A diffusion approach to Stein's method on Riemannian manifolds

Le, Huiling; Lewis, Alexander; Bharath, Karthik; Fallaize, Christopher J.

doi:10.48550/arxiv.2003.11497

Cited by 9 publications

(18 citation statements)

References 34 publications

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“…Remark. This is the sufficient condition initially presented in [7]. An identical condition is put forward in [19] which redefines the Bakry-Émery-Ricci tensor.…”

Section: Motivation and Backgroundmentioning

confidence: 80%

“…We point the reader towards the exposition [16] which provides a comprehensive guide of Stein's method. More recently, Stein's method has greatly expanded in scope to many distributional types; general univariate [9], multivariate [10,15], and manifold valued [7,19]. One can split Stein's method into two distinct approaches: the classical Stein density approach, and the diffusion approach.…”

Section: Formulate and Solve The So-called Stein Equationmentioning

confidence: 99%

“…As described in the Introduction, recent development within the area of Stein's method [7] and [19] have led to the ability to extend the diffusion method originally presented in [10] to general manifolds. This particular construction of Stein's method, however, relies upon a sufficient condition which links the geometry of the manifold with the probability density p = e −φ , namely that Ric + Hess φ ≥ 2κg (1) for some κ > 0 with Ric the Ricci curvature on the manifold and Hess φ the Hessian of φ.…”

Section: Motivation and Backgroundmentioning

confidence: 99%

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Stein's Method for Probability Distributions on $\mathbb{S}^1$

Lewis

2021

Preprint

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In this paper, we propose a modification to the density approach to Stein's method for intervals for the unit circle S 1 which is motivated by the differing geometry of S 1 to Euclidean space. We provide an upper bound to the Wasserstein metric for circular distributions and exhibit a variety of different bounds between distributions; particularly, between the von-Mises and wrapped normal distributions, and the wrapped normal and wrapped Cauchy distributions.

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“…Remark. This is the sufficient condition initially presented in [7]. An identical condition is put forward in [19] which redefines the Bakry-Émery-Ricci tensor.…”

Section: Motivation and Backgroundmentioning

confidence: 80%

Section: Formulate and Solve The So-called Stein Equationmentioning

confidence: 99%

Section: Motivation and Backgroundmentioning

confidence: 99%

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Stein's Method for Probability Distributions on $\mathbb{S}^1$

Lewis

2021

Preprint

Self Cite

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“…To address distributions defined on Riemannian manifolds, Stein operators involving second-order differential operators have been studied [6,37]. For smooth Riemannian manifold M and scalar-valued function f : M → R, the second-order Stein operator for density q on M is defined as…”

Section: Second-order Stein Operatormentioning

confidence: 99%

“…It is worth to note that for the above-mentioned works on goodness-of-fit tests, specific Stein operators are required to be developed independently to address the statistical inference problem for data scenarios and the Stein operators can have diverse forms and seem to be unconnected. Beyond goodness-of-fit testing, KSD have recently been studied in the context of numerical integration [6], Bayesian inferences [43], density estimations [7,37] and measure transport [19].…”

Section: Introductionmentioning

confidence: 99%

Standardisation-function Kernel Stein Discrepancy: A Unifying View on Kernel Stein Discrepancy Tests for Goodness-of-fit

Xu¹

2021

Preprint

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Non-parametric goodness-of-fit testing procedures based on kernel Stein discrepancies (KSD) are promising approaches to validate general unnormalised distributions in various scenarios. Existing works have focused on studying optimal kernel choices to boost test performances. However, the Stein operators are generally non-unique, while different choices of Stein operators can also have considerable effect on the test performances. In this work, we propose a unifying framework, the generalised kernel Stein discrepancy (GKSD), to theoretically compare and interpret different Stein operators in performing the KSD-based goodness-of-fit tests. We derive explicitly that how the proposed GKSD framework generalises existing Stein operators and their corresponding tests. In addition, we show that GKSD framework can be used as a guide to develop kernel-based non-parametric goodness-of-fit tests for complex new data scenarios, e.g. truncated distributions or compositional data. Experimental results demonstrate that the proposed tests control type-I error well and achieve higher test power than existing approaches, including the test based on maximum-mean-discrepancy (MMD).Preprint. Under review.

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Optimal Thinning of MCMC Output

Riabiz

Chen

Cockayne

et al. 2022

Journal of the Royal Statistical Society Series B: Statistical Methodology

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The use of heuristics to assess the convergence and compress the output of Markov chain Monte Carlo can be sub-optimal in terms of the empirical approximations that are produced. Typically a number of the initial states are attributed to 'burn in' and removed, while the remainder of the chain is 'thinned' if compression is also required. In this paper, we consider the problem of retrospectively selecting a subset of states, of fixed cardinality, from the sample path such that the approximation provided by their empirical distribution is close to optimal. A novel method is proposed, based on greedy minimisation of a kernel Stein discrepancy, that is suitable when the gradient of the log-target can be evaluated and approximation using a small number of states is required. Theoretical results guarantee consistency of the method and its effectiveness is demonstrated in the challenging context of parameter inference for ordinary differential equations. Software is available in the Stein Thinning package in Python, R and MATLAB.

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A diffusion approach to Stein's method on Riemannian manifolds

Cited by 9 publications

References 34 publications

Stein's Method for Probability Distributions on $\mathbb{S}^1$

Stein's Method for Probability Distributions on $\mathbb{S}^1$

Standardisation-function Kernel Stein Discrepancy: A Unifying View on Kernel Stein Discrepancy Tests for Goodness-of-fit

Optimal Thinning of MCMC Output

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