Universal probabilistic programming systems (PPSs) provide a powerful and expressive framework for specifying rich and complex probabilistic models. However, this expressiveness comes at the cost of substantially complicating the process of drawing inferences from the model. In particular, inference can become challenging when the support of the model varies between executions. Though general-purpose inference engines have been designed to operate in such settings, they are typically highly inefficient, often relying on proposing from the prior to make transitions. To address this, we introduce a new inference framework: Divide, Conquer, and Combine (DCC). DCC divides the program into separate straight-line sub-programs, each of which has a fixed support allowing more powerful inference algorithms to be run locally, before recombining their outputs in a principled fashion. We show how DCC can be implemented as an automated and general-purpose PPS inference engine, and empirically confirm that it can provide substantial performance improvements over previous approaches.

I consider two problems in machine learning and statistics: the problem of estimating the joint probability density of a collection of random variables, known as density estimation, and the problem of inferring model parameters when their likelihood is intractable, known as likelihood-free inference. The contribution of the thesis is a set of new methods for addressing these problems that are based on recent advances in neural networks and deep learning.

In this work we study the problem of inferring a discrete probability distribution using both expert knowledge and empirical data. This is an important issue for many applications where the scarcity of data prevents a purely empirical approach. In this context, it is common to rely first on an initial domain knowledge a priori before proceeding to an online data acquisition. We are particularly interested in the intermediate regime where we do not have enough data to do without the initial expert a priori of the experts, but enough to correct it if necessary. We present here a novel way to tackle this issue with a method providing an objective way to choose the weight to be given to experts compared to data. We show, both empirically and theoretically, that our proposed estimator is always more efficient than the best of the two models (expert or data) within a constant.

This paper tackles the challenge presented by small-data to the task of Bayesian inference. A novel methodology, based on manifold learning and manifold sampling, is proposed for solving this computational statistics problem under the following assumptions: 1) neither the prior model nor the likelihood function are Gaussian and neither can be approximated by a Gaussian measure; 2) the number of functional input (system parameters) and functional output (quantity of interest) can be large; 3) the number of available realizations of the prior model is small, leading to the small-data challenge typically associated with expensive numerical simulations; the number of experimental realizations is also small; 4) the number of the posterior realizations required for decision is much larger than the available initial dataset. The method and its mathematical aspects are detailed. Three applications are presented for validation: The first two involve mathematical constructions aimed to develop intuition around the method and to explore its performance. The third example aims to demonstrate the operational value of the method using a more complex application related to the statistical inverse identification of the non-Gaussian matrix-valued random elasticity field of a damaged biological tissue (osteoporosis in a cortical bone) using ultrasonic waves.

In this paper, we consider the sigmoid Gaussian Hawkes process model: the baseline intensity and triggering kernel of Hawkes process are both modeled as the sigmoid transformation of random trajectories drawn from Gaussian processes (GP). By introducing auxiliary latent random variables (branching structure, P\'{o}lya-Gamma random variables and latent marked Poisson processes), the likelihood is converted to two decoupled components with a Gaussian form which allows for an efficient conjugate analytical inference. Using the augmented likelihood, we derive an expectation-maximization (EM) algorithm to obtain the maximum a posteriori (MAP) estimate. Furthermore, we extend the EM algorithm to an efficient approximate Bayesian inference algorithm: mean-field variational inference. We demonstrate the performance of two algorithms on simulated fictitious data. Experiments on real data show that our proposed inference algorithms can recover well the underlying prompting characteristics efficiently.

The main way for defining equivalence among acyclic directed graphs is based on the conditional independencies of the distributions that they can generate. However, it is known that when cycles are allowed in the structure, conditional independence is not a suitable notion for equivalence of two structures, as it does not reflect all the information in the distribution that can be used for identification of the underlying structure. In this paper, we present a general, unified notion of equivalence for linear Gaussian directed graphs. Our proposed definition for equivalence is based on the set of distributions that the structure is able to generate. We take a first step towards devising methods for characterizing the equivalence of two structures, which may be cyclic or acyclic. Additionally, we propose a score-based method for learning the structure from observational data.

This paper presents the Poisson-randomized gamma dynamical system (PRGDS), a model for sequentially observed count tensors that encodes a strong inductive bias toward sparsity and burstiness. The PRGDS is based on a new motif in Bayesian latent variable modeling, an alternating chain of discrete Poisson and continuous gamma latent states that is analytically convenient and computationally tractable. This motif yields closed-form complete conditionals for all variables by way of the Bessel distribution and a novel discrete distribution that we call the shifted confluent hypergeometric distribution. We draw connections to closely related models and compare the PRGDS to these models in studies of real-world count data sets of text, international events, and neural spike trains. We find that a sparse variant of the PRGDS, which allows the continuous gamma latent states to take values of exactly zero, often obtains better predictive performance than other models and is uniquely capable of inferring latent structures that are highly localized in time.

Both the median-based classifier and the quantile-based classifier are useful for discriminating high-dimensional data with heavy-tailed or skewed inputs. But these methods are restricted as they assign equal weight to each variable in an unregularized way. The ensemble quantile classifier is a more flexible regularized classifier that provides better performance with high-dimensional data, asymmetric data or when there are many irrelevant extraneous inputs. The improved performance is demonstrated by a simulation study as well as an application to text categorization. It is proven that the estimated parameters of the ensemble quantile classifier consistently estimate the minimal population loss under suitable general model assumptions. It is also shown that the ensemble quantile classifier is Bayes optimal under suitable assumptions with asymmetric Laplace distribution inputs.

Approximate Bayesian Computation (ABC) is a popular method for approximate inference in generative models with intractable but easy-to-sample likelihood. It constructs an approximate posterior distribution by finding parameters for which the simulated data are close to the observations in terms of summary statistics. These statistics are defined beforehand and might induce a loss of information, which has been shown to deteriorate the quality of the approximation. To overcome this problem, Wasserstein-ABC has been recently proposed, and compares the datasets via the Wasserstein distance between their empirical distributions, but does not scale well to the dimension or the number of samples. We propose a new ABC technique, called Sliced-Wasserstein ABC and based on the Sliced-Wasserstein distance, which has better computational and statistical properties. We derive two theoretical results showing the asymptotical consistency of our approach, and we illustrate its advantages on synthetic data and an image denoising task.

The use of electronic cigarette (e-cigarette) is increasing among adolescents. This is problematic since consuming nicotine at an early age can cause harmful effects in developing teenager's brain and health. Additionally, the use of e-cigarette has a possibility of leading to the use of cigarettes, which is more severe. There were many researches about e-cigarette and cigarette that mostly focused on finding and analyzing causes of smoking using conventional statistics. However, there is a lack of research on developing prediction models, which is more applicable to anti-smoking campaign, about e-cigarette and cigarette. In this paper, we research the prediction models that can be used to predict an individual e-cigarette user's (including non-e-cigarette users) intention to smoke cigarettes, so that one can be early informed about the risk of going down the path of smoking cigarettes. To construct the prediction models, five machine learning (ML) algorithms are exploited and tested for their accuracy in predicting the intention to smoke cigarettes among never smokers using data from the 2018 National Youth Tobacco Survey (NYTS). In our investigation, the Gradient Boosting Classifier, one of the prediction models, shows the highest accuracy out of all the other models. Also, with the best prediction model, we made a public website that enables users to input information to predict their intentions of smoking cigarettes.