This document aims to provide a review on learning with deep generative models (DGMs), which is an highly-active area in machine learning and more generally, artificial intelligence. This review is not meant to be a tutorial, but when necessary, we provide self-contained derivations for completeness. This review has two features. First, though there are different perspectives to classify DGMs, we choose to organize this review from the perspective of graphical modeling, because the learning methods for directed DGMs and undirected DGMs are fundamentally different. Second, we differentiate model definitions from model learning algorithms, since different learning algorithms can be applied to solve the learning problem on the same model, and an algorithm can be applied to learn different models. We thus separate model definition and model learning, with more emphasis on reviewing, differentiating and connecting different learning algorithms. We also discuss promising future research directions. This review is by no means comprehensive as the field is evolving rapidly. The authors apologize in advance for any missed papers and inaccuracies in descriptions. Corrections and comments are highly welcome.

In recent years, we've seen a resurgence in AI, or artificial intelligence, and machine learning. Machine learning has led to some amazing results, like being able to analyze medical images and predict diseases on-par with human experts. Google's AlphaGo program was able to beat a world champion in the strategy game go using deep reinforcement learning. Machine learning is even being used to program self driving cars, which is going to change the automotive industry forever. Imagine a world with drastically reduced car accidents, simply by removing the element of human error.

Alzheimer's Disease (AD) is characterized by a cascade of biomarkers becoming abnormal, the pathophysiology of which is very complex and largely unknown. Event-based modeling (EBM) is a data-driven technique to estimate the sequence in which biomarkers for a disease become abnormal based on cross-sectional data. It can help in understanding the dynamics of disease progression and facilitate early diagnosis and prognosis. In this work we propose a novel discriminative approach to EBM, which is shown to be more accurate than existing state-of-the-art EBM methods. The method first estimates for each subject an approximate ordering of events. Subsequently, the central ordering over all subjects is estimated by fitting a generalized Mallows model to these approximate subject-specific orderings. We also introduce the concept of relative distance between events which helps in creating a disease progression timeline. Subsequently, we propose a method to stage subjects by placing them on the estimated disease progression timeline. We evaluated the proposed method on Alzheimer's Disease Neuroimaging Initiative (ADNI) data and compared the results with existing state-of-the-art EBM methods. We also performed extensive experiments on synthetic data simulating the progression of Alzheimer's disease. The event orderings obtained on ADNI data seem plausible and are in agreement with the current understanding of progression of AD. The proposed patient staging algorithm performed consistently better than that of state-of-the-art EBM methods. Event orderings obtained in simulation experiments were more accurate than those of other EBM methods and the estimated disease progression timeline was observed to correlate with the timeline of actual disease progression. The results of these experiments are encouraging and suggest that discriminative EBM is a promising approach to disease progression modeling.

The multi-armed bandit is a sequential allocation task where an agent must learn a policy that maximizes long term payoff, where only the reward of the played arm is observed at each iteration. In the stochastic setting, the reward for each action is generated from an unknown distribution, which depends on a given 'context', available at each interaction with the world. Thompson sampling is a generative, interpretable multi-armed bandit algorithm that has been shown both to perform well in practice, and to enjoy optimality properties for certain reward functions. Nevertheless, Thompson sampling requires sampling from parameter posteriors and calculation of expected rewards, which are possible for a very limited choice of distributions. We here extend Thompson sampling to more complex scenarios by adopting a very flexible set of reward distributions: nonparametric Gaussian mixture models. The generative process of Bayesian nonparametric mixtures naturally aligns with the Bayesian modeling of multi-armed bandits. This allows for the implementation of an efficient and flexible Thompson sampling algorithm: the nonparametric model autonomously determines its complexity in an online fashion, as it observes new rewards for the played arms. We show how the proposed method sequentially learns the nonparametric mixture model that best approximates the true underlying reward distribution. Our contribution is valuable for practical scenarios, as it avoids stringent model specifications, and yet attains reduced regret.

The multi-armed bandit (MAB) problem is a sequential allocation task where the goal is to learn a policy that maximizes long term payoff, where only the reward of the executed action is observed; i.e., sequential optimal decisions are made, while simultaneously learning how the world operates. In the stochastic setting, the reward for each action is generated from an unknown distribution. To decide the next optimal action to take, one must compute sufficient statistics of this unknown reward distribution, e.g. upper-confidence bounds (UCB), or expectations in Thompson sampling. Closed-form expressions for these statistics of interest are analytically intractable except for simple cases. We here propose to leverage Monte Carlo estimation and, in particular, the flexibility of (sequential) importance sampling (IS) to allow for accurate estimation of the statistics of interest within the MAB problem. IS methods estimate posterior densities or expectations in probabilistic models that are analytically intractable. We first show how IS can be combined with state-of-the-art MAB algorithms (Thompson sampling and Bayes-UCB) for classic (Bernoulli and contextual linear-Gaussian) bandit problems. Furthermore, we leverage the power of sequential IS to extend the applicability of these algorithms beyond the classic settings, and tackle additional useful cases. Specifically, we study the dynamic linear-Gaussian bandit, and both the static and dynamic logistic cases too. The flexibility of (sequential) importance sampling is shown to be fundamental for obtaining efficient estimates of the key sufficient statistics in these challenging scenarios.

Approximate Bayesian computation (ABC) is a method for Bayesian inference when the likelihood is unavailable but simulating from the model is possible. However, many ABC algorithms require a large number of simulations, which can be costly. To reduce the computational cost, Bayesian optimisation (BO) and surrogate models such as Gaussian processes have been proposed. Bayesian optimisation enables one to intelligently decide where to evaluate the model next but common BO strategies are not designed for the goal of estimating the posterior distribution. Our paper addresses this gap in the literature. We propose to compute the uncertainty in the ABC posterior density, which is due to a lack of simulations to estimate this quantity accurately, and define a loss function that measures this uncertainty. We then propose to select the next evaluation location to minimise the expected loss. Experiments show that the proposed method often produces the most accurate approximations as compared to common BO strategies.

When learning from positive and unlabelled data, it is a strong assumption that the positive observations are randomly sampled from the distribution of $X$ conditional on $Y = 1$, where X stands for the feature and Y the label. Most existing algorithms are optimally designed under the assumption. However, for many real-world applications, the observed positive examples are dependent on the conditional probability $P(Y = 1|X)$ and should be sampled biasedly. In this paper, we assume that a positive example with a higher $P(Y = 1|X)$ is more likely to be labelled and propose a probabilistic-gap based PU learning algorithms. Specifically, by treating the unlabelled data as noisy negative examples, we could automatically label a group positive and negative examples whose labels are identical to the ones assigned by a Bayesian optimal classifier with a consistency guarantee. The relabelled examples have a biased domain, which is remedied by the kernel mean matching technique. The proposed algorithm is model-free and thus do not have any parameters to tune. Experimental results demonstrate that our method works well on both generated and real-world datasets.

Mastering the dynamics of social influence requires separating, in a database of information propagation traces, the genuine causal processes from temporal correlation, homophily and other spurious causes. However, most of the studies to characterize social influence and, in general, most data-science analyses focus on correlations, statistical independence, conditional independence etc.; only recently, there has been a resurgence of interest in "causal data science", e.g., grounded on causality theories. In this paper we adopt a principled causal approach to the analysis of social influence from information-propagation data, rooted in probabilistic causal theory. Our approach develops around two phases. In the first step, in order to avoid the pitfalls of misinterpreting causation when the data spans a mixture of several subtypes ("Simpson's paradox"), we partition the set of propagation traces in groups, in such a way that each group is as less contradictory as possible in terms of the hierarchical structure of information propagation. For this goal we borrow from the literature the notion of "agony" and define the Agony-bounded Partitioning problem, which we prove being hard, and for which we develop two efficient algorithms with approximation guarantees. In the second step, for each group from the first phase, we apply a constrained MLE approach to ultimately learn a minimal causal topology. Experiments on synthetic data show that our method is able to retrieve the genuine causal arcs w.r.t. a known ground-truth generative model. Experiments on real data show that, by focusing only on the extracted causal structures instead of the whole social network, we can improve the effectiveness of predicting influence spread.

Missing data imputation can help improve the performance of prediction models in situations where missing data hide useful information. This paper compares methods for imputing missing categorical data for supervised classification tasks. We experiment on two machine learning benchmark datasets with missing categorical data, comparing classifiers trained on non-imputed (i.e., one-hot encoded) or imputed data with different levels of additional missing-data perturbation. We show imputation methods can increase predictive accuracy in the presence of missing-data perturbation, which can actually improve prediction accuracy by regularizing the classifier. We achieve the state-of-the-art on the Adult dataset with missing-data perturbation and k-nearest-neighbors (k-NN) imputation.

Missing values widely exist in many real-world datasets, which hinders the performing of advanced data analytics. Properly filling these missing values is crucial but challenging, especially when the missing rate is high. Many approaches have been proposed for missing value imputation (MVI), but they are mostly heuristics-based, lacking a principled foundation and do not perform satisfactorily in practice. In this paper, we propose a probabilistic framework based on deep generative models for MVI. Under this framework, imputing the missing entries amounts to seeking a fixed-point solution between two conditional distributions defined on the missing entries and latent variables respectively. These distributions are parameterized by deep neural networks (DNNs) which possess high approximation power and can capture the nonlinear relationships between missing entries and the observed values. The learning of weight parameters of DNNs is performed by maximizing an approximation of the log-likelihood of observed values. We conducted extensive evaluation on 13 datasets and compared with 11 baselines methods, where our methods largely outperforms the baselines.