Related papers: Bernstein - von Mises theorems for statistical inv…
We study the inverse problem of recovering the order and the diffusion coefficient of an elliptic fractional partial differential equation from a finite number of noisy observations of the solution. We work in a Bayesian framework and show…
We provide a comprehensive semi-parametric study of Bayesian partially identified econometric models. While the existing literature on Bayesian partial identification has mostly focused on the structural parameter, our primary focus is on…
We show global uniqueness in an inverse problem for the fractional Schr\"odinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness in the partial…
This note considers a finite dimensional statistical model for the Calder\'on problem with piecewise constant conductivities. In this setting it is shown that injectivity of the forward map and its linearisation suffice to prove the…
For $D$ a bounded domain in $\mathbb R^d, d \ge 2,$ with smooth boundary $\partial D$, the non-linear inverse problem of recovering the unknown conductivity $\gamma$ determining solutions $u=u_{\gamma, f}$ of the partial differential…
In the recent Bayesian nonparametric literature, many examples have been reported in which Bayesian estimators and posterior distributions do not achieve the optimal convergence rate, indicating that the Bernstein-von Mises theorem does not…
A functional Bernstein - von Mises theorem is proved for posterior measures arising in a data assimilation problem with the two-dimensional Navier-Stokes equation where a Gaussian process prior is assigned to the initial condition of the…
We consider the recovery of an unknown function $f$ from a noisy observation of the solution $u_f$ to a partial differential equation that can be written in the form $\mathcal{L} u_f=c(f,u_f)$, for a differential operator $\mathcal{L}$ that…
In the bayesian analysis of Inverse Problems most relevant cases the forward maps (FM, or regressor function) are defined in terms of a system of (O, P)DE's with intractable solutions. These necessarily involve a numerical method to find…
In a smooth semiparametric model, the marginal posterior distribution of the finite dimensional parameter of interest is expected to be asymptotically equivalent to the sampling distribution of frequentist's efficient estimators. This is…
The inferential model (IM) framework offers alternatives to the familiar probabilistic (e.g., Bayesian and fiducial) uncertainty quantification in statistical inference. Allowing this uncertainty quantification to be imprecise makes it…
The problem of interest in this article is to study the (nonlocal) inverse problem of recovering a potential based on the boundary measurement associated with the fractional Schr\"{o}dinger equation. Let $0<a<1$, and $u$ solves…
Volatility estimation based on high-frequency data is key to accurately measure and control the risk of financial assets. A L\'{e}vy process with infinite jump activity and microstructure noise is considered one of the simplest, yet…
The celebrated Bernstein von-Mises theorem ensures that credible regions from Bayesian posterior are well-calibrated when the model is correctly-specified, in the frequentist sense that their coverage probabilities tend to the nominal…
I propose a semiparametric Bayesian inference framework for conditional moment equalities. The core idea is that these models deterministically map a conditional distribution of data to a structural parameter via the restriction that a…
We develop a general mathematical framework for variational problems where the unknown function assumes values in the space of probability measures on some metric space. We study weak and strong topologies and define a total variation…
Solving inverse problems involving measurement noise and modeling errors requires regularization in order to avoid data overfit. Geophysical inverse problems, in which the Earth's highly heterogeneous structure is unknown, present a…
We consider the heat equation with absorption in a bounded domain of $\mathbb{R}^d$, where both the scalar diffusivity and the absorption function are unknown. We investigate a Bayesian approach for recovering the diffusivity from a noisy…
We consider the Bayesian nonparametric estimation of a nonlinear reaction function in a reaction-diffusion stochastic partial differential equation (SPDE). The likelihood is well-defined and tractable by the infinite-dimensional Girsanov…
We review the Bayesian theory of semiparametric inference following Bickel and Kleijn (2012) and Kleijn and Knapik (2013). After an overview of efficiency in parametric and semiparametric estimation problems, we consider the Bernstein-von…