Lower bounds for posterior rates with Gaussian process priors

By I. Castillo. 2008

97 citations recorded by the source corpus.

Upper bounds for rates of convergence of posterior distributions associated to Gaussian process priors are obtained by van der Vaart and van Zanten in [14] and expressed in terms of a concentration function involving the Reproducing Kernel Hilbert Space of the Gaussian prior. Here lower-bound counterparts are obtained. As a corollary, we obtain the precise rate of convergence of posteriors for Gaussian priors in various settings. Additionally, we extend the upper-bound results of [14] about Riemann-Liouville priors to a continuous family of parameters.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/a51f3683a1dd38ed9bbaa3b94d64ab99f983e926

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Citations in the corpus

Posterior concentration rates for infinite dimensional exponential families (2012, methodology): “Also, randomizing as in the prior model (PH) leads to adaptation, which is not possible with purely Gaussian priors (see Theorem 1 of Castillo [3]).”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, methodology): “This follows from the results on the concentration function for Riemann–Louville-type processes obtained in Theorem 4 in [9].”
Thomas Bayes’ walk on manifolds (2012, methodology): “Proof We use a general approach to prove lower bounds for posterior measures introduced in [6] (see [6], [7] for examples).”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, background): “If f0 ∈ H and α = β , it has been shown [2, 7, 27] that the posterior contracts at the minimax rate of convergence, while if α = β , then strictly suboptimal rates are achieved.”
Bayesian Nonparametric Causal Inference: Information Rates and Learning Algorithms (2017, result): “Using the results of the paper by Castillo in [49], the upper bound in Theorem 4 can be shown to be tight.”
Bayes procedures for adaptive inference in inverse problems for the white noise model (2012, unknown): “If α (cid:16)= β , only sub-optimal rates are attained in general (cf. [7]).”
Information Rates of Nonparametric Gaussian Process Methods (2011, background): “The proof, following an idea of Castillo (2008) and given in Section 4.4, is based on the fact that balls of this type also receive very littleprior mass, essentially because the inequality of the preceding lemma can be reversed.”
Adaptive Bayesian Procedures Using Random Series Priors (2014, methodology): “structure of its reproducing kernel Hilbert space, were studied by Tokdar and Ghosh [2007], Ghosal and Roy [2006], Choi and Schervish [2007], van der Vaart and van Zanten [2007, 2008, 2009], Castillo [2008, 2012].”
On Bayes Risk Lower Bounds (2014, background): “, where H ( P || Q ) = ( p − q ) 2 1 / 2 is the Hellinger distance between P and Q .”
Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator (2015, background): “Type (T2) and (T3) priors have been studied with fixed hyper-parameters by [5, 7, 15, 31, 34] or using a prior on λ = τ and λ = α in [4, 14, 18, 29].”
Needles and Straw in a Haystack: Posterior concentration for possibly sparse sequences (2012, background): “8 follows the approach to get lower bound type results introduced in [8], which uses the principle that sets with very little prior mass receive no posterior mass, see also Figure 2.”
BAYESIAN LINEAR REGRESSION WITH SPARSE PRIORS (2014, unknown): “1 in [10] (based on [9]) implies that E0Π ( ‖β‖2 < sn |Y ) → 0, for any sequence sn such that for some rn → ∞”
Limits of Estimating Heterogeneous Treatment Effects: Guidelines for Practical Algorithm Design (2018, background): “The bound in Theorem 2 can be shown to be tight using the results in (Castillo, 2008).”
Noname manuscript No. (will be inserted by the editor) A semiparametric Bernstein- von Mises theorem for Gaussian process priors (unknown): “Its properties for Bayesian estimation are studied in [5,29] and summarized in Sect.”
On the Frequentist Properties of Bayesian Nonparametric Methods (2016, background): “Although (7) gives only an upper bound on the posterior concentration rate, some lower bounds have been derived in the literature showing that it is often a sharp upper bound, see (Castillo, 2008).”
On Bayesian supremum norm contraction rates (2013, background): “Ghosal, Ghosh and van der Vaart [12], Scricciolo [28], Tokdar and Ghosh [34], van der Vaart and van Zanten [37], [3], Rivoirard and Rousseau [26], and random dyadic histograms, for random histogram priors see e.”