Recent progress in machine learning methods, and the emerging availability of programmable interfaces for scanning probe microscopes (SPMs), have propelled automated and autonomous microscopies to the forefront of attention of the scientific community. However, enabling automated microscopy requires the development of task-specific machine learning methods, understanding the interplay between physics discovery and machine learning, and fully defined discovery workflows. This, in turn, requires balancing the physical intuition and prior knowledge of the domain scientist with rewards that define experimental goals and machine learning algorithms that can translate these to specific experimental protocols. Here, we discuss the basic principles of Bayesian active learning and illustrate its applications for SPM. We progress from the Gaussian Process as a simple data-driven method and Bayesian inference for physical models as an extension of physics-based functional fits to more complex deep kernel learning methods, structured Gaussian Processes, and hypothesis learning. These frameworks allow for the use of prior data, the discovery of specific functionalities as encoded in spectral data, and exploration of physical laws manifesting during the experiment. The discussed framework can be universally applied to all techniques combining imaging and spectroscopy, SPM methods, nanoindentation, electron microscopy and spectroscopy, and chemical imaging methods, and can be particularly impactful for destructive or irreversible measurements.

Normalizing Flows (NFs) are flexible explicit generative models that have been shown to accurately model complex real-world data distributions. However, their invertibility constraint imposes limitations on data distributions that reside on lower dimensional manifolds embedded in higher dimensional space. Practically, this shortcoming is often bypassed by adding noise to the data which impacts the quality of the generated samples. In contrast to prior work, we approach this problem by generating samples from the original data distribution given full knowledge about the perturbed distribution and the noise model. To this end, we establish that NFs trained on perturbed data implicitly represent the manifold in regions of maximum likelihood. Then, we propose an optimization objective that recovers the most likely point on the manifold given a sample from the perturbed distribution. Finally, we focus on 3D point clouds for which we utilize the explicit nature of NFs, i.e. surface normals extracted from the gradient of the log-likelihood and the log-likelihood itself, to apply Poisson surface reconstruction to refine generated point sets.

Sharing medical data between institutions is difficult in practice due to data protection laws and official procedures within institutions. Therefore, most existing algorithms are trained on relatively small electroencephalogram (EEG) data sets which is likely to be detrimental to prediction accuracy. In this work, we simulate a case when the data can not be shared by splitting the publicly available data set into disjoint sets representing data in individual institutions. We propose to train a (local) detector in each institution and aggregate their individual predictions into one final prediction. Four aggregation schemes are compared, namely, the majority vote, the mean, the weighted mean and the Dawid-Skene method. The method was validated on an independent data set using only a subset of EEG channels. The ensemble reaches accuracy comparable to a single detector trained on all the data when sufficient amount of data is available in each institution. The weighted mean aggregation scheme showed best performance, it was only marginally outperformed by the Dawid--Skene method when local detectors approach performance of a single detector trained on all available data.

Bonus-Malus Systems (BMSs) are nowadays widely employed in automobile insurance to dynamically adjust a premium based on a customer's claims experience. The intuition behind these posterior ratemaking systems is that as we observe more claiming behavior, we learn more about the underlying risk profile. These systems are therefore a commercially attractive form of experience rating, in which we correct the prior premium for past claims to reflect our updated beliefs about a customer's risk profile. Moreover, they traditionally consider a customer's number of claims irrespective of their sizes and thus implicitly assume independence between the claim counts and sizes (Hey, 1970; Denuit et al., 2007; Boucher and Inoussa, 2014; Verschuren, 2021). Alternative Bayesian forms of experience rating typically depend only on the frequency component as well or consider the two components separately (see, e.g., Denuit and Lang (2004); Bühlmann and Gisler (2005); Mahmoudvand and Hassani (2009); Bermúdez and Karlis (2011, 2017)).

Tensor decomposition is a powerful tool for extracting physically meaningful latent factors from multi-dimensional nonnegative data, and has been an increasing interest in a variety of fields such as image processing, machine learning, and computer vision. In this paper, we propose a sparse nonnegative Tucker decomposition and completion method for the recovery of underlying nonnegative data under noisy observations. Here the underlying nonnegative data tensor is decomposed into a core tensor and several factor matrices with all entries being nonnegative and the factor matrices being sparse. The loss function is derived by the maximum likelihood estimation of the noisy observations, and the $\ell_0$ norm is employed to enhance the sparsity of the factor matrices. We establish the error bound of the estimator of the proposed model under generic noise scenarios, which is then specified to the observations with additive Gaussian noise, additive Laplace noise, and Poisson observations, respectively. Our theoretical results are better than those by existing tensor-based or matrix-based methods. Moreover, the minimax lower bounds are shown to be matched with the derived upper bounds up to logarithmic factors. Numerical examples on both synthetic and real-world data sets demonstrate the superiority of the proposed method for nonnegative tensor data completion.

In second-order uncertain Bayesian networks, the conditional probabilities are only known within distributions, i.e., probabilities over probabilities. The delta-method has been applied to extend exact first-order inference methods to propagate both means and variances through sum-product networks derived from Bayesian networks, thereby characterizing epistemic uncertainty, or the uncertainty in the model itself. Alternatively, second-order belief propagation has been demonstrated for polytrees but not for general directed acyclic graph structures. In this work, we extend Loopy Belief Propagation to the setting of second-order Bayesian networks, giving rise to Second-Order Loopy Belief Propagation (SOLBP). For second-order Bayesian networks, SOLBP generates inferences consistent with those generated by sum-product networks, while being more computationally efficient and scalable.

When sampling for Bayesian inference, one popular approach is to use Hamiltonian Monte Carlo (HMC) and specifically the No-U-Turn Sampler (NUTS) which automatically decides the end time of the Hamiltonian trajectory. However, HMC and NUTS can require numerous numerical gradients of the target density, and can prove slow in practice. We propose Hamiltonian neural networks (HNNs) with HMC and NUTS for solving Bayesian inference problems. Once trained, HNNs do not require numerical gradients of the target density during sampling. Moreover, they satisfy important properties such as perfect time reversibility and Hamiltonian conservation, making them well-suited for use within HMC and NUTS because stationarity can be shown. We also propose an HNN extension called latent HNNs (L-HNNs), which are capable of predicting latent variable outputs. Compared to HNNs, L-HNNs offer improved expressivity and reduced integration errors. Finally, we employ L-HNNs in NUTS with an online error monitoring scheme to prevent sample degeneracy in regions of low probability density. We demonstrate L-HNNs in NUTS with online error monitoring on several examples involving complex, heavy-tailed, and high-local-curvature probability densities. Overall, L-HNNs in NUTS with online error monitoring satisfactorily inferred these probability densities. Compared to traditional NUTS, L-HNNs in NUTS with online error monitoring required 1--2 orders of magnitude fewer numerical gradients of the target density and improved the effective sample size (ESS) per gradient by an order of magnitude.

This paper proposes a novel adaptive sample space-based Viterbi algorithm for target localization in an online manner. The method relies on discretizing the target's motion space into cells representing a finite number of hidden states. Then, the most probable trajectory of the tracked target is computed via dynamic programming in a Hidden Markov Model (HMM) framework. The proposed method uses a Bayesian estimation framework which is neither limited to Gaussian noise models nor requires a linearized target motion model or sensor measurement models. However, an HMM-based approach to localization can suffer from poor computational complexity in scenarios where the number of hidden states increases due to high-resolution modeling or target localization in a large space. To improve this poor computational complexity, this paper proposes a belief propagation in the most probable belief space with a low to high-resolution sequentially, reducing the required resources significantly. The proposed method is inspired by the k-d Tree algorithm (e.g., quadtree) commonly used in the computer vision field. Experimental tests using an ultra-wideband (UWB) sensor network demonstrate our results.

Inferring linear relationships lies at the heart of many empirical investigations. A measure of linear dependence should correctly evaluate the strength of the relationship as well as qualify whether it is meaningful for the population. Pearson's correlation coefficient (PCC), the \textit{de-facto} measure for bivariate relationships, is known to lack in both regards. The estimated strength $r$ maybe wrong due to limited sample size, and nonnormality of data. In the context of statistical significance testing, erroneous interpretation of a $p$-value as posterior probability leads to Type I errors -- a general issue with significance testing that extends to PCC. Such errors are exacerbated when testing multiple hypotheses simultaneously. To tackle these issues, we propose a machine-learning-based predictive data calibration method which essentially conditions the data samples on the expected linear relationship. Calculating PCC using calibrated data yields a calibrated $p$-value that can be interpreted as posterior probability together with a calibrated $r$ estimate, a desired outcome not provided by other methods. Furthermore, the ensuing independent interpretation of each test might eliminate the need for multiple testing correction. We provide empirical evidence favouring the proposed method using several simulations and application to real-world data.

Understanding the spread of COVID-19 has been the subject of numerous studies, highlighting the significance of reliable epidemic models. Here, we introduce a novel epidemic model using a latent Hawkes process with temporal covariates for modelling the infections. Unlike other models, we model the reported cases via a probability distribution driven by the underlying Hawkes process. Modelling the infections via a Hawkes process allows us to estimate by whom an infected individual was infected. We propose a Kernel Density Particle Filter (KDPF) for inference of both latent cases and reproduction number and for predicting the new cases in the near future. The computational effort is proportional to the number of infections making it possible to use particle filter type algorithms, such as the KDPF. We demonstrate the performance of the proposed algorithm on synthetic data sets and COVID-19 reported cases in various local authorities in the UK, and benchmark our model to alternative approaches.