We prove semi-empirical concentration inequalities for random variables which are given as possibly nonlinear functions of independent random variables. These inequalities characterize the concentration of the random variable in terms of the data/distribution-dependent Efron-Stein (ES) estimate of its variance and they do not require any additional assumptions on the moments. In particular, this allows us to state semi-empirical Bernstein inequalities for general functions of unbounded random variables, which gives user-friendly concentration bounds for cases where related methods (entropy method / bounded differences) might be more challenging to apply. We extend these results to Efron-Stein PAC-Bayesian inequalities which hold for arbitrary probability kernels that define a random, data-dependent choice of the function of interest. Finally, we demonstrate a number of applications, including PAC-Bayesian generalization bounds for unbounded loss functions, empirical Bernstein-type generalization bounds, new truncation-free bounds for off-policy evaluation with Weighted Importance Sampling (WIS), and off-policy PAC-Bayesian learning with WIS.

Knowing when a graphical model is perfect to a distribution is essential in order to relate separation in the graph to conditional independence in the distribution, and this is particularly important when performing inference from data. When the model is perfect, there is a one-to-one correspondence between conditional independence statements in the distribution and separation statements in the graph. Previous work has shown that almost all models based on linear directed acyclic graphs as well as Gaussian chain graphs are perfect, the latter of which subsumes Gaussian graphical models (i.e., the undirected Gaussian models) as a special case. However, the complexity of chain graph models leads to a proof of this result which is indirect and mired by the complications of parameterizing this general class. In this paper, we directly approach the problem of perfectness for the Gaussian graphical models, and provide a new proof, via a more transparent parametrization, that almost all such models are perfect. Our approach is based on, and substantially extends, a construction of Ln\v{e}ni\v{c}ka and Mat\'u\v{s} showing the existence of a perfect Gaussian distribution for any graph.

In this paper we present a novel quasi-Newton algorithm for use in stochastic optimisation. Quasi-Newton methods have had an enormous impact on deterministic optimisation problems because they afford rapid convergence and computationally attractive algorithms. In essence, this is achieved by learning the second-order (Hessian) information based on observing first-order gradients. We extend these ideas to the stochastic setting by employing a highly flexible model for the Hessian and infer its value based on observing noisy gradients. In addition, we propose a stochastic counterpart to standard line-search procedures and demonstrate the utility of this combination on maximum likelihood identification for general nonlinear state space models.

Neural networks provide state-of-the-art performance on a variety of tasks. However, they are often overconfident when making predictions. This inability to properly account for uncertainty limits their application to high-risk decision making, active learning and Bayesian optimisation. To address this, Bayesian inference has been proposed as a framework for improving uncertainty estimates. In practice, Bayesian neural networks rely on poorly understood approximations for computational tractability. We prove that two commonly used approximation methods, the factorised Gaussian assumption and Monte Carlo dropout, lead to pathological estimates of the predictive uncertainty in single hidden layer ReLU networks. This indicates that more flexible approximations are needed to obtain reliable uncertainty estimates.

In many scientific contexts, different investigators experiment with or observe different variables with data from a domain in which the distinct variable sets might well be related. This sort of fragmentation sometimes occurs in molecular biology, whether in studies of RNA expression or studies of protein interaction, and it is common in the social sciences. Models are built on the diverse data sets, but combining them can provide a more unified account of the causal processes in the domain. On the other hand, this problem is made challenging by the fact that a variable in one data set may influence variables in another although neither data set contains all of the variables involved. Several authors have proposed using conditional independence properties of fragmentary (marginal) data collections to form unified causal explanations when it is assumed that the data have a common causal explanation but cannot be merged to form a unified dataset. These methods typically return a large number of alternative causal models. The first part of the thesis shows that marginal datasets contain extra information that can be used to reduce the number of possible models, in some cases yielding a unique model.

Power load forecast with Machine Learning is a fairly mature application of artificial intelligence and it is indispensable in operation, control and planning. Data selection techniqies have been hardly used in this application. However, the use of such techniques could be beneficial provided the assumption that the data is identically distributed is clearly not true in load forecasting, but it is cyclostationary. In this work we present a fully automatic methodology to determine what are the most adequate data to train a predictor which is based on a full Bayesian probabilistic model. We assess the performance of the method with experiments based on real publicly available data recorded from several years in the United States of America.

Link: Bayesian Machine Learning in Python: A/B Testing Udemy In this course, while we will do traditional A/B testing in order to appreciate its complexity, what we will eventually get to is the Bayesian machine learning way of doing things. First, we'll see if we can improve on traditional A/B testing with adaptive methods. These all help you solve the explore-exploit dilemma. Bestseller Created by Lazy Programmer Inc What you'll learn Use adaptive algorithms to improve A/B testing performance Understand the difference between Bayesian and frequentist statistics Apply Bayesian methods to A/B testing In this course, while we will do traditional A/B testing in order to appreciate its complexity, what we will eventually get to is the Bayesian machine learning way of doing things. First, we'll see if we can improve on traditional A/B testing with adaptive methods.

Reasoning based on causality, instead of association has been considered as a key ingredient towards real machine intelligence. However, it is a challenging task to infer causal relationship/structure among variables. In recent years, an Independent Mechanism (IM) principle was proposed, stating that the mechanism generating the cause and the one mapping the cause to the effect are independent. As the conjecture, it is argued that in the causal direction, the conditional distributions instantiated at different value of the conditioning variable have less variation than the anti-causal direction. Existing state-of-the-arts simply compare the variance of the RKHS mean embedding norms of these conditional distributions. In this paper, we prove that this norm-based approach sacrifices important information of the original conditional distributions. We propose a Kernel Intrinsic Invariance Measure (KIIM) to capture higher order statistics corresponding to the shapes of the density functions. We show our algorithm can be reduced to an eigen-decomposition task on a kernel matrix measuring intrinsic deviance/invariance. Causal directions can then be inferred by comparing the KIIM scores of two hypothetic directions. Experiments on synthetic and real data are conducted to show the advantages of our methods over existing solutions.

Probabilistic modeling and inference are core tools in diverse fields including statistics, machine learning, computer vision, cognitive science, robotics, natural language processing, and artificial intelligence. To meet the functional requirements of applications, practitioners use a broad range of modeling techniques and approximate inference algorithms. However, implementing inference algorithms is often difficult and error prone. Gen simplifies the use of probabilistic modeling and inference, by providing modeling languages in which users express models, and high-level programming constructs that automate aspects of inference.

It is well known in astronomy that propagating non-Gaussian prediction uncertainty in photometric redshift estimates is key to reducing bias in downstream cosmological analyses. Similarly, likelihood-free inference approaches, which are beginning to emerge as a tool for cosmological analysis, require the full uncertainty landscape of the parameters of interest given observed data. However, most machine learning (ML) based methods with open-source software target point prediction or classification, and hence fall short in quantifying uncertainty in complex regression and parameter inference settings such as the applications mentioned above. As an alternative to methods that focus on predicting the response (or parameters) $\mathbf{y}$ from features $\mathbf{x}$, we provide nonparametric conditional density estimation (CDE) tools for approximating and validating the entire probability density $\mathrm{p}(\mathbf{y} \mid \mathbf{x})$ given training data for $\mathbf{x}$ and $\mathbf{y}$. As there is no one-size-fits-all CDE method, the goal of this work is to provide a comprehensive range of statistical tools and open-source software for nonparametric CDE and method assessment which can accommodate different types of settings and which in addition can easily be fit to the problem at hand. Specifically, we introduce CDE software packages in $\texttt{Python}$ and $\texttt{R}$ based on four ML prediction methods adapted and optimized for CDE: $\texttt{NNKCDE}$, $\texttt{RFCDE}$, $\texttt{FlexCode}$, and $\texttt{DeepCDE}$. Furthermore, we present the $\texttt{cdetools}$ package, which includes functions for computing a CDE loss function for model selection and tuning of parameters, together with diagnostics functions. We provide sample code in $\texttt{Python}$ and $\texttt{R}$ as well as examples of applications to photometric redshift estimation and likelihood-free cosmology via CDE.