Bayesian ODE solvers: the maximum a posteriori estimate

There is a growing interest in probabilistic numerical solutions to ordinary differential equations. In this paper, the maximum a posteriori estimate is studied under the class of $$\nu $$ ν times differentiable linear time-invariant Gauss–Markov priors, which can be computed with an iterated extended Kalman smoother. The maximum a posteriori estimate corresponds to an optimal interpolant in the reproducing kernel Hilbert space associated with the prior, which in the present case is equivalent to a Sobolev space of smoothness $$\nu +1$$ ν + 1 . Subject to mild conditions on the vector field, convergence rates of the maximum a posteriori estimate are then obtained via methods from nonlinear analysis and scattered data approximation. These results closely resemble classical convergence results in the sense that a $$\nu $$ ν times differentiable prior process obtains a global order of $$\nu $$ ν , which is demonstrated in numerical examples.

Information Collecting and Dissemination in the Network of Taxpayers: Bayesian Approach

The current research is based on the assumption that the result of tax inspections is not only collection of taxes and fines. The information about audited taxpayers is also collected and helps to renew a priori knowledge of each agent's evasion propensity and obtain new a posteriori estimate of this propensity. In the beginning of the following tax period the fiscal authority can correct auditing strategy using updated information on every taxpayer. Each inspection is considered as a repeated game, in which the choice of agents to audit is associated with their revealed tendency to evade. Taxpayers also renew the information on the number of inspected neighbors using their social connections, represented by networks of various con gurations, and estimate the probability of auditing before the next tax period. Thus, the application of the Bayesian approach to the process of collecting and disseminating information in the network of taxpayers allows to optimize the audit scheme, reducing unnecessary expenses of tax authority and eventually increasing net tax revenue. To illustrate the application of the approach described above to the indicated problem, numerical simulation and scenario analysis were carried out.

Incorporating word embeddings in unsupervised morphological segmentation

We investigate the usage of semantic information for morphological segmentation since words that are derived from each other will remain semantically related. We use mathematical models such as maximum likelihood estimate (MLE) and maximum a posteriori estimate (MAP) by incorporating semantic information obtained from dense word vector representations. Our approach does not require any annotated data which make it fully unsupervised and require only a small amount of raw data together with pretrained word embeddings for training purposes. The results show that using dense vector representations helps in morphological segmentation especially for low-resource languages. We present results for Turkish, English, and German. Our semantic MLE model outperforms other unsupervised models for Turkish language. Our proposed models could be also used for any other low-resource language with concatenative morphology.

Does optimal estimation perform adequately for hyperspectral surface reflectance retrieval?

<p>The current and coming imaging spectroscopy missions (EMIT, ECOSTRESS, AVIRIS-NG), and observables for potential future missions studying Surface Biology and Geology (SBG) observe a wide range of spectral bands, which can be used to infer about surface properties. The current state of the art approach for performing the retrieval of surface reflectance is optimal estimation (OE), which amounts to finding the maximum a posteriori estimate of the surface reflectance, after which the posterior covariance is approximated by linearizing the forward model (Rodgers, 2001). While this method has a principled basis and often performs well, with challenging atmospheres the optimization may fall into local minima, or the estimated posterior mean and covariance may be wrong.  Addressing these failures under realistic observing conditions is particularly important to realize the full potential of upcoming global observations.                                                                                                                                                                        </p><p><br>As a preparation to improving the quality of future retrievals, we evaluate the performance of OE against posteriors generated with advanced Bayesian techniques.  We present results from comparing the OE posterior mean and covariance to the true posterior, as computed by MCMC, for moderately challenging atmospheric conditions, and an instrument configuration consistent with AVIRIS-NG. </p>

Computable upper error bounds for Krylov approximations to matrix exponentials and associated $${\varvec{\varphi }}$$-functions

An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. This means that the deviation of the error estimate from the true error tends to zero faster than the error itself. Furthermore, this result is extended to Krylov approximations of $$\varphi $$φ-functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Alternative error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.

Bayesian Compress Sensing Based Countermeasure Scheme Against the Interrupted Sampling Repeater Jamming

The interrupted sampling repeater jamming (ISRJ) is considered an efficient deception method of jamming for coherent radar detection. However, current countermeasure methods against ISRJ interference may fail in detecting weak echoes, particularly when the transmitting power of the jammer is relatively high. In this paper, we propose a novel countermeasure scheme against ISRJ based on Bayesian compress sensing (BCS), where stable target signal can be reconstructed over a relatively large range of signal-to-noise ratio (SNR) for both single target and multi-target scenarios. By deriving the ISRJ jamming strategy, only the unjammed discontinuous time segments are extracted to build a sparse target model for the reconstruction algorithm. An efficient alternate iteration is applied to optimize and solve the maximum a posteriori estimate (MAP) of the sparse targets model. Simulation results demonstrate the robustness of the proposed scheme with low SNR or large jammer ratio. Moreover, when compared with traditional FFT or greedy sparsity adaptive matching pursuit algorithm (SAMP), the proposed algorithm significantly improves on the aspects of both the grating lobe level and target detection/false detection probability.

Application of a Priori and a Posteriori Estimate on Risk Assessment

The problem of setting the values and interconnections between elements of the models in the safety, protection and security field, appears as the biggest obstacle in taking crisis management decisions. The article attempts to represent a mathematical approach to modify the expected values and interconnections that can occur in the models describing the protected system in order to minimize errors caused by subjectivity. Here presented procedures are described in the examples of their potential use. The main idea is to focus on improving estimates for better response to reality, then to find new estimates, since those would still be weighed down by the subjectivity caused errors. Based on this premise this article attempts to characterize application of mathematical methods on minimizing the subjectivity caused errors in the models in risk assessment.

Bayesian Framework to Quantify Uncertainties in Piezoelectric Energy Harvesters

The interest of this work is to describe a framework that allows the use of the well-known dynamic estimators in piezoelectric harvester (deterministic performance estimators) but taking into account the random error associated to the mathematical model and the uncertainties on the model parameters. The framework presented could be employed to perform Posterior Robust Stochastic Analysis, which is the case when the harvester can be tested or it is already installed and the experimental data is available. In particular, it is introduced a procedure to update the electromechanical properties of PEHs based on Bayesian updating techniques. The mean of the updated electromechanical properties are identified adopting a Maximum a Posteriori estimate while the probability density function associated is obtained by applying a Laplaces asymptotic approximation (updated properties could be expressed as a mean value together a band of confidence). The procedure is exemplified using the experimental characterization of 20 PEHs, all of them with same nominal characteristics. Results show the capability of the procedure to update not only the electromechanical properties of each PEH (mandatory information for the prediction of a particular PEH) but also the characteristics of the whole sample of harvesters (mandatory information for design purposes). The results reveal the importance to include the model parameter uncertainties in order to generate robust predictive tools in energy harvesting. In that sense, the present framework constitutes a powerful tool in the robust design and prediction of piezoelectric energy harvesters performance.