A Bernstein–von Mises theorem in the nonparametric right-censoring model

By Yongdai Kim, Jaeyong Lee. 2004

69 citations recorded by the source corpus.

In the recent Bayesian nonparametric literature, many examples have been reported in which Bayesian estimators and posterior distributions do not achieve the optimal convergence rate, indicating that the Bernstein-von Mises theorem does not hold. In this article, we give a positive result in this direction by showing that the Bernstein-von Mises theorem holds in survival models for a large class of prior processes neutral to the right. We also show that, for an arbitrarily given convergence rate n -α with 0 < α ≤ 1/2, a prior process neutral to the right can be chosen so that its posterior distribution achieves the convergence rate n -α .

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

Finite Sample Bernstein -- von Mises Theorem for Semiparametric Problems (2013, background): “symptotic normality of the posterior measure for these classes of models is even more challenging; see e.g. Shen (2002). Some results for particular semi and nonparametric problems are available from Kim and Lee (2004); Kim (2006); Leahu (2011); Castillo and Nickl (2013). Cheng and Kosorok (2008) ob”
The Bernstein–von Mises theorem for the proportional hazard model (2006, methodology): “For right-censored data, Kim and Lee [13] showed that not all neutral to the right prior processes have consistent posteriors and gave sufficient conditions for consistency.”
Noname manuscript No. (will be inserted by the editor) A semiparametric Bernstein- von Mises theorem for Gaussian process priors (unknown): “However, positive BVM results are known in particular nonparametric settings, see [10, 17].”
On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures (2013, background): “There exists a Donsker-type BvM theorem for the cumulative distribution function based on Dirichlet process priors, see Lo [31], and this carries over to a variety of closely related nonparametric situations, including quantile inference and censoring models, where Bernstein–von Mises results are av”
High dimensional Bernstein-von Mises: simple examples (2010, methodology): “In survival analysis, Bernstein-von Mises theorems for the cumulative hazard function are established by Kim and Lee [16] and for the cumulative hazard and fixed dimensional covariate regression parameter in a proportional hazards model in [15].”
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].”
Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors (2010, methodology): “[12] and [11] study the nonparametric right-censoring model and the proportional hazard model.”
Large sample asymptotics for the two-parameter Poisson–Dirichlet process (2007, result): “This is in line with nonparametric Bernstein Von Mises results of for instance, (31), (30),(28), (5) and (22).”
BERNSTEIN-VON MISES THEOREM FOR LINEAR FUNCTIONALS OF THE DENSITY (2009, background): “Nice positive results are obtained in [14] and [15], however they rely heavily on a conjugacy property and on the fact that their priors put mass one on discrete probabilities which makes the comparison with the empirical distribution more tractable.”
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.”
The semiparametric Bernstein-von Mises theorem (2010, background): “n studied in various semiparametric models: several papers have provided studies of asymptotic normality of posterior distributions for models from survival analysis. Particularly, Kim and Lee (2004) [23] show that the innite-dimensional posterior for the cumulative hazard function in right-censored”