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.

Although much of the success of Deep Learning builds on learning good representations, a rigorous method to evaluate their quality is lacking. In this paper, we treat the evaluation of representations as a model selection problem and propose to use the Minimum Description Length (MDL) principle to devise an evaluation metric. Contrary to the established practice of limiting the capacity of the readout model, we design a hybrid discrete and continuous-valued model space for the readout models and employ a switching strategy to combine their predictions. The MDL score takes model complexity, as well as data efficiency into account. As a result, the most appropriate model for the specific task and representation will be chosen, making it a unified measure for comparison. The proposed metric can be efficiently computed with an online method and we present results for pre-trained vision encoders of various architectures (ResNet and ViT) and objective functions (supervised and self-supervised) on a range of downstream tasks. We compare our methods with accuracy-based approaches and show that the latter are inconsistent when multiple readout models are used. Finally, we discuss important properties revealed by our evaluations such as model scaling, preferred readout model, and data efficiency.

Learning with noisy labels has become imperative in the Big Data era, which saves expensive human labors on accurate annotations. Previous noise-transition-based methods have achieved theoretically-grounded performance under the Class-Conditional Noise model (CCN). However, these approaches builds upon an ideal but impractical anchor set available to pre-estimate the noise transition. Even though subsequent works adapt the estimation as a neural layer, the ill-posed stochastic learning of its parameters in back-propagation easily falls into undesired local minimums. We solve this problem by introducing a Latent Class-Conditional Noise model (LCCN) to parameterize the noise transition under a Bayesian framework. By projecting the noise transition into the Dirichlet space, the learning is constrained on a simplex characterized by the complete dataset, instead of some ad-hoc parametric space wrapped by the neural layer. We then deduce a dynamic label regression method for LCCN, whose Gibbs sampler allows us efficiently infer the latent true labels to train the classifier and to model the noise. Our approach safeguards the stable update of the noise transition, which avoids previous arbitrarily tuning from a mini-batch of samples. We further generalize LCCN to different counterparts compatible with open-set noisy labels, semi-supervised learning as well as cross-model training. A range of experiments demonstrate the advantages of LCCN and its variants over the current state-of-the-art methods.

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.

Learning the underlying Bayesian Networks (BNs), represented by directed acyclic graphs (DAGs), of the concerned events from purely-observational data is a crucial part of evidential reasoning. This task remains challenging due to the large and discrete search space. A recent flurry of developments followed NOTEARS[1] recast this combinatorial problem into a continuous optimization problem by leveraging an algebraic equality characterization of acyclicity. However, the continuous optimization methods suffer from obtaining non-spare graphs after the numerical optimization, which leads to the inflexibility to rule out the potentially cycle-inducing edges or false discovery edges with small values. To address this issue, in this paper, we develop a completely data-driven DAG structure learning method without a predefined value to post-threshold small values. We name our method NOTEARS with adaptive Lasso (NOTEARS-AL), which is achieved by applying the adaptive penalty method to ensure the sparsity of the estimated DAG. Moreover, we show that NOTEARS-AL also inherits the oracle properties under some specific conditions. Extensive experiments on both synthetic and a real-world dataset demonstrate that our method consistently outperforms NOTEARS.

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.