Related papers: Bernstein - von Mises theorems for statistical inv…

We consider the statistical linear inverse problem of making inference on an unknown source function in an elliptic partial differential equation from noisy observations of its solution. We employ nonparametric Bayesian procedures based on…

The inferential model (IM) framework offers an alternative to the classical probabilistic (e.g., Bayesian and fiducial) uncertainty quantification in statistical inference. A key distinction is that classical uncertainty quantification…

A standard inverse problem is to determine a source which is supported in an unknown domain $D$ from external boundary measurements. Here we consider the case of a time-dependent situation where the source is equal to unity in an unknown…

This paper studies quasi Bayesian estimation and uncertainty quantification for an unknown function that is identified by a nonparametric conditional moment restriction. We derive contraction rates for a class of Gaussian process priors.…

We study full Bayesian procedures for high-dimensional linear regression. We adopt data-dependent empirical priors introduced in [1]. In their paper, these priors have nice posterior contraction properties and are easy to compute. Our paper…

A fundamental problem of statistical data analysis, distribution density estimation by experimental data, is considered. A new method with optimal asymptotic behavior, the root density estimator, is developed. The method proposed may be…

I prove a semiparametric Bernstein-von Mises theorem for a partially linear regression model with independent priors for the low-dimensional parameter of interest and the infinite-dimensional nuisance parameters. My result avoids a…

By defining a prepotential function for the stationary Schr\"odinger equation we derive an inversion formula for the space variable $x$ as a function of the wave-function $\psi$. The resulting equation is a Legendre transform that relates…

We establish that the potential appearing in a fractional Schr\"odinger operator is uniquely determined by an internal spectral data.

Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy and incomplete data or inherent non-uniqueness. A prominent example is full waveform inversion, which estimates Earth's subsurface…

In this work, we develop a Bayesian framework for solving inverse problems in which the unknown parameter belongs to a space of Radon measures taking values in a separable Hilbert space. The inherent ill-posedness of such problems is…

We consider Bayesian nonparametric inference in the right-censoring survival model, where modeling is made at the level of the hazard rate. We derive posterior limiting distributions for linear functionals of the hazard, and then for `many'…

We study the problem of nonparametric estimation of the fractional derivative of unknown distribution function and of spectral function and show that these problems are well posed when the order of derivative is less than 0.5. We prove also…

The inverse potential problem consists in determining the density of the volume potential from measurements outside the sources. Its ill-posedness is due both to the non-uniqueness of the solution and to the instability of the solution with…

This study investigates the variational posterior convergence rates of inverse problems for partial differential equations (PDEs) with parameters in Besov spaces $B_{pp}^\alpha$ ($p \geq 1$) which are modeled naturally in a Bayesian manner…

We investigate Bernstein-von Mises theorems for adaptive nonparametric Bayesian procedures in the canonical Gaussian white noise model. We consider both a Hilbert space and multiscale setting with applications in $L^2$ and $L^\infty$…

In this paper, we present an inverse problem of identifying the reaction coefficient for time fractional diffusion equations in two dimensional spaces by using boundary Neumann data. It is proved that the forward operator is continuous with…

Bayesian inverse problem on an infinite dimensional separable Hilbert space with the whole state observed is well posed when the prior state distribution is a Gaussian probability measure and the data error covariance is a cylindric…

We study the following fractional Schr\"{o}dinger equation \begin{equation}\label{eq0.1} \varepsilon^{2s}(-\Delta)^s u + Vu = |u|^{p - 2}u,\ \ x\in\,\,\mathbb{R}^N. \end{equation} We show that if the external potential $V\in…

Deep learning methods continue to have a decided impact on machine learning, both in theory and in practice. Statistical theoretical developments have been mostly concerned with approximability or rates of estimation when recovering…