Multivariate Stein factors for a class of strongly log-concave distributions

Mackey, Lester; Gorham, Jackson

doi:10.1214/16-ecp15

Cited by 29 publications

(58 citation statements)

References 9 publications

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“…Using independence and the same argument as for R 31 , we have |R 32 | ≤ Cdβη ǫ √ n . 24 Now we bound R 33 . Using integration by parts, and combining two independent Gaussians into one, we have…”

Section: (B8)mentioning

confidence: 98%

Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula

Fang

Shao

2018

Probab. Theory Relat. Fields

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Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second-or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance. We apply the theorem to the unadjusted Langevin algorithm.

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Section: (B8)mentioning

confidence: 98%

Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula

Fang

Shao

2018

Probab. Theory Relat. Fields

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“…In particular, if p is c-strongly log-concave, meaning that ϕ ′′ ≥ c > 0 on R, then the Stein kernel is uniformly bounded and τ ν ∞ ≤ c −1 . For more about the Stein method related to (strongly) log-concave measures, see for instance [MG16]. Furthermore, by differentiating (15), we obtain for any x ∈ I (ν),…”

Section: On Covariance Identities and The Stein Kernelmentioning

confidence: 99%

Weighted Poincaré inequalities, concentration inequalities and tail bounds related to Stein kernels in dimension one

Saumard¹

2019

Bernoulli

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We investigate links between the so-called Stein's density approach in dimension one and some functional and concentration inequalities. We show that measures having a finite first moment and a density with connected support satisfy a weighted Poincaré inequality with the weight being the Stein kernel, that indeed exists and is unique in this case. Furthermore, we prove weighted log-Sobolev and asymmetric Brascamp-Lieb type inequalities related to Stein kernels. We also show that existence of a uniformly bounded Stein kernel is sufficient to ensure a positive Cheeger isoperimetric constant. Then we derive new concentration inequalities. In particular, we prove generalized Mills' type inequalities when a Stein kernel is uniformly bounded and sub-gamma concentration for Lipschitz functions of a variable with a sub-linear Stein kernel. When some exponential moments are finite, a general concentration inequality is then expressed in terms of Legendre-Fenchel transform of the Laplace transform of the Stein kernel. Along the way, we prove a general lemma for bounding the Laplace transform of a random variable, that should be useful in many other contexts when deriving concentration inequalities. Finally, we provide density and tail formulas as well as tail bounds, generalizing previous results that where obtained in the context of Malliavin calculus.

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“…The first quantity |f | k 0 ,F in ( 18) can be approximated by |f n | k 0 ,F when f n is a reasonable approximation for f and this can in turn can be bounded as |f n | k 0 ,F ≤ (a K 0 a) 1/2 . The finiteness of |f | k 0 ,F ensures the existence of a solution to the Stein equation, sufficient conditions for which are discussed in Mackey & Gorham (2016); Si et al (2020). The second quantity (w K 0 w) 1/2 in (18) is computable and is recognized as a kernel Stein discrepancy between the empirical measure n i=1 w i δ(x (i) ) and the distribution whose density is p, based on the Stein operator L (Chwialkowski et al, 2016;Liu et al, 2016).…”

Section: Theoretical Properties and Convergence Assessmentmentioning

confidence: 99%

Semi-exact control functionals from Sard’s method

et al. 2021

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A novel control variate technique is proposed for post-processing of Markov chain Monte Carlo output, based both on Stein’s method and an approach to numerical integration due to Sard. The resulting estimators of posterior expected quantities of interest are proven to be polynomially exact in the Gaussian context, while empirical results suggest the estimators approximate a Gaussian cubature method near the Bernstein-von-Mises limit. The main theoretical result establishes a bias-correction property in settings where the Markov chain does not leave the posterior invariant. Empirical results are presented across a selection of Bayesian inference tasks. All methods used in this paper are available in the R package ZVCV.

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Multivariate Stein factors for a class of strongly log-concave distributions

Cited by 29 publications

References 9 publications

Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula

Multivariate approximations in Wasserstein distance by Stein’s method and Bismut’s formula

Weighted Poincaré inequalities, concentration inequalities and tail bounds related to Stein kernels in dimension one

Semi-exact control functionals from Sard’s method

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