Probabilistic Integration: A Role in Statistical Computation?

Briol, François‐Xavier; Oates, Chris J.; Osborne, Michael A.; Sejdinović, Dino

doi:10.48550/arxiv.1512.00933

Cited by 24 publications

(67 citation statements)

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“…The slope of the trend line was ≈ −1/4, in close agreement with the theoretical analysis. This does not resemble the rapid posterior contraction results established in BQ when p(dx) is a priori known, which can be exponential for the Gaussian kernel [5]. This reflects the more challenging nature of the estimation problem when p(dx) is unavailable in closed-form.…”

Section: Detailed Resultsmentioning

confidence: 79%

“…From Lemmas 1 and 3 in [5], we have that the idealised BQ estimate based on the bounded kernel k satisfies µ 0 − µ 0,n H = O P (n −1/2 ). Indeed, µ 0 − µ 0,n H ≤ µ 0 − μ0,n H , where…”

Section: A1 Proof Of Theoremmentioning

confidence: 99%

“…This induces both a prior and a posterior over the value of p(f ) as push-forward measures under the projection operator f → p(f ). Since its introduction, several authors have related BQ to other methods such as the 'herding' approach from machine learning [24,4], random feature approximations used in kernel methods [1], classical quadrature rules [42] and Quasi Monte Carlo (QMC) methods [5]. Most recently, [28] extended theoretical results for BQ to misspecified prior models, and [29] who provided efficient matrix algebraic methods for the implementation of BQ.…”

Section: Motivation: Predictive Assessment Of Computer Modelsmentioning

confidence: 99%

“…The posterior in Eqn. 2 was studied in [5], where rates of posterior contraction were established under further assumptions on the smoothness of the covariance function k and the smoothness of the integrand. Note that the matrix inverse of k(X, X) incurs a (naive) computational cost of O(n 3 ); however this cost is post-hoc and decoupled from (more expensive) computation that involves the computer model.…”

Section: Probabilistic Integration For Tractable Distributions (Bq)mentioning

confidence: 99%

“…The proof is provided in the Supplemental Information. Assumption (1) derives from results on consistent BQ [5] and can be relaxed further with the results in [28] (not discussed here), while assumptions (2-5) derive from previous work on consistent estimation with DPM priors [19]. For the case of BQ when p(dx) is known and H a Sobolev space of order s > 1/2 on Ω = [0, 1], the corresponding posterior contraction rate is exp(−Cn 2s− ) [5, Thm.…”

Section: Kernel Means Via Stick Breakingmentioning

confidence: 99%

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Probabilistic Models for Integration Error in the Assessment of Functional Cardiac Models

Oates¹,

Niederer²,

Lee³

et al. 2016

Preprint

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This paper studies the numerical computation of integrals, representing estimates or predictions, over the output f (x) of a computational model with respect to a distribution p(dx) over uncertain inputs x to the model. For the functional cardiac models that motivate this work, neither f nor p possess a closed-form expression and evaluation of either requires ≈ 100 CPU hours, precluding standard numerical integration methods. Our proposal is to treat integration as an estimation problem, with a joint model for both the a priori unknown function f and the a priori unknown distribution p. The result is a posterior distribution over the integral that explicitly accounts for dual sources of numerical approximation error due to a severely limited computational budget. This construction is applied to account, in a statistically principled manner, for the impact of numerical errors that (at present) are confounding factors in functional cardiac model assessment.

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Section: Detailed Resultsmentioning

confidence: 79%

Section: A1 Proof Of Theoremmentioning

confidence: 99%

Section: Motivation: Predictive Assessment Of Computer Modelsmentioning

confidence: 99%

Section: Probabilistic Integration For Tractable Distributions (Bq)mentioning

confidence: 99%

Section: Kernel Means Via Stick Breakingmentioning

confidence: 99%

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Probabilistic Models for Integration Error in the Assessment of Functional Cardiac Models

Oates¹,

Niederer²,

Lee³

et al. 2016

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

2019

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This paper presents convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We provide convergence guarantees based on two different assumptions on a quadrature rule: one on quadrature weights, and the other on design points. More precisely, we show that convergence rates can be derived (i) if the sum of absolute weights remains constant (or does not increase quickly), or (ii) if the minimum distance between design points does not decrease very quickly. As a consequence of the latter result, we derive a rate of convergence for Bayesian quadrature in misspecified settings. We reveal a condition on design points to make Bayesian quadrature robust to misspecification, and show that, under this condition, it may adaptively achieve the optimal rate of convergence in the Sobolev space of a lesser order (i.e., of the unknown smoothness of a test integrand), under a slightly stronger regularity condition on the integrand. MSC 2010 subject classification: Primary: 65D30, Secondary: 65D32, 65D05, 46E35, 46E22.

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De-noising by thresholding operator adapted wavelets

Yoo

Owhadi

2019

Stat Comput

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Donoho and Johnstone [13] proposed a method from reconstructing an unknown smooth function u from noisy data u + ζ by translating the empirical wavelet coefficients of u + ζ towards zero. We consider the situation where the prior information on the unknown function u may not be the regularity of u but that of Lu where L is a linear operator (such as a PDE or a graph Laplacian). We show that the approximation of u obtained by thresholding the gamblet (operator adapted wavelet) coefficients of u + ζ is near minimax optimal (up to a multiplicative constant), and with high probability, its energy norm (defined by the operator) is bounded by that of u up to a constant depending on the amplitude of the noise. Since gamblets can be computed in O(N polylog N ) complexity and are localized both in space and eigenspace, the proposed method is of near-linear complexity and generalizable to non-homogeneous noise.

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Probabilistic Integration: A Role in Statistical Computation?

Cited by 24 publications

References 0 publications

Probabilistic Models for Integration Error in the Assessment of Functional Cardiac Models

Probabilistic Models for Integration Error in the Assessment of Functional Cardiac Models

Convergence Analysis of Deterministic Kernel-Based Quadrature Rules in Misspecified Settings

De-noising by thresholding operator adapted wavelets

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