This book is devoted to the problem of sequential probability forecasting, that is, predicting the probabilities of the next outcome of a growing sequence of observations given the past. This problem is considered in a very general setting that unifies commonly used probabilistic and non-probabilistic settings, trying to make as few as possible assumptions on the mechanism generating the observations. A common form that arises in various formulations of this problem is that of mixture predictors, which are formed as a combination of a finite or infinite set of other predictors attempting to combine their predictive powers. The main subject of this book are such mixture predictors, and the main results demonstrate the universality of this method in a very general probabilistic setting, but also show some of its limitations. While the problems considered are motivated by practical applications, involving, for example, financial, biological or behavioural data, this motivation is left implicit and all the results exposed are theoretical. The book targets graduate students and researchers interested in the problem of sequential prediction, and, more generally, in theoretical analysis of problems in machine learning and non-parametric statistics, as well as mathematical and philosophical foundations of these fields. The material in this volume is presented in a way that presumes familiarity with basic concepts of probability and statistics, up to and including probability distributions over spaces of infinite sequences. Familiarity with the literature on learning or stochastic processes is not required.

We give methods for Bayesian inference of directed acyclic graphs, DAGs, and the induced causal effects from passively observed complete data. Our methods build on a recent Markov chain Monte Carlo scheme for learning Bayesian networks, which enables efficient approximate sampling from the graph posterior, provided that each node is assigned a small number K of candidate parents. We present algorithmic tricks to significantly reduce the space and time requirements of the method, making it feasible to use substantially larger values of K. Furthermore, we investigate the problem of selecting the candidate parents per node so as to maximize the covered posterior mass. Finally, we combine our sampling method with a novel Bayesian approach for estimating causal effects in linear Gaussian DAG models. Numerical experiments demonstrate the performance of our methods in detecting ancestor-descendant relations, and in effect estimation our Bayesian method is shown to outperform existing approaches.

Reinforcement learning methods for traffic signal control has gained increasing interests recently and achieved better performances compared with traditional transportation methods. However, reinforcement learning based methods usually requires heavy training data and computational resources which largely limit its application in real-world traffic signal control. This makes meta-learning, which enables data-efficient and fast-adaptation training by leveraging the knowledge of previous learning experiences, catches attentions in traffic signal control. In this paper, we propose a novel value-based Bayesian meta-reinforcement learning framework BM-DQN to robustly speed up the learning process in new scenarios by utilizing well-trained prior knowledge learned from existing scenarios. This framework based on our proposed fast-adaptation variation to Gradient-EM Bayesian Meta-learning and the fast update advantage of DQN, which allows fast adaptation to new scenarios with continual learning ability and robustness to uncertainty. The experiments on 2D navigation and traffic signal control show that our proposed framework adapts more quickly and robustly in new scenarios than previous methods, and specifically, much better continual learning ability in heterogeneous scenarios.

Online Social Networks (OSNs) provide a platform for users to share their thoughts and opinions with their community of friends or to the general public. In order to keep the platform safe for all users, as well as to keep it compliant with local laws, OSNs typically create a set of community standards organized into policy groups, and use Machine Learning (ML) models to identify and remove content that violates any of the policies. However, out of the billions of content that is uploaded on a daily basis only a small fraction is so unambiguously violating that it can be removed by the automated models. Prevalence estimation is the task of estimating the fraction of violating content in the residual items by sending a small sample of these items to human labelers to get ground truth labels. This task is exceedingly hard because even though we can easily get the ML scores or features for all of the billions of items we can only get ground truth labels on a few thousands of these items due to practical considerations. Indeed the prevalence can be so low that even after a judicious choice of items to be labeled there can be many days in which not even a single item is labeled violating. A pragmatic choice for such low prevalence, $10^{-4}$ to $10^{-5}$, regimes is to report the upper bound, or $97.5\%$ confidence interval, prevalence (UBP) that takes the uncertainties of the sampling and labeling processes into account and gives a smoothed estimate. In this work we present two novel techniques Bucketed-Beta-Binomial and a Bucketed-Gaussian Process for this UBP task and demonstrate on real and simulated data that it has much better coverage than the commonly used bootstrapping technique.

In this article, I will show how missing values can lead to biased estimates by working through a common dataset in 4 scenarios where the missing value mechanism differs. The content in this article is based on Chapter 15 in [1]. The code to reproduce the results in this article can be found in this notebook. I assume the reader is familiar with building generalized linear models (GLMs) and using directed acyclic graphs (DAGs) to illustrate causality. Suppose we want to study the effect of student diligence and homework quality. Let's imagine that we were somehow able to assign a real number to measure a student's diligence and that homework quality is measured on a 10-point scale, 0 to 10.

The emergence of Big Data raises the question of how to model economic relations when there is a large number of possible explanatory variables. We revisit the issue by comparing the possibility of using dense or sparse models in a Bayesian approach, allowing for variable selection and shrinkage. More specifically, we discuss the results reached by Giannone, Lenza, and Primiceri (2020) through a "Spike-and-Slab" prior, which suggest an "illusion of sparsity" in economic data, as no clear patterns of sparsity could be detected. We make a further revision of the posterior distributions of the model, and propose three experiments to evaluate the robustness of the adopted prior distribution. We find that the pattern of sparsity is sensitive to the prior distribution of the regression coefficients, and present evidence that the model indirectly induces variable selection and shrinkage, which suggests that the "illusion of sparsity" could be, itself, an illusion. Code is available on github.com/bfava/IllusionOfIllusion.

We develop in this paper a framework of empirical gain maximization (EGM) to address the robust regression problem where heavy-tailed noise or outliers may present in the response variable. The idea of EGM is to approximate the density function of the noise distribution instead of approximating the truth function directly as usual. Unlike the classical maximum likelihood estimation that encourages equal importance of all observations and could be problematic in the presence of abnormal observations, EGM schemes can be interpreted from a minimum distance estimation viewpoint and allow the ignorance of those observations. Furthermore, it is shown that several well-known robust nonconvex regression paradigms, such as Tukey regression and truncated least square regression, can be reformulated into this new framework. We then develop a learning theory for EGM, by means of which a unified analysis can be conducted for these well-established but not fully-understood regression approaches. Resulting from the new framework, a novel interpretation of existing bounded nonconvex loss functions can be concluded. Within this new framework, the two seemingly irrelevant terminologies, the well-known Tukey's biweight loss for robust regression and the triweight kernel for nonparametric smoothing, are closely related. More precisely, it is shown that the Tukey's biweight loss can be derived from the triweight kernel. Similarly, other frequently employed bounded nonconvex loss functions in machine learning such as the truncated square loss, the Geman-McClure loss, and the exponential squared loss can also be reformulated from certain smoothing kernels in statistics. In addition, the new framework enables us to devise new bounded nonconvex loss functions for robust learning.

ParaMonte (standing for Parallel Monte Carlo) is a serial and MPI/Coarray-parallelized library of Monte Carlo routines for sampling mathematical objective functions of arbitrary-dimensions, in particular, the posterior distributions of Bayesian models in data science, Machine Learning, and scientific inference. The ParaMonte library has been developed with the design goal of unifying the **automation**, **accessibility**, **high-performance**, **scalability**, and **reproducibility** of Monte Carlo simulations. The current implementation of the library includes **ParaDRAM**, a **Para**llel **D**elyaed-**R**ejection **A**daptive **M**etropolis Markov Chain Monte Carlo sampler, accessible from a wide range of programming languages including C, C++, Fortran, with a unified Application Programming Interface and simulation environment across all supported programming languages. The ParaMonte library is MIT-licensed and is permanently located and maintained at [https://github.com/cdslaborg/paramonte](https://github.com/cdslaborg/paramonte).

In the present work, we consider variable selection and shrinkage for the Gaussian dynamic linear regression within a Bayesian framework. In particular, we propose a novel method that allows for time-varying sparsity, based on an extension of spike-and-slab priors for dynamic models. This is done by assigning appropriate Markov switching priors for the time-varying coefficients' variances, extending the previous work of Ishwaran and Rao (2005). Furthermore, we investigate different priors, including the common Inverted gamma prior for the process variances, and other mixture prior distributions such as Gamma priors for both the spike and the slab, which leads to a mixture of Normal-Gammas priors (Griffin ad Brown, 2010) for the coefficients. In this sense, our prior can be view as a dynamic variable selection prior which induces either smoothness (through the slab) or shrinkage towards zero (through the spike) at each time point. The MCMC method used for posterior computation uses Markov latent variables that can assume binary regimes at each time point to generate the coefficients' variances. In that way, our model is a dynamic mixture model, thus, we could use the algorithm of Gerlach et al (2000) to generate the latent processes without conditioning on the states. Finally, our approach is exemplified through simulated examples and a real data application.

Recent advances in artificial intelligence have propelled the development of innovative computational materials modeling and design techniques. In particular, generative deep learning models have been used for molecular representation, discovery and design with applications ranging from drug discovery to solar cell development. In this work, we assess the predictive capabilities of a molecular generative model developed based on variational inference and graph theory. The encoder network is based on the scattering transform, which allows for a better generalization of the model in the presence of limited training data. The scattering layers incorporate adaptive spectral filters which are tailored to the training dataset based on the molecular graphs' spectra. The decoding network is a one-shot graph generative model that conditions atom types on molecular topology. We present a quantitative assessment of the latent space in terms of its predictive ability for organic molecules in the QM9 dataset. To account for the limited size training data set, a Bayesian formalism is considered that allows us capturing the uncertainties in the predicted properties.