In sponsored search advertising (SSA), advertisers need to select keywords and determine matching types for selected keywords simultaneously, i.e., keyword targeting. An optimal keyword targeting strategy guarantees reaching the right population effectively. This paper aims to address the keyword targeting problem, which is a challenging task because of the incomplete information of historical advertising performance indices and the high uncertainty in SSA environments. First, we construct a data distribution estimation model and apply a Markov Chain Monte Carlo method to make inference about unobserved indices (i.e., impression and click-through rate) over three keyword matching types (i.e., broad, phrase and exact). Second, we formulate a stochastic keyword targeting model (BB-KSM) combining operations of keyword selection and keyword matching to maximize the expected profit under the chance constraint of the budget, and develop a branch-and-bound algorithm incorporating a stochastic simulation process for our keyword targeting model. Finally, based on a realworld dataset collected from field reports and logs of past SSA campaigns, computational experiments are conducted to evaluate the performance of our keyword targeting strategy. Experimental results show that, (a) BB-KSM outperforms seven baselines in terms of profit; (b) BB-KSM shows its superiority as the budget increases, especially in situations with more keywords and keyword combinations; (c) the proposed data distribution estimation approach can effectively address the problem of incomplete performance indices over the three matching types and in turn significantly promotes the performance of keyword targeting decisions. This research makes important contributions to the SSA literature and the results offer critical insights into keyword management for SSA advertisers.

Multimodal distributions of some physics based model parameters are often encountered in engineering due to different situations such as a change in some environmental conditions, and the presence of some types of damage and nonlinearity. In statistical model updating, for locally identifiable parameters, it can be anticipated that multi-modal posterior distributions would be found. The full characterization of these multi-modal distributions is important as methodologies for structural condition monitoring in structures are frequently based in the comparison of the damaged and healthy models of the structure. The characterization of posterior multi-modal distributions using state-of-the-art sampling techniques would require a large number of simulations of expensive to run physics-based models. Therefore, when a limited number of simulations can be run, as it often occurs in engineering, the traditional sampling techniques would not be able to capture accurately the multimodal distributions. This could potentially lead to large numerical errors when assessing the performance of an engineering structure under uncertainty.

Complex high-dimensional co-occurrence data are increasingly popular from a complex system of interacting physical, biological and social processes in discretely indexed modifiable areal units or continuously indexed locations of a study region for landscape-based mechanism. Modeling, predicting and interpreting complex co-occurrences are very general and fundamental problems of statistical and machine learning in a broad variety of real-world modern applications. Probability and conditional probability of co-occurrence are introduced by being defined in a general setting with set functions to develop a rigorous measure-theoretic foundation for the inherent challenge of data sparseness. The data sparseness is a main challenge inherent to probabilistic modeling and reasoning of co-occurrence in statistical inference. The behavior of a class of natural integrals called E-integrals is investigated based on the defined conditional probability of co-occurrence. The results on the properties of E-integral are presented. The paper offers a novel measure-theoretic framework where E-integral as a basic measure-theoretic concept can be the starting point for the expectation functional approach preferred by Whittle (1992) and Pollard (2001) to the development of probability theory for the inherent challenge of co-occurrences emerging in modern high-dimensional co-occurrence data problems and opens the doors to more sophisticated and interesting research in complex high-dimensional co-occurrence data science.

Machine learning (ML) formalizes the problem of getting computers to learn from experience as optimization of performance according to some metric(s) on a set of data examples. This is in contrast to requiring behaviour specified in advance (e.g. by hard-coded rules). Formalization of this problem has enabled great progress in many applications with large real-world impact, including translation, speech recognition, self-driving cars, and drug discovery. But practical instantiations of this formalism make many assumptions - for example, that data are i.i.d.: independent and identically distributed - whose soundness is seldom investigated. And in making great progress in such a short time, the field has developed many norms and ad-hoc standards, focused on a relatively small range of problem settings. As applications of ML, particularly in artificial intelligence (AI) systems, become more pervasive in the real world, we need to critically examine these assumptions, norms, and problem settings, as well as the methods that have become de-facto standards. There is much we still do not understand about how and why deep networks trained with stochastic gradient descent are able to generalize as well as they do, why they fail when they do, and how they will perform on out-of-distribution data. In this thesis I cover some of my work towards better understanding deep net generalization, identify several ways assumptions and problem settings fail to generalize to the real world, and propose ways to address those failures in practice.

The problem of classifying turbulent environments from partial observation is key for some theoretical and applied fields, from engineering to earth observation and astrophysics, e.g. to precondition searching of optimal control policies in different turbulent backgrounds, to predict the probability of rare events and/or to infer physical parameters labelling different turbulent set-ups. To achieve such goal one can use different tools depending on the system's knowledge and on the quality and quantity of the accessible data. In this context, we assume to work in a model-free setup completely blind to all dynamical laws, but with a large quantity of (good quality) data for training. As a prototype of complex flows with different attractors, and different multi-scale statistical properties we selected 10 turbulent 'ensembles' by changing the rotation frequency of the frame of reference of the 3d domain and we suppose to have access to a set of partial observations limited to the instantaneous kinetic energy distribution in a 2d plane, as it is often the case in geophysics and astrophysics. We compare results obtained by a Machine Learning (ML) approach consisting of a state-of-the-art Deep Convolutional Neural Network (DCNN) against Bayesian inference which exploits the information on velocity and enstrophy moments. First, we discuss the supremacy of the ML approach, presenting also results at changing the number of training data and of the hyper-parameters. Second, we present an ablation study on the input data aimed to perform a ranking on the importance of the flow features used by the DCNN, helping to identify the main physical contents used by the classifier. Finally, we discuss the main limitations of such data-driven methods and potential interesting applications.

The goal of this thesis is to develop the optimisation and generalisation theoretic foundations of learning in artificial neural networks. On optimisation, a new theoretical framework is proposed for deriving architecture-dependent first-order optimisation algorithms. The approach works by combining a "functional majorisation" of the loss function with "architectural perturbation bounds" that encode an explicit dependence on neural architecture. The framework yields optimisation methods that transfer hyperparameters across learning problems. On generalisation, a new correspondence is proposed between ensembles of networks and individual networks. It is argued that, as network width and normalised margin are taken large, the space of networks that interpolate a particular training set concentrates on an aggregated Bayesian method known as a "Bayes point machine". This correspondence provides a route for transferring PAC-Bayesian generalisation theorems over to individual networks. More broadly, the correspondence presents a fresh perspective on the role of regularisation in networks with vastly more parameters than data.

Identifying unfamiliar inputs, also known as out-of-distribution (OOD) detection, is a crucial property of any decision making process. A simple and empirically validated technique is based on deep ensembles where the variance of predictions over different neural networks acts as a substitute for input uncertainty. Nevertheless, a theoretical understanding of the inductive biases leading to the performance of deep ensemble's uncertainty estimation is missing. To improve our description of their behavior, we study deep ensembles with large layer widths operating in simplified linear training regimes, in which the functions trained with gradient descent can be described by the neural tangent kernel. We identify two sources of noise, each inducing a distinct inductive bias in the predictive variance at initialization. We further show theoretically and empirically that both noise sources affect the predictive variance of non-linear deep ensembles in toy models and realistic settings after training. Finally, we propose practical ways to eliminate part of these noise sources leading to significant changes and improved OOD detection in trained deep ensembles.

Extreme multi-label classification (XMC), or extreme multi-label learning, aims to find the relevant labels for a data input from an enormous label space. With increasingly growing information in the era of big data, XMC has become more and more important, and has been widely applied to various real-world applications, such as advertising [37], product search [9], and document retrieval [6]. However, for domains with potential high risks from mistakes like public health and medicine, it is crucial to model the predictive uncertainty for their downstream XMC applications like food classification [54] and medical diagnosis [2]. In particular, an input sometimes could have only few or even no matches in the label space, so the outputs could be noisy without uncertainty quantification. It is also insufficient to only model uncertainty for the entire input since XMC models could have different confidence for each label among the whole enormous space. To estimate predictive uncertainty, Bayesian and probabilistic models [20] are inherently applicable because variance can intrinsically be viewed as an uncertainty measurement. However, although Bayesian approaches are mathematically grounded to model uncertainty, their computational costs are usually exorbitant for large-scale data. To address this issue, the most popular solution is to approximate Bayesian inference by sampling models as an ensemble [17].

In this article, we have seen the Bayesian approach in action with the help of a small example. It uses prior knowledge and updates it with observed data to create a posterior, exactly like humans intuitively do. This approach is better than discarding the data and just proceeding with some prior, obviously. It is even more powerful than the maximum likelihood method: you can see this by choosing a flat prior, i.e. the prior gives the same probability (or density) to every possible value θ and is essentially a constant. Furthermore, the Bayes method even gives you a distribution of the parameters, while the maximum likelihood method does not.

Parameter estimation and associated uncertainty quantification is an important problem in dynamical systems characterized by ordinary differential equation (ODE) models that are often nonlinear. Typically, such models have analytically intractable trajectories which result in likelihoods and posterior distributions that are similarly intractable. Bayesian inference for ODE systems via simulation methods require numerical approximations to produce inference with high accuracy at a cost of heavy computational power and slow convergence. At the same time, Artificial Neural Networks (ANN) offer tractability that can be utilized to construct an approximate but tractable likelihood and posterior distribution. In this paper we propose a hybrid approach, where Laplace-based Bayesian inference is combined with an ANN architecture for obtaining approximations to the ODE trajectories as a function of the unknown initial values and system parameters. Suitable choices of a collocation grid and customized loss functions are proposed to fine tune the ODE trajectories and Laplace approximation. The effectiveness of our proposed methods is demonstrated using an epidemiological system with non-analytical solutions, the Susceptible-Infectious-Removed (SIR) model for infectious diseases, based on simulated and real-life influenza datasets. The novelty and attractiveness of our proposed approach include (i) a new development of Bayesian inference using ANN architectures for ODE based dynamical systems, and (ii) a computationally fast posterior inference by avoiding convergence issues of benchmark Markov Chain Monte Carlo methods. These two features establish the developed approach as an accurate alternative to traditional Bayesian computational methods, with improved computational cost.