13,280 research outputs found
Hierarchical Bayesian level set inversion
The level set approach has proven widely successful in the study of inverse problems for inter- faces, since its systematic development in the 1990s. Re- cently it has been employed in the context of Bayesian inversion, allowing for the quantification of uncertainty within the reconstruction of interfaces. However the Bayesian approach is very sensitive to the length and amplitude scales in the prior probabilistic model. This paper demonstrates how the scale-sensitivity can be cir- cumvented by means of a hierarchical approach, using a single scalar parameter. Together with careful con- sideration of the development of algorithms which en- code probability measure equivalences as the hierar- chical parameter is varied, this leads to well-defined Gibbs based MCMC methods found by alternating Metropolis-Hastings updates of the level set function and the hierarchical parameter. These methods demon- strably outperform non-hierarchical Bayesian level set methods
Hierarchical Bayesian auto-regressive models for large space time data with applications to ozone concentration modelling
Increasingly large volumes of space-time data are collected everywhere by mobile computing applications, and in many of these cases temporal data are obtained by registering events, for example telecommunication or web traffic data. Having both the spatial and temporal dimensions adds substantial complexity to data analysis and inference tasks. The computational complexity increases rapidly for fitting Bayesian hierarchical models, as such a task involves repeated inversion of large matrices. The primary focus of this paper is on developing space-time auto-regressive models under the hierarchical Bayesian setup. To handle large data sets, a recently developed Gaussian predictive process approximation method (Banerjee et al. [1]) is extended to include auto-regressive terms of latent space-time processes. Specifically, a space-time auto-regressive process, supported on a set of a smaller number of knot locations, is spatially interpolated to approximate the original space-time process. The resulting model is specified within a hierarchical Bayesian framework and Markov chain Monte Carlo techniques are used to make inference. The proposed model is applied for analysing the daily maximum 8-hour average ground level ozone concentration data from 1997 to 2006 from a large study region in the eastern United States. The developed methods allow accurate spatial prediction of a temporally aggregated ozone summary, known as the primary ozone standard, along with its uncertainty, at any unmonitored location during the study period. Trends in spatial patterns of many features of the posterior predictive distribution of the primary standard, such as the probability of non-compliance with respect to the standard, are obtained and illustrated
Bayesian Coronal Seismology
In contrast to the situation in a laboratory, the study of the solar
atmosphere has to be pursued without direct access to the physical conditions
of interest. Information is therefore incomplete and uncertain and inference
methods need to be employed to diagnose the physical conditions and processes.
One of such methods, solar atmospheric seismology, makes use of observed and
theoretically predicted properties of waves to infer plasma and magnetic field
properties. A recent development in solar atmospheric seismology consists in
the use of inversion and model comparison methods based on Bayesian analysis.
In this paper, the philosophy and methodology of Bayesian analysis are first
explained. Then, we provide an account of what has been achieved so far from
the application of these techniques to solar atmospheric seismology and a
prospect of possible future extensions.Comment: 19 pages, accepted in Advances in Space Researc
A variational Bayesian method for inverse problems with impulsive noise
We propose a novel numerical method for solving inverse problems subject to
impulsive noises which possibly contain a large number of outliers. The
approach is of Bayesian type, and it exploits a heavy-tailed t distribution for
data noise to achieve robustness with respect to outliers. A hierarchical model
with all hyper-parameters automatically determined from the given data is
described. An algorithm of variational type by minimizing the Kullback-Leibler
divergence between the true posteriori distribution and a separable
approximation is developed. The numerical method is illustrated on several one-
and two-dimensional linear and nonlinear inverse problems arising from heat
conduction, including estimating boundary temperature, heat flux and heat
transfer coefficient. The results show its robustness to outliers and the fast
and steady convergence of the algorithm.Comment: 20 pages, to appear in J. Comput. Phy
Multi-scale uncertainty quantification in geostatistical seismic inversion
Geostatistical seismic inversion is commonly used to infer the spatial
distribution of the subsurface petro-elastic properties by perturbing the model
parameter space through iterative stochastic sequential
simulations/co-simulations. The spatial uncertainty of the inferred
petro-elastic properties is represented with the updated a posteriori variance
from an ensemble of the simulated realizations. Within this setting, the
large-scale geological (metaparameters) used to generate the petro-elastic
realizations, such as the spatial correlation model and the global a priori
distribution of the properties of interest, are assumed to be known and
stationary for the entire inversion domain. This assumption leads to
underestimation of the uncertainty associated with the inverted models. We
propose a practical framework to quantify uncertainty of the large-scale
geological parameters in seismic inversion. The framework couples
geostatistical seismic inversion with a stochastic adaptive sampling and
Bayesian inference of the metaparameters to provide a more accurate and
realistic prediction of uncertainty not restricted by heavy assumptions on
large-scale geological parameters. The proposed framework is illustrated with
both synthetic and real case studies. The results show the ability retrieve
more reliable acoustic impedance models with a more adequate uncertainty spread
when compared with conventional geostatistical seismic inversion techniques.
The proposed approach separately account for geological uncertainty at
large-scale (metaparameters) and local scale (trace-by-trace inversion)
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
Brain Activity Mapping from MEG Data via a Hierarchical Bayesian Algorithm with Automatic Depth Weighting
A recently proposed iterated alternating sequential (IAS) MEG inverse solver algorithm, based on the coupling of a hierarchical Bayesian model with computationally efficient Krylov subspace linear solver, has been shown to perform well for both superficial and deep brain sources. However, a systematic study of its ability to correctly identify active brain regions is still missing. We propose novel statistical protocols to quantify the performance of MEG inverse solvers, focusing in particular on how their accuracy and precision at identifying active brain regions. We use these protocols for a systematic study of the performance of the IAS MEG inverse solver, comparing it with three standard inversion methods, wMNE, dSPM, and sLORETA. To avoid the bias of anecdotal tests towards a particular algorithm, the proposed protocols are Monte Carlo sampling based, generating an ensemble of activity patches in each brain region identified in a given atlas. The performance in correctly identifying the active areas is measured by how much, on average, the reconstructed activity is concentrated in the brain region of the simulated active patch. The analysis is based on Bayes factors, interpreting the estimated current activity as data for testing the hypothesis that the active brain region is correctly identified, versus the hypothesis of any erroneous attribution. The methodology allows the presence of a single or several simultaneous activity regions, without assuming that the number of active regions is known. The testing protocols suggest that the IAS solver performs well with both with cortical and subcortical activity estimation
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