On Bayesian supremum norm contraction rates

By I. Castillo. 2013

77 citations recorded by the source corpus.

Building on ideas from Castillo and Nickl (6), a method is pro- vided to study nonparametric Bayesian posterior convergence rates when 'strong' measures of distances, such as the sup-norm, are considered. In particular, we show that likelihood methods can achieve optimal minimax sup-norm rates in density estimation on the unit interval. The introduced methodology is used to prove that commonly used families of prior distri- butions on densities, namely log-density priors and dyadic random density histograms, can indeed achieve optimal sup-norm rates of convergence. New results are also derived in the Gaussian white noise model as a further il- lustration of the presented techniques. AMS 2000 subject classications: Primary 62G20; secondary 62G05, 62G07.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7c7b54933fe62e9737657a7b8f1578c8287c0b03

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Citations in the corpus

Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, methodology): “The sub-optimality of our rate for d > 5 is related to the presence of common semiparametric ‘bias terms’: the approximation-theoretic Lemma 6 below, which quantitatively improves on previous bounds in [9, 33] of a similar kind to apply also when d > 1, gives the desired rate only when d ≤ 4.”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, methodology): “Bickel and Kleijn [3], Castillo [8], Rivoirard and Rousseau [41] and Castillo and Rousseau [13] provide sufficient conditions for BvMs for semiparametric functionals.”
Frequentist coverage and sup-norm convergence rate in Gaussian process regression (2017, background): “The Hellinger and total variation norms are by far the most common metrics for establishing posterior contraction rates and there is only a recent small literature for stronger norms [9, 22, 23, 53].”
Using Stacking to Average Bayesian Predictive Distributions (with Discussion) (2017, background): “…have appealing frequentist asymptotic properties, such as appropriate notions of consistency (Schwartz, 1965) and optimal rates of convergence (van der Vaart and van Zanten, 2009; Bhattacharya et al., 2014; Castillo, 2014; Shen and Ghosal, 2015; Li and Ghosal, 2017; Ghosal and van der Vaart, 2017).”
On adaptive posterior concentration rates (2013, background): “These studies include in particular density estimation in the case of independent and identically distributed observations as in Ghosal et al. [16], nonparametric regression (Ghosal and van der Vaart [17]) and the white noise model (Zhao [32], Belitser and Ghosal [3]), Markov models (Tang and Ghosal”
On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures (2013, methodology): “(I) For the frequencies l ≤ jn one proves, as in Lemma 1 in [3], for some constant C > 0, the sub-Gaussian bound”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, methodology): “Castillo and Nickl [3] showed that for the Gaussian white noise model a BvM theorem can hold in weaker topologies for some natural priors, and the resulting credible sets appropriately modified will have asymptotically the correct coverage and optimal size.”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, methodology): “Finally, although there is no immediate analogue of the BvM property for infinite dimensional parameters, as pointed out by Cox [16] and Freedman [19], some recent contributions have introduced possible notions of nonparametric BvM; see Castillo and Nickl [13] and also Leahu [29].”