Symmetry exploits for Bayesian cubature methods

Karvonen, Toni; Särkkä, Simo; Oates, Chris J.

doi:10.1007/s11222-019-09896-8

Cited by 7 publications

(7 citation statements)

References 45 publications

(98 reference statements)

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“…Our approach is closely related to Bayesian quadrature (BQ), see e.g. O'Hagan (1991); Hennig et al (2015); Karvonen et al (2018). In particular, BQ methods have been used by Osborne et al (2012); Gunter et al (2014); Chai and Garnett (2019) to compute the marginal likelihood and to quantify the numerical error of this integral probabilistically.…”

Section: Introductionmentioning

confidence: 99%

Parallel Gaussian Process Surrogate Bayesian Inference with Noisy Likelihood Evaluations

et al. 2021

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We consider Bayesian inference when only a limited number of noisy log-likelihood evaluations can be obtained. This occurs for example when complex simulator-based statistical models are fitted to data, and synthetic likelihood (SL) method is used to form the noisy log-likelihood estimates using computationally costly forward simulations. We frame the inference task as a sequential Bayesian experimental design problem, where the log-likelihood function is modelled with a hierarchical Gaussian process (GP) surrogate model, which is used to efficiently select additional log-likelihood evaluation locations. Motivated by recent progress in the related problem of batch Bayesian optimisation, we develop various batch-sequential design strategies which allow to run some of the potentially costly simulations in parallel. We analyse the properties of the resulting method theoretically and empirically. Experiments with several toy problems and simulation models suggest that our method is robust, highly parallelisable, and sample-efficient.

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Section: Introductionmentioning

confidence: 99%

Parallel Gaussian Process Surrogate Bayesian Inference with Noisy Likelihood Evaluations

et al. 2021

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“…After transformation into Fourier space, solutions were computed using a fourthorder Runge-Kutta numerical integrator ETDRK4 [72] for some user defined length scale L (which we take to be L = 1/2π in our simulation). See [72] for a complete description of the fourth-order ETDRK4 scheme, as well as example MATLAB code used to compute solutions to (17).…”

Section: C5 Kuramoto-sivashinsky Equationmentioning

confidence: 99%

“…non-probabilistic) numerical procedure as data, which can be used to constrain a random variable model for the quantity of interest [3]. Conjugate Gaussian inference has been widely exploited, with an arsenal of PN methods developed for linear algebra [4][5][6][7][8][9][10][11], cubature [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], optimisation [31][32][33][34][35][36], and differential equations [37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52]…”

Section: Introductionmentioning

confidence: 99%

Black Box Probabilistic Numerics

Teymur¹,

Breen²,

Karvonen³

et al. 2021

Preprint

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Probabilistic numerics casts numerical tasks, such the numerical solution of differential equations, as inference problems to be solved. One approach is to model the unknown quantity of interest as a random variable, and to constrain this variable using data generated during the course of a traditional numerical method. However, data may be nonlinearly related to the quantity of interest, rendering the proper conditioning of random variables difficult and limiting the range of numerical tasks that can be addressed. Instead, this paper proposes to construct probabilistic numerical methods based only on the final output from a traditional method. A convergent sequence of approximations to the quantity of interest constitute a dataset, from which the limiting quantity of interest can be extrapolated, in a probabilistic analogue of Richardson's deferred approach to the limit. This black box approach (1) massively expands the range of tasks to which probabilistic numerics can be applied, (2) inherits the features and performance of state-of-the-art numerical methods, and (3) enables provably higher orders of convergence to be achieved. Applications are presented for nonlinear ordinary and partial differential equations, as well as for eigenvalue problems-a setting for which no probabilistic numerical methods have yet been developed.

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“…Strategies to ensure analytic expressions for the integrals in ( 7) and (8) were proposed in Briol et al (2019); Jagadeeswaran and Hickernell (2019). For large n, techniques have been put forward to facilitate the efficient inversion of the matrix K XX Karvonen et al, 2019;Jagadeeswaran and Hickernell, 2019). Proposals that go beyond the standard BC method were outlined in Section 1.…”

Section: Standard Bayesian Cubaturementioning

confidence: 99%

A Locally Adaptive Bayesian Cubature Method

Fisher¹,

Oates²,

Powell³

et al. 2019

Preprint

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Bayesian cubature (BC) is a popular inferential perspective on the cubature of expensive integrands, wherein the integrand is emulated using a stochastic process model. Several approaches have been put forward to encode sequential adaptation (i.e. dependence on previous integrand evaluations) into this framework. However, these proposals have been limited to either estimating the parameters of a stationary covariance model or focusing computational resources on regions where large values are taken by the integrand. In contrast, many classical adaptive cubature methods focus computational resources on spatial regions in which local error estimates are largest. The contributions of this work are three-fold: First, we present a theoretical result that suggests there does not exist a direct Bayesian analogue of the classical adaptive trapezoidal method. Then we put forward a novel BC method that has empirically similar behaviour to the adaptive trapezoidal method. Finally we present evidence that the novel method provides improved cubature performance, relative to standard BC, in a detailed empirical assessment.

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Symmetry exploits for Bayesian cubature methods

Cited by 7 publications

References 45 publications

Parallel Gaussian Process Surrogate Bayesian Inference with Noisy Likelihood Evaluations

Parallel Gaussian Process Surrogate Bayesian Inference with Noisy Likelihood Evaluations

Black Box Probabilistic Numerics

A Locally Adaptive Bayesian Cubature Method

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