Algorithms for Kullback-Leibler Approximation of Probability Measures in Infinite Dimensions

In this paper we study algorithms to find a Gaussian approximation to a target measure defined on a Hilbert space of functions; the target measure itself is defined via its density with respect to a reference Gaussian measure. We employ the Kullback-Leibler divergence as a distance and find the best Gaussian approximation by minimizing this distance. It then follows that the approximate Gaussian must be equivalent to the Gaussian reference measure, defining a natural function space setting for the underlying calculus of variations problem. We introduce a computational algorithm which is well-adapted to the required minimization, seeking to find the mean as a function, and parameterizing the covariance in two different ways: through low rank perturbations of the reference covariance; and through Schr\"odinger potential perturbations of the inverse reference covariance. Two applications are shown: to a nonlinear inverse problem in elliptic PDEs, and to a conditioned diffusion process. We also show how the Gaussian approximations we obtain may be used to produce improved pCN-MCMC methods which are not only well-adapted to the high-dimensional setting, but also behave well with respect to small observational noise (resp. small temperatures) in the inverse problem (resp. conditioned diffusion).Comment: 28 page

Gaussian Approximations for Probability Measures on R^d

This paper concerns the approximation of probability measures on R^d with respect to the Kullback-Leibler divergence. Given an admissible target measure, we show the existence of the best approximation, with respect to this divergence, from certain sets of Gaussian measures and Gaussian mixtures. The asymptotic behavior of such best approximations is then studied in the small parameter limit where the measure concentrates; this asympotic behavior is characterized using Γ-convergence. The theory developed is then applied to understand the frequentist consistency of Bayesian inverse problems in finite dimensions. For a fixed realization of additive observational noise, we show the asymptotic normality of the posterior measure in the small noise limit. Taking into account the randomness of the noise, we prove a Bernstein-Von Mises type result for the posterior measure

Representation in stochastic search for phylogenetictree reconstruction

Phylogenetic tree reconstruction is a process in which the ancestral relationships among a group of organisms are inferred from their DNA sequences. For all but trivial sized data sets, finding the optimal tree is computationally intractable. Many heuristic algorithms exist, but the branch-swapping algorithm used in the software package PAUP is the most popular. This method performs a stochastic search over the space of trees, using a branch-swapping operation to construct neighboring trees in the search space. This study introduces a new stochastic search algorithm that operates over an alternative representation of trees, namely as permutations of taxa giving the order in which they are processed during stepwise addition. Experiments on several data sets suggest that this algorithm for generating an initial tree, when followed by branch-swapping, can produce better trees for a given total amount of time.Engineering and Applied Science

Gaussian approximations for transition paths in molecular dynamics

This paper is concerned with transition paths within the framework of the overdamped Langevin dynamics model of chemical reactions. We aim to give an efficient description of typical transition paths in the small temperature regime. We adopt a variational point of view and seek the best Gaussian approximation, with respect to Kullback-Leibler divergence, of the non-Gaussian distribution of the diffusion process. We interpret the mean of this Gaussian approximation as the "most likely path" and the covariance operator as a means to capture the typical fluctuations around this most likely path. We give an explicit expression for the Kullback-Leibler divergence in terms of the mean and the covariance operator for a natural class of Gaussian approximations and show the existence of minimisers for the variational problem. Then the low temperature limit is studied via Γ-convergence of the associated variational problem. The limiting functional consists of two parts: The first part only depends on the mean and coincides with the Γ-limit of the Freidlin-Wentzell rate functional. The second part depends on both, the mean and the covariance operator and is minimized if the dynamics are given by a time-inhomogenous Ornstein-Uhlenbeck process found by linearization of the Langevin dynamics around the Freidlin-Wentzell minimizer

Kullback--Leibler approximation for probability measures on infinite dimensional spaces

In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure $\nu$, from within a simple class of measures, which approximates $\mu$. This problem is studied in the case where the Kullback--Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted, and the particular case where $\nu$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimizing sequences. Furthermore, parameterization of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularization is explained, and a regularized minimization is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms

Kullback--Leibler approximation for probability measures on infinite dimensional spaces

In a variety of applications it is important to extract information from a probability measure $\mu$ on an infinite dimensional space. Examples include the Bayesian approach to inverse problems and (possibly conditioned) continuous time Markov processes. It may then be of interest to find a measure $\nu$, from within a simple class of measures, which approximates $\mu$. This problem is studied in the case where the Kullback--Leibler divergence is employed to measure the quality of the approximation. A calculus of variations viewpoint is adopted, and the particular case where $\nu$ is chosen from the set of Gaussian measures is studied in detail. Basic existence and uniqueness theorems are established, together with properties of minimizing sequences. Furthermore, parameterization of the class of Gaussians through the mean and inverse covariance is introduced, the need for regularization is explained, and a regularized minimization is studied in detail. The calculus of variations framework resulting from this work provides the appropriate underpinning for computational algorithms

A zone of preferential ion heating extends tens of solar radii from Sun

The extreme temperatures and non-thermal nature of the solar corona and solar wind arise from an unidentified physical mechanism that preferentially heats certain ion species relative to others. Spectroscopic indicators of unequal temperatures commence within a fraction of a solar radius above the surface of the Sun, but the outer reach of this mechanism has yet to be determined. Here we present an empirical procedure for combining interplanetary solar wind measurements and a modeled energy equation including Coulomb relaxation to solve for the typical outer boundary of this zone of preferential heating. Applied to two decades of observations by the Wind spacecraft, our results are consistent with preferential heating being active in a zone extending from the transition region in the lower corona to an outer boundary 20-40 solar radii from the Sun, producing a steady state super-mass-proportional $\alpha$-to-proton temperature ratio of $5.2-5.3$. Preferential ion heating continues far beyond the transition region and is important for the evolution of both the outer corona and the solar wind. The outer boundary of this zone is well below the orbits of spacecraft at 1 AU and even closer missions such as Helios and MESSENGER, meaning it is likely that no existing mission has directly observed intense preferential heating, just residual signatures. We predict that {Parker Solar Probe} will be the first spacecraft with a perihelia sufficiently close to the Sun to pass through the outer boundary, enter the zone of preferential heating, and directly observe the physical mechanism in action.Comment: 11 pages, 7 figures, accepted for publication in the Astrophysical Journal on 1 August 201

Evaluation of the angiotensin II receptor blocker azilsartan medoxomil in African-American patients with hypertension

The efficacy and safety of azilsartan medoxomil (AZL-M) were evaluated in African-American patients with hypertension in a 6-week, double-blind, randomized, placebo-controlled trial, for which the primary end point was change from baseline in 24-hour mean systolic blood pressure (BP). There were 413 patients, with a mean age of 52years, 57% women, and baseline 24-hour BP of 146/91mmHg. Treatment differences in 24-hour systolic BP between AZL-M 40mg and placebo (-5.0mmHg; 95% confidence interval, -8.0 to -2.0) and AZL-M 80mg and placebo (-7.8mmHg; 95% confidence interval, -10.7 to -4.9) were significant (P.001 vs placebo for both comparisons). Changes in the clinic BPs were similar to the ambulatory BP results. Incidence rates of adverse events were comparable among the treatment groups, including those of a serious nature. In African-American patients with hypertension, AZL-M significantly reduced ambulatory and clinic BPs in a dose-dependent manner and was well tolerated