In most relevant cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior deriving from the error in the numerical solver. To compare a numerical and the theoretical posterior distributions we propose to use Bayes Factors (BF), considering both of them as models for the data at hand. We prove that the theoretical vs a numerical posterior BF tends to 1, in the same order (of the step size used) as the numerical forward map solver does. For higher order solvers (eg. Runge-Kutta) the Bayes Factor is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error free posteriors. Two examples are presented where nearly 90% CPU time is saved while all inference results are identical to using a solver with a much finer time step.

Bayesian Analysis of ODE's: solver optimal accuracy and Bayes factors

In this article, we address the parameter estimation of micromotion targets in synthetic aperture radar (SAR), where scattering parameters and micromotion parameters of targets are coupled resulting in a nonlinear parameter estimation problem. The conventional methods address this nonlinear problem by matched filter, which are computationally expensive and of lower resolutions. In contrast, we address this problem by linearizing the forward model as a linear combination of elements of an over-complete dictionary. The essential idea of sparse signal representation models comes from the fact that SAR micromotion targets are sparsely distributed in the observation scene. Accordingly, we propose to jointly estimate the target micromotion and scattering parameters via a Bayesian approach with sparsity-inducing priors. In addition, we present a variational approximation framework for Bayesian computation. Numerical simulations demonstrate the proposed sparsity-inducing reconstruction method achieves higher resolution and better performance with smaller measures compared to conventional methods.

/pdf/parameter-estimation-for-sar-micromotion-target-based-on-2jv0m60iz5.pdf

Parameter estimation for SAR micromotion target based on sparse signal representation

In most cases in the Bayesian analysis of ODE inverse problems, a numerical solver needs to be used. Therefore, we cannot work with the exact theoretical posterior distribution but only with an approximate posterior derived from the error in the numerical solver. To compare an approximate posterior distribution with the theoretical one, we propose using Bayes factors (BFs), considering both of them as models for the data at hand. From a theoretical point of view, we prove that the theoretical vs. numerical posterior BF tends to 1, in the same order as the numerical solver used. In practice, we illustrate the fact that for higher order solvers (e.g., Runge--Kutta) the BF is already nearly 1 for step sizes that would take far less computational effort. Considerable CPU time may be saved by using coarser solvers that nevertheless produce practically error-free posteriors. Two examples are presented where nearly 90% CPU time is saved, with all inference results being identical to those obtained using a solver...

Bayesian Analysis of ODEs: Solver Optimal Accuracy and Bayes Factors

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 approximate versions of such solutions and lead to a numerical/approximate posterior distribution. Recently several results have been published on the regularity conditions required on such numerical methods to ensure converge of the numerical to the theoretical posterior. However, more practical guidelines are needed to ensure a suitable working numerical posterior. ]Capistran2016] prove for ODE's that the Bayes Factor of the approximate vs the theoretical model tends to 1 in the same order as the numerical method order. In this work we generalize the latter paper in that we consider 1) also PDE's, 2) correlated observations, 3) practical guidelines in a multidimensional setting and 4) explore the use of expected Bayes Factors. This permits us to obtain bounds on the absolute global errors to be tolerated by the FM numerical solver, which we illustrate with some examples. Since the Bayes Factor is kept above 0.95 we expect that the resulting numerical posterior is basically indistinguishable from the theoretical posterior, even though we are using an approximate numerical FM. The method is illustrated with some examples using synthetic data.

Numerical posterior distribution error control and expected Bayes Factors in the bayesian Uncertainty Quantification of Inverse Problems

In this paper, we address the numerical posterior error control problem for the Bayesian approach to inverse problems or recently known as Bayesian Uncertainty Quantification (UQ). We generalize the results of () to (a priori) expected Bayes factors (BF) and in a more general, infinite-dimensional setting. In this inverse problem, the unavoidable numerical approximation of the Forward Map (FM, i.e., the regressor function), arising from the numerical solution of a system of differential equations, demands error estimates of the corresponding approximate numerical posterior distribution. Our approach is to make such comparisons in the setting of Bayesian model selection and BFs. The main result of this paper is a bound on the absolute global error tolerated by the numerical solver of the FM in order to keep the BF of the numerical versus the theoretical posterior near one. For two examples, we provide a detailed analysis of the computation and implementation of the introduced bound. Furthermore, we show that the resulting numerical posterior turns out to be nearly identical from the theoretical posterior, given the control of the BF near one.

/pdf/error-control-of-the-numerical-posterior-with-bayes-factors-3vvceko3ak.pdf

Error Control of the Numerical Posterior with Bayes Factors in Bayesian Uncertainty Quantification

We develop a recognition-oriented SAR image formation framework from a Bayesian perspective In our framework, the prior knowledge of the targets and scene will be treated as restriction to the expected image based on Bayesian inference within an inverse problem frame As a result, an image beneficial for recognition with background suppression and the important feature enhancement can be achieved Results on both simulated and MSTAR data validate the performance of our proposed approach

http://ieeexplore.ieee.org/iel5/6082084/6086911/06087035.pdf

Recognition-oriented Bayesian SAR imaging

Signal parameter estimation is a crucial issue in SAR/ISAR imaging, especially for multicomponent linear frequency modulated (LFM) signal with single degree of freedom. A new method of parameter estimation based on sparse signal representation is presented in this paper, which expands signal on a set of over-complete basis. The method is analyzed and validated for performance through simulation, with three commonly used signal sparse representation algorithms compared, including BP, FOCUSS and Sparse Bayesian Learning. The result shows that Sparse Bayesian Learning performs better in sparse components than the other two algorithms, which can estimate signal parameters more efficiently.

A new method for parameter estimation of multicomponent LFM signal based on sparse signal representation

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Recognition-oriented Bayesian SAR imaging

A new method for parameter estimation of multicomponent LFM signal based on sparse signal representation