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.

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.

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.