Asymptotics of Infinite-Dimensional Integrals With Respect to Smooth Measures I

Albeverio, Sergio; Steblovskaya, Victoria

doi:10.1142/s021902579900031x

Cited by 14 publications

(12 citation statements)

References 17 publications

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“…Differently from the standard finite-dimensional setting, the literature on the Laplace approximation method in the infinite-dimensional setting appears to be not well developed. That is, to the best of our knowledge, infinite-dimensional Laplace approximations are limited to the case in which the measure π is a Gaussian measure (Albeverio and Steblovskaya [3,2]). Unfortunately, this literature does not cover the case in which the Hessian of the map θ → K(θ | θ 0 ) at θ 0 is not coercive (uniformly elliptic), which is precisely the case of interest in our specific problem.…”

Section: First Termmentioning

confidence: 99%

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Strong posterior contraction rates via Wasserstein dynamics

Dolera¹,

Favaro²,

Mainini³

2022

Preprint

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Posterior contractions rates (PCRs) quantify the speed at which the posterior distribution concentrates on arbitrarily small neighborhoods of the true model, in a suitable way, as the sample size goes to infinity. In this paper, we develop a novel approach to PCRs, for both finitedimensional (parametric) and infinite-dimensional (nonparametric) Bayesian models. Critical to our approach is the combination of an assumption of local Lipschitz-continuity for the posterior distribution with a dynamic formulation of the Wasserstein distance, here referred to as Wasserstein dynamics, which allows to set forth a connection between the problem of establishing PCRs and some classical problems in mathematical analysis, probability theory and mathematical statistics: Laplace methods for approximating integrals, Sanov's large deviation principle under the Wasserstein distance, rates of convergence of mean Glivenko-Cantelli theorems, and estimates of weighted Poincaré-Wirtinger constants. Under dominated Bayesian models, we present two main results: i) a theorem on PCRs for the regular infinite-dimensional exponential family of statistical models; ii) a theorem on PCRs for a general dominated statistical models. Some applications of our results are presented for the regular parametric model, the multinomial model, the finite-dimensional and the infinite-dimensional logistic-Gaussian model and the infinite-dimensional linear regression. It turns out that our approach leads to optimal PCRs in finite dimension, whereas in infinite dimension it is shown explicitly how prior distributions affect the corresponding PCRs. In general, with regards to infinite-dimensional Bayesian models for density estimation, our approach to PCRs is the first to consider strong norm distances on parameter spaces of functions, such as Sobolev-like norms, as most of the literature deals with spaces of density functions endowed with L p norms or the Hellinger distance.

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Section: First Termmentioning

confidence: 99%

“…2 , B(M 2 )) with i-the marginal γ i , for i = 1, 2. See Ambrosio et al[5, Chapter 7] and Ambrosio et al [5, Proposition 7.1.5].…”

mentioning

confidence: 99%

Strong posterior contraction rates via Wasserstein dynamics

Dolera¹,

Favaro²,

Mainini³

2022

Preprint

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show abstract

“…Moreover it is a representation of the solution of the Schrödinger equation (27) evaluated at x ∈ R d at time t.…”

Section: The Schrödinger Equationmentioning

confidence: 99%

Theory and Applications of Infinite Dimensional Oscillatory Integrals

Albeverio

2007

Stochastic Analysis and Applications

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“…Closing this introduction we would like to mention also the recent work [2], where the case of quadratic phase function has been considered for Laplace-type integrals with respect to a smooth measure on a rigged Hilbert space. For this more general setting and under stronger analytic assumptions on the amplitude function an asymptotic expansion similar to (4.2) has been obtained.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of Infinite-Dimensional Integrals With Respect to Smooth Measures I

Cited by 14 publications

References 17 publications

Strong posterior contraction rates via Wasserstein dynamics

Strong posterior contraction rates via Wasserstein dynamics

Theory and Applications of Infinite Dimensional Oscillatory Integrals

asymptotics of oscillatory integrals with quadratic phase function on wiener space

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