This paper is a review of a particular approach to the method of maximum entropy as a general framework for inference. The discussion emphasizes the pragmatic elements in the derivation. An epistemic notion of information is defined in terms of its relation to the Bayesian beliefs of ideally rational agents. The method of updating from a prior to a posterior probability distribution is designed through an eliminative induction process. The logarithmic relative entropy is singled out as the unique tool for updating that (a) is of universal applicability; (b) that recognizes the value of prior information; and (c) that recognizes the privileged role played by the notion of independence in science. The resulting framework -- the ME method -- can handle arbitrary priors and arbitrary constraints. It includes MaxEnt and Bayes' rule as special cases and, therefore, it unifies entropic and Bayesian methods into a single general inference scheme. The ME method goes beyond the mere selection of a single posterior, but also addresses the question of how much less probable other distributions might be, which provides a direct bridge to the theories of fluctuations and large deviations.

Our evolution as a species made a huge step forward when we understood the relationships between causes and effects. These associations may be trivial for some events, but they are not in complex scenarios. To rigorously prove that some occurrences are caused by others, causal theory and causal inference were formalized, introducing the do-operator and its associated rules. The main goal of this report is to review and implement in Python some algorithms to compute conditional and non-conditional causal queries from observational data. To this end, we first present some basic background knowledge on probability and graph theory, before introducing important results on causal theory, used in the construction of the algorithms. We then thoroughly study the identification algorithms presented by Shpitser and Pearl in 2006 [SP 2006a, SP 2006b], explaining our implementation in Python alongside. The main identification algorithm can be seen as a repeated application of the rules of do-calculus, and it eventually either returns an expression for the causal query from experimental probabilities or fails to identify the causal effect, in which case the effect is non-identifiable. We introduce our newly developed Python library and give some usage examples. Keywords DAG, do-calculus, causality, causal model, identifiability, graph, C-component, hedge, d-separation.

However, Our contributions in this work are: instead of computing the curvature matrix, we show that, under some regularity conditions, - We show that under some regularity conditions, a diagonal the Laplace approximation can be easily constructed Laplace approximation can be constructed without using the gradient second moment. This computing anything besides what is already being quantity is already estimated by many exponential computed by widely used optimizers; moving average variants of Adagrad such as Adam and RMSprop, but is traditionally discarded - We qualitatively compare this approximation with after training. We show that our method (L2M) methods such as deep ensembles (Lakshminarayanan does not require changes in models or optimization, et al., 2017), MC Dropout (Gal & Ghahramani, 2016), can be implemented in a few lines of code Hamiltonian Monte Carlo (HMC) (Cobb & Jalaian, to yield reasonable results, and it does not require 2020), among others; any extra computational steps besides what is already - We also show that our approximation is orthogonal being computed by optimizers, without introducing to methods such as ensembling (Lakshminarayanan any new hyperparameter. We hope our et al., 2017) and does not require changing training method can open new research directions on using procedures, estimating new quantities, or adding new quantities already computed by optimizers for hyperparameters.

Subspace-valued functions arise in a wide range of problems, including parametric reduced order modeling (PROM). In PROM, each parameter point can be associated with a subspace, which is used for Petrov-Galerkin projections of large system matrices. Previous efforts to approximate such functions use interpolations on manifolds, which can be inaccurate and slow. To tackle this, we propose a novel Bayesian nonparametric model for subspace prediction: the Gaussian Process Subspace regression (GPS) model. This method is extrinsic and intrinsic at the same time: with multivariate Gaussian distributions on the Euclidean space, it induces a joint probability model on the Grassmann manifold, the set of fixed-dimensional subspaces. The GPS adopts a simple yet general correlation structure, and a principled approach for model selection. Its predictive distribution admits an analytical form, which allows for efficient subspace prediction over the parameter space. For PROM, the GPS provides a probabilistic prediction at a new parameter point that retains the accuracy of local reduced models, at a computational complexity that does not depend on system dimension, and thus is suitable for online computation. We give four numerical examples to compare our method to subspace interpolation, as well as two methods that interpolate local reduced models. Overall, GPS is the most data efficient, more computationally efficient than subspace interpolation, and gives smooth predictions with uncertainty quantification.

We show that many machine-learning algorithms are specific instances of a single algorithm called the Bayesian learning rule. The rule, derived from Bayesian principles, yields a wide-range of algorithms from fields such as optimization, deep learning, and graphical models. This includes classical algorithms such as ridge regression, Newton's method, and Kalman filter, as well as modern deep-learning algorithms such as stochastic-gradient descent, RMSprop, and Dropout. The key idea in deriving such algorithms is to approximate the posterior using candidate distributions estimated by using natural gradients. Different candidate distributions result in different algorithms and further approximations to natural gradients give rise to variants of those algorithms. Our work not only unifies, generalizes, and improves existing algorithms, but also helps us design new ones.

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.