Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency

The Bayesian formulation of inverse problems is attractive for three primary reasons: it provides a clear modelling framework; means for uncertainty quantification; and it allows for principled learning of hyperparameters. The posterior distribution may be explored by sampling methods, but for many problems it is computationally infeasible to do so. In this situation maximum a posteriori (MAP) estimators are often sought. Whilst these are relatively cheap to compute, and have an attractive variational formulation, a key drawback is their lack of invariance under change of parameterization. This is a particularly significant issue when hierarchical priors are employed to learn hyperparameters. In this paper we study the effect of the choice of parameterization on MAP estimators when a conditionally Gaussian hierarchical prior distribution is employed. Specifically we consider the centred parameterization, the natural parameterization in which the unknown state is solved for directly, and the noncentred parameterization, which works with a whitened Gaussian as the unknown state variable, and arises when considering dimension-robust MCMC algorithms; MAP estimation is well-defined in the nonparametric setting only for the noncentred parameterization. However, we show that MAP estimates based on the noncentred parameterization are not consistent as estimators of hyperparameters; conversely, we show that limits of finite-dimensional centred MAP estimators are consistent as the dimension tends to infinity. We also consider empirical Bayesian hyperparameter estimation, show consistency of these estimates, and demonstrate that they are more robust with respect to noise than centred MAP estimates. An underpinning concept throughout is that hyperparameters may only be recovered up to measure equivalence, a well-known phenomenon in the context of the Ornstein-Uhlenbeck process.Comment: 36 pages, 8 figure

Earth-Approaching Asteroids as Targets for Exploration

A mission strategy focused on the earth approaching asteroids is presented. The opportunities for sample return and manned visits favor the closer Apollos and Amors over the main belt objects

Earth-approaching asteroids: Populations, origin, and compositional types

Origin, physical properties, and discovery history of smaller asteroids are reviewed. They appear to link the main belt objects, namely the comets and meteorites. Physical observations suggest that a wide variety of compositional types are represented among the near-earth asteroids; the apparent rarity of carbonaceous objects is stated

Limits of turbulence and outer scale profiling with non-Kolmogorov statistics

SLODAR (SLOpe Detection And Ranging) methods recover the atmospheric turbulence profile from cross-correlations of wavefront sensor (WFS) measurements, based on known turbulence models. Our work grows out of several experiments showing that turbulence statistics can deviate significantly from the classical Kolmogorov/ von Kármán models, especially close to the ground. We present a novel SLODAR-type method which simultaneously recovers both the turbulence profile in the atmosphere and the turbulence statistics at the ground layer - namely the slope of the spatial frequency power law. We consider its application to outer scale (L0)- reconstruction and investigate the limits of the joint estimation of such parameters.Peer reviewe

A TV-Gaussian prior for infinite-dimensional Bayesian inverse problems and its numerical implementations

Many scientific and engineering problems require to perform Bayesian inferences in function spaces, in which the unknowns are of infinite dimension. In such problems, choosing an appropriate prior distribution is an important task. In particular we consider problems where the function to infer is subject to sharp jumps which render the commonly used Gaussian measures unsuitable. On the other hand, the so-called total variation (TV) prior can only be defined in a finite dimensional setting, and does not lead to a well-defined posterior measure in function spaces. In this work we present a TV-Gaussian (TG) prior to address such problems, where the TV term is used to detect sharp jumps of the function, and the Gaussian distribution is used as a reference measure so that it results in a well-defined posterior measure in the function space. We also present an efficient Markov Chain Monte Carlo (MCMC) algorithm to draw samples from the posterior distribution of the TG prior. With numerical examples we demonstrate the performance of the TG prior and the efficiency of the proposed MCMC algorithm

Fast Gibbs sampling for high-dimensional Bayesian inversion

Solving ill-posed inverse problems by Bayesian inference has recently attracted considerable attention. Compared to deterministic approaches, the probabilistic representation of the solution by the posterior distribution can be exploited to explore and quantify its uncertainties. In applications where the inverse solution is subject to further analysis procedures, this can be a significant advantage. Alongside theoretical progress, various new computational techniques allow to sample very high dimensional posterior distributions: In [Lucka2012], a Markov chain Monte Carlo (MCMC) posterior sampler was developed for linear inverse problems with $\ell_1$-type priors. In this article, we extend this single component Gibbs-type sampler to a wide range of priors used in Bayesian inversion, such as general $\ell_p^q$ priors with additional hard constraints. Besides a fast computation of the conditional, single component densities in an explicit, parameterized form, a fast, robust and exact sampling from these one-dimensional densities is key to obtain an efficient algorithm. We demonstrate that a generalization of slice sampling can utilize their specific structure for this task and illustrate the performance of the resulting slice-within-Gibbs samplers by different computed examples. These new samplers allow us to perform sample-based Bayesian inference in high-dimensional scenarios with certain priors for the first time, including the inversion of computed tomography (CT) data with the popular isotropic total variation (TV) prior.Comment: submitted to "Inverse Problems

Platelet function and filamin A expression in two families with novel FLNA gene mutations associated with periventricular nodular heterotopia and panlobular emphysema

Pathogenic variants of the X-linked FLNA gene encoding filamin A protein have been associated with a wide spectrum of symptoms, including the recently described pulmonary phenotype with childhood-onset panlobular emphysema. We describe three female patients from two families with novel heterozygous FLNA variants c.5837_2del and c.508C > T. Analysis of immunofluorescence of peripheral blood smears and platelet function was performed for all patients. FLNA-negative platelets were observed, suggesting that these variants result in the loss of a functional protein product. All three patients also had periventricular nodular heterotopia and panlobular emphysema. However, they had considerably milder symptoms and later age of onset than in the previously reported cases. Therefore, patients with pathogenic FLNA variants should be studied actively for lung involvement even in the absence of pronounced respiratory symptoms. Conversely, any patient with unexplained panlobular emphysema should be analyzed for pathogenic FLNA variants. We also suggest that immunofluorescence analysis is a useful tool for investigating the pathogenicity of novel FLNA variants.Peer reviewe