure of the Bernstein-von Mises theorem in a simple sequence model of function estimation in Gaussian white noise.”
Philosophy and the practice of Bayesian statistics (2010, unknown): “Using an informative prior distribution (even if only weakly informative, as in Gelman et al. (2008a)) makes our estimates less sensitive to the data than, say, maximum-likelihood estimates would be, which can be a net gain.”
On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures (2013, background): “Notably, Freedman [14] has shown that in a basic Gaussian conjugate 2-sequence space setting, the BvM theorem does not hold true in generality; see also the related recent contributions [24, 28].”
On the Bernstein-von Mises phenomenon in the Gaussian white noise model (2011, background): “, [1, 11, 12] for semiparamatric and [5, 6] for nonparametric, the results in [2] and [3] led to the widely accepted belief that the BvM phenomenon does not occur in the nonparametric framework.”