We investigate the benefit of treating all the parameters in a Bayesian neural network stochastically and find compelling theoretical and empirical evidence that this standard construction may be unnecessary. To this end, we prove that expressive predictive distributions require only small amounts of stochasticity. In particular, partially stochastic networks with only $n$ stochastic biases are universal probabilistic predictors for $n$-dimensional predictive problems. In empirical investigations, we find no systematic benefit of full stochasticity across four different inference modalities and eight datasets; partially stochastic networks can match and sometimes even outperform fully stochastic networks, despite their reduced memory costs.

While machine learning models rapidly advance the state-of-the-art on various real-world tasks, out-of-domain (OOD) generalization remains a challenging problem given the vulnerability of these models to spurious correlations. We propose a balanced mini-batch sampling strategy to transform a biased data distribution into a spurious-free balanced distribution, based on the invariance of the underlying causal mechanisms for the data generation process. We argue that the Bayes optimal classifiers trained on such balanced distribution are minimax optimal across a diverse enough environment space. We also provide an identifiability guarantee of the latent variable model of the proposed data generation process, when utilizing enough train environments. Experiments are conducted on DomainBed, demonstrating empirically that our method obtains the best performance across 20 baselines reported on the benchmark.

Unsupervised meta-learning aims to learn the meta knowledge from unlabeled data and rapidly adapt to novel tasks. However, existing approaches may be misled by the context-bias (e.g. background) from the training data. In this paper, we abstract the unsupervised meta-learning problem into a Structural Causal Model (SCM) and point out that such bias arises due to hidden confounders. To eliminate the confounders, we define the priors are \textit{conditionally} independent, learn the relationships between priors and intervene on them with casual factorization. Furthermore, we propose Causal Meta VAE (CMVAE) that encodes the priors into latent codes in the causal space and learns their relationships simultaneously to achieve the downstream few-shot image classification task. Results on toy datasets and three benchmark datasets demonstrate that our method can remove the context-bias and it outperforms other state-of-the-art unsupervised meta-learning algorithms because of bias-removal. Code is available at \url{https://github.com/GuodongQi/CMVAE}

(Neal and Hinton, 1998) recast maximum likelihood estimation of any given latent variable model as the minimization of a free energy functional $F$, and the EM algorithm as coordinate descent applied to $F$. Here, we explore alternative ways to optimize the functional. In particular, we identify various gradient flows associated with $F$ and show that their limits coincide with $F$'s stationary points. By discretizing the flows, we obtain practical particle-based algorithms for maximum likelihood estimation in broad classes of latent variable models. The novel algorithms scale to high-dimensional settings and perform well in numerical experiments.

Bayesian methods are a popular choice for statistical inference in small-data regimes due to the regularization effect induced by the prior. In the context of density estimation, the standard nonparametric Bayesian approach is to target the posterior predictive of the Dirichlet process mixture model. In general, direct estimation of the posterior predictive is intractable and so methods typically resort to approximating the posterior distribution as an intermediate step. The recent development of quasi-Bayesian predictive copula updates, however, has made it possible to perform tractable predictive density estimation without the need for posterior approximation. Although these estimators are computationally appealing, they tend to struggle on non-smooth data distributions. This is due to the comparatively restrictive form of the likelihood models from which the proposed copula updates were derived. To address this shortcoming, we consider a Bayesian nonparametric model with an autoregressive likelihood decomposition and a Gaussian process prior. While the predictive update of such a model is typically intractable, we derive a quasi-Bayesian predictive update that achieves state-of-the-art results in small-data regimes.

The sole aim of this book is to give a self-contained introduction to concepts and mathematical tools in Bayesian matrix decomposition in order to seamlessly introduce matrix decomposition techniques and their applications in subsequent sections. However, we clearly realize our inability to cover all the useful and interesting results concerning Bayesian matrix decomposition and given the paucity of scope to present this discussion, e.g., the separated analysis of variational inference for conducting the optimization. We refer the reader to literature in the field of Bayesian analysis for a more detailed introduction to the related fields. This book is primarily a summary of purpose, significance of important Bayesian matrix decomposition methods, e.g., real-valued decomposition, nonnegative matrix factorization, Bayesian interpolative decomposition, and the origin and complexity of the methods which shed light on their applications. The mathematical prerequisite is a first course in statistics and linear algebra. Other than this modest background, the development is self-contained, with rigorous proof provided throughout.

In this paper, we study the problem of transferring the available Markov Decision Process (MDP) models to learn and plan efficiently in an unknown but similar MDP. We refer to it as \textit{Model Transfer Reinforcement Learning (MTRL)} problem. First, we formulate MTRL for discrete MDPs and Linear Quadratic Regulators (LQRs) with continuous state actions. Then, we propose a generic two-stage algorithm, MLEMTRL, to address the MTRL problem in discrete and continuous settings. In the first stage, MLEMTRL uses a \textit{constrained Maximum Likelihood Estimation (MLE)}-based approach to estimate the target MDP model using a set of known MDP models. In the second stage, using the estimated target MDP model, MLEMTRL deploys a model-based planning algorithm appropriate for the MDP class. Theoretically, we prove worst-case regret bounds for MLEMTRL both in realisable and non-realisable settings. We empirically demonstrate that MLEMTRL allows faster learning in new MDPs than learning from scratch and achieves near-optimal performance depending on the similarity of the available MDPs and the target MDP.

The problems of selecting partial correlation and causality graphs for count data are considered. A parameter driven generalized linear model is used to describe the observed multivariate time series of counts. Partial correlation and causality graphs corresponding to this model explain the dependencies between each time series of the multivariate count data. In order to estimate these graphs with tunable sparsity, an appropriate likelihood function maximization is regularized with an l1-type constraint. A novel MCEM algorithm is proposed to iteratively solve this regularized MLE. Asymptotic convergence results are proved for the sequence generated by the proposed MCEM algorithm with l1-type regularization. The algorithm is first successfully tested on simulated data. Thereafter, it is applied to observed weekly dengue disease counts from each ward of Greater Mumbai city. The interdependence of various wards in the proliferation of the disease is characterized by the edges of the inferred partial correlation graph. On the other hand, the relative roles of various wards as sources and sinks of dengue spread is quantified by the number and weights of the directed edges originating from and incident upon each ward. From these estimated graphs, it is observed that some special wards act as epicentres of dengue spread even though their disease counts are relatively low.

Population synthesis consists of generating synthetic but realistic representations of a target population of micro-agents for the purpose of behavioral modeling and simulation. We introduce a new framework based on copulas to generate synthetic data for a target population of which only the empirical marginal distributions are known by using a sample from another population sharing similar marginal dependencies. This makes it possible to include a spatial component in the generation of population synthesis and to combine various sources of information to obtain more realistic population generators. Specifically, we normalize the data and treat them as realizations of a given copula, and train a generative model on the normalized data before injecting the information on the marginals. We compare the copulas framework to IPF and to modern probabilistic approaches such as Bayesian networks, variational auto-encoders, and generative adversarial networks. We also illustrate on American Community Survey data that the method proposed allows to study the structure of the data at different geographical levels in a way that is robust to the peculiarities of the marginal distributions.

Understanding how different classes are distributed in an unlabeled data set is an important challenge for the calibration of probabilistic classifiers and uncertainty quantification. Approaches like adjusted classify and count, black-box shift estimators, and invariant ratio estimators use an auxiliary (and potentially biased) black-box classifier trained on a different (shifted) data set to estimate the class distribution and yield asymptotic guarantees under weak assumptions. We demonstrate that all these algorithms are closely related to the inference in a particular Bayesian model, approximating the assumed ground-truth generative process. Then, we discuss an efficient Markov Chain Monte Carlo sampling scheme for the introduced model and show an asymptotic consistency guarantee in the large-data limit. We compare the introduced model against the established point estimators in a variety of scenarios, and show it is competitive, and in some cases superior, with the state of the art.