Reproducing kernel Hilbert spaces of Gaussian priors

By Van der Vaart, V. Zanten. 2008

234 citations recorded by the source corpus.

We review definitions and properties of reproducing kernel Hilbert spaces attached to Gaussian variables and processes, with a view to applications in nonparametric Bayesian statistics using Gaussian priors. The rate of contraction of posterior distributions based on Gaussian priors can be described through a concentration function that is expressed in the reproducing Hilbert space. Absolute continuity of Gaussian measures and concentration inequalities play an important role in understanding and deriving this result. Series expansions of Gaussian variables and transformations of their reproducing kernel Hilbert spaces under linear maps are useful tools to compute the concentration function.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f24cfefdfedce43bb575d9d229dbaaa56a57705d

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Citations in the corpus

Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, methodology): “In fact, it is strictly increasing, using the results on pages 213–214 in [37] combined with the fact that the RKHS of W ∗ f0, equal to L2 ∗ f0, contains functions of arbitrary supremum norm.”
Bayesian Nonparametric Causal Inference: Information Rates and Learning Algorithms (2017, methodology): “For instance, using the squared exponential kernel would lead to a suboptimal information rate of (log(n))−(α0 ∧α1 )/2+d/4 , whereas spline kernels achieved a rate of (n/ log(n))−2(α0 ∧α1 )/(2(α0 ∧α1 )+d) , which is optimal up to a logarithmic factor [48].”
Rates of contraction of posterior distributions based on Gaussian process priors (2008, background): “Van der Vaart and Van Zanten [30] reviews facts that are relevant to the present applications.”
Convergence Rates of Efficient Global Optimization Algorithms (2011, background): “Vazquez and Bect (2010) show that when π is a fixed Gaussian process prior of finite smoothness, expected improvement converges on the minimum of any f ∈ H, and almost surely for f drawn from π.”
Thomas Bayes’ walk on manifolds (2012, background): “The term Atw0(ε) is finite for all ε > 0 if and only if w0 lies in the closure in B of Ht (which can be checked to coincide with the support of the Gaussian prior, see [30], Lemma 5.”
Representation formulae for the fractional Brownian motion (2009, methodology): “This technique can also be used for non continuous processes, see for instance [ 36 ].”
Noname manuscript No. (will be inserted by the editor) A semiparametric Bernstein- von Mises theorem for Gaussian process priors (unknown): “If the Gaussian process is given as a Borel measurable map in a Banach space, H is generally defined in a more abstract way through the Pettis integral, but both definition coincide in general, see [30], Sect.”
Lower bounds for posterior rates with Gaussian process priors (2008, background): “k≥1 hkσkεk‖(2)Hα = ∑ k≥1 h 2 k, see for instance [15], Theorem 4.”
On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures (2013, background): “The proof is similar to Theorem 1 in [4], replacing H(δ) there by C([0,1]) (a separable Banach space): the function in that proof is strictly increasing: any shell {g ∈ C([0,1]) : s < ‖g‖∞ < t}, 0 ≤ s < t , contains an element of the RKHS (see [36]) of Brownian motion [in the case of the white noise”
On Bayesian supremum norm contraction rates (2013, unknown): “ﬁes, for C>0 large enough, En f0Π[kTk∞ ≤ C √ nεn|X(n)] → 1. Proof. For any given c>0, Borell’s inequality implies that Π[kTk∞ >C √ nεn] ≤ e−cnε 2 n for large enough C: apply Corollary 5.1 in [24], where σ(T) is bounded using σl≤ 2−l(1/2+r). Next, one applies Lemma 1 in [9]. To do so, one ne”
Convergence Types and Rates in Generic Karhunen-Loève Expansions with Applications to Sample Path Properties (2014, methodology): “prior process is then important for the mathematical analysis, as these spaces determine both the approximation properties of the non-parametric method as well as its statistical properties, see e.g. [52, 74, 73]. In this regard note that one of the strengths of these methods is that they can be con”
Bayesian inference with rescaled Gaussian process priors (2007, background): “[ 21 ], Section 10) we have that the reproducing kernel Hilbert space of the process (2.5) viewed as a map in C[0,1] is the Sobolev space Hk+1[0,1] of functions h : [0,1] → R that are k times continuously dif-”