The increasing integration of intermittent renewable generation, especially at the distribution level,necessitates advanced planning and optimisation methodologies contingent on the knowledge of thegrid, specifically the admittance matrix capturing the topology and line parameters of an electricnetwork. However, a reliable estimate of the admittance matrix may either be missing or quicklybecome obsolete for temporally varying grids. In this work, we propose a data-driven identificationmethod utilising voltage and current measurements collected from micro-PMUs. More precisely,we first present a maximum likelihood approach and then move towards a Bayesian framework,leveraging the principles of maximum a posteriori estimation. In contrast with most existing con-tributions, our approach not only factors in measurement noise on both voltage and current data,but is also capable of exploiting available a priori information such as sparsity patterns and knownline parameters. Simulations conducted on benchmark cases demonstrate that, compared to otheralgorithms, our method can achieve significantly greater accuracy.

Consider an ensemble of $k$ individual classifiers whose accuracies are known. Upon receiving a test point, each of the classifiers outputs a predicted label and a confidence in its prediction for this particular test point. In this paper, we address the question of whether we can determine the accuracy of the ensemble. Surprisingly, even when classifiers are combined in the statistically optimal way in this setting, the accuracy of the resulting ensemble classifier cannot be computed from the accuracies of the individual classifiers-as would be the case in the standard setting of confidence weighted majority voting. We prove tight upper and lower bounds on the ensemble accuracy. We explicitly construct the individual classifiers that attain the upper and lower bounds: specialists and generalists. Our theoretical results have very practical consequences: (1) If we use ensemble methods and have the choice to construct our individual (independent) classifiers from scratch, then we should aim for specialist classifiers rather than generalists. (2) Our bounds can be used to determine how many classifiers are at least required to achieve a desired ensemble accuracy. Finally, we improve our bounds by considering the mutual information between the true label and the individual classifier's output.

This work was developed aiming to employ Statistical techniques to the field of Music Emotion Recognition, a well-recognized area within the Signal Processing world, but hardly explored from the statistical point of view. Here, we opened several possibilities within the field, applying modern Bayesian Statistics techniques and developing efficient algorithms, focusing on the applicability of the results obtained. Although the motivation for this project was the development of a emotion-based music recommendation system, its main contribution is a highly adaptable multivariate model that can be useful interpreting any database where there is an interest in applying regularization in an efficient manner. Broadly speaking, we will explore what role a sound theoretical statistical analysis can play in the modeling of an algorithm that is able to understand a well-known database and what can be gained with this kind of approach.

Learning disentangled and interpretable representations is an important step towards accomplishing comprehensive data representations on the manifold. In this paper, we propose a novel representation learning algorithm which combines the inference abilities of Variational Autoencoders (VAE) with the generalization capability of Generative Adversarial Networks (GAN). The proposed model, called InfoVAEGAN, consists of three networks~: Encoder, Generator and Discriminator. InfoVAEGAN aims to jointly learn discrete and continuous interpretable representations in an unsupervised manner by using two different data-free log-likelihood functions onto the variables sampled from the generator's distribution. We propose a two-stage algorithm for optimizing the inference network separately from the generator training. Moreover, we enforce the learning of interpretable representations through the maximization of the mutual information between the existing latent variables and those created through generative and inference processes.

For a multivariate normal distribution, the sparsity of the covariance and precision matrices encodes complete information about independence and conditional independence properties. For general distributions, the covariance and precision matrices reveal correlations and so-called partial correlations between variables, but these do not, in general, have any correspondence with respect to independence properties. In this paper, we prove that, for a certain class of non-Gaussian distributions, these correspondences still hold, exactly for the covariance and approximately for the precision. The distributions -- sometimes referred to as "nonparanormal" -- are given by diagonal transformations of multivariate normal random variables. We provide several analytic and numerical examples illustrating these results.

Gaussian processes with derivative information are useful in many settings where derivative information is available, including numerous Bayesian optimization and regression tasks that arise in the natural sciences. Incorporating derivative observations, however, comes with a dominating $O(N^3D^3)$ computational cost when training on $N$ points in $D$ input dimensions. This is intractable for even moderately sized problems. While recent work has addressed this intractability in the low-$D$ setting, the high-$N$, high-$D$ setting is still unexplored and of great value, particularly as machine learning problems increasingly become high dimensional. In this paper, we introduce methods to achieve fully scalable Gaussian process regression with derivatives using variational inference. Analogous to the use of inducing values to sparsify the labels of a training set, we introduce the concept of inducing directional derivatives to sparsify the partial derivative information of a training set. This enables us to construct a variational posterior that incorporates derivative information but whose size depends neither on the full dataset size $N$ nor the full dimensionality $D$. We demonstrate the full scalability of our approach on a variety of tasks, ranging from a high dimensional stellarator fusion regression task to training graph convolutional neural networks on Pubmed using Bayesian optimization. Surprisingly, we find that our approach can improve regression performance even in settings where only label data is available.

Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and semi-supervised learning. Here we present new approach to manifold hypothesis checking and underlying manifold dimension estimation. In order to do it we use two very different methods simultaneously - one geometric, another probabilistic - and check whether they give the same result. Our geometrical method is a modification for sparse data of a well-known box-counting algorithm for Minkowski dimension calculation. The probabilistic method is new. Although it exploits standard nearest neighborhood distance, it is different from methods which were previously used in such situations. This method is robust, fast and includes special preliminary data transformation. Experiments on real datasets show that the suggested approach based on two methods combination is powerful and effective.

The belief function in Dempster Shafer evidence theory can express more information than the traditional Bayesian distribution. It is widely used in approximate reasoning, decision-making and information fusion. However, its power exponential explosion characteristics leads to the extremely high computational complexity when handling large amounts of elements in classic computers. In order to solve the problem, we encode the basic belief assignment (BBA) into quantum states, which makes each qubit correspond to control an element. Besides the high efficiency, this quantum expression is very conducive to measure the similarity between two BBAs, and the measuring quantum algorithm we come up with has exponential acceleration theoretically compared to the corresponding classical algorithm. In addition, we simulate our quantum version of BBA on Qiskit platform, which ensures the rationality of our algorithm experimentally. We believe our results will shed some light on utilizing the characteristic of quantum computation to handle belief function more conveniently.

Agent Based Modelling (ABM) is a computational framework for simulating the behaviours and interactions of autonomous agents. As Agent Based Models are usually representative of complex systems, obtaining a likelihood function of the model parameters is nearly always intractable. There is a necessity to conduct inference in a likelihood free context in order to understand the model output. Approximate Bayesian Computation is a suitable approach for this inference. It can be applied to an Agent Based Model to both validate the simulation and infer a set of parameters to describe the model. Recent research in ABC has yielded increasingly efficient algorithms for calculating the approximate likelihood. These are investigated and compared using a pedestrian model in the Hamilton CBD.

Due to their increasing spread, confidence in neural network predictions became more and more important. However, basic neural networks do not deliver certainty estimates or suffer from over or under confidence. Many researchers have been working on understanding and quantifying uncertainty in a neural network's prediction. As a result, different types and sources of uncertainty have been identified and a variety of approaches to measure and quantify uncertainty in neural networks have been proposed. This work gives a comprehensive overview of uncertainty estimation in neural networks, reviews recent advances in the field, highlights current challenges, and identifies potential research opportunities. It is intended to give anyone interested in uncertainty estimation in neural networks a broad overview and introduction, without presupposing prior knowledge in this field. A comprehensive introduction to the most crucial sources of uncertainty is given and their separation into reducible model uncertainty and not reducible data uncertainty is presented. The modeling of these uncertainties based on deterministic neural networks, Bayesian neural networks, ensemble of neural networks, and test-time data augmentation approaches is introduced and different branches of these fields as well as the latest developments are discussed. For a practical application, we discuss different measures of uncertainty, approaches for the calibration of neural networks and give an overview of existing baselines and implementations. Different examples from the wide spectrum of challenges in different fields give an idea of the needs and challenges regarding uncertainties in practical applications. Additionally, the practical limitations of current methods for mission- and safety-critical real world applications are discussed and an outlook on the next steps towards a broader usage of such methods is given.