We study the posterior contraction rates of a Bayesian method with Gaussian process priors in nonparametric regression and its plug-in property for differential operators. For a general class of kernels, we establish convergence rates of the posterior measure of the regression function and its derivatives, which are both minimax optimal up to a logarithmic factor for functions in certain classes. Our calculation shows that the rate-optimal estimation of the regression function and its derivatives share the same choice of hyperparameter, indicating that the Bayes procedure remarkably adapts to the order of derivatives and enjoys a generalized plug-in property that extends real-valued functionals to function-valued functionals. This leads to a practically simple method for estimating the regression function and its derivatives, whose finite sample performance is assessed using simulations. Our proof shows that, under certain conditions, to any convergence rate of Bayes estimators there corresponds the same convergence rate of the posterior distributions (i.e., posterior contraction rate), and vice versa. This equivalence holds for a general class of Gaussian processes and covers the regression function and its derivative functionals, under both the $L_2$ and $L_{\infty}$ norms. In addition to connecting these two fundamental large sample properties in Bayesian and non-Bayesian regimes, such equivalence enables a new routine to establish posterior contraction rates by calculating convergence rates of nonparametric point estimators. At the core of our argument is an operator-theoretic framework for kernel ridge regression and equivalent kernel techniques. We derive a range of sharp non-asymptotic bounds that are pivotal in establishing convergence rates of nonparametric point estimators and the equivalence theory, which may be of independent interest.

Distributed inference/estimation in Bayesian framework in the context of sensor networks has recently received much attention due to its broad applicability. The variational Bayesian (VB) algorithm is a technique for approximating intractable integrals arising in Bayesian inference. In this paper, we propose two novel distributed VB algorithms for general Bayesian inference problem, which can be applied to a very general class of conjugate-exponential models. In the first approach, the global natural parameters at each node are optimized using a stochastic natural gradient that utilizes the Riemannian geometry of the approximation space, followed by an information diffusion step for cooperation with the neighbors. In the second method, a constrained optimization formulation for distributed estimation is established in natural parameter space and solved by alternating direction method of multipliers (ADMM). An application of the distributed inference/estimation of a Bayesian Gaussian mixture model is then presented, to evaluate the effectiveness of the proposed algorithms. Simulations on both synthetic and real datasets demonstrate that the proposed algorithms have excellent performance, which are almost as good as the corresponding centralized VB algorithm relying on all data available in a fusion center.

In this paper, we consider a novel framework of positive-unlabeled data in which as positive data survival times are observed for subjects who have events during the observation time as positive data and as unlabeled data censoring times are observed but whether the event occurs or not are unknown for some subjects. We consider two cases: (1) when censoring time is observed in positive data, and (2) when it is not observed. For both cases, we developed parametric models, nonparametric models, and machine learning models and the estimation strategies for these models. Simulation studies show that under this data setup, traditional survival analysis may yield severely biased results, while the proposed estimation method can provide valid results.

The Bayesian treatment of neural networks dictates that a prior distribution is specified over their weight and bias parameters. This poses a challenge because modern neural networks are characterized by a large number of parameters, and the choice of these priors has an uncontrolled effect on the induced functional prior, which is the distribution of the functions obtained by sampling the parameters from their prior distribution. We argue that this is a hugely limiting aspect of Bayesian deep learning, and this work tackles this limitation in a practical and effective way. Our proposal is to reason in terms of functional priors, which are easier to elicit, and to "tune" the priors of neural network parameters in a way that they reflect such functional priors. Gaussian processes offer a rigorous framework to define prior distributions over functions, and we propose a novel and robust framework to match their prior with the functional prior of neural networks based on the minimization of their Wasserstein distance. We provide vast experimental evidence that coupling these priors with scalable Markov chain Monte Carlo sampling offers systematically large performance improvements over alternative choices of priors and state-of-the-art approximate Bayesian deep learning approaches. We consider this work a considerable step in the direction of making the long-standing challenge of carrying out a fully Bayesian treatment of neural networks, including convolutional neural networks, a concrete possibility.

Imagery is frequently used to model, represent and communicate knowledge. In particular, graphs are one of the most powerful tools, being able to represent relations between objects. Causal relations are frequently represented by directed graphs, with nodes denoting causes and links denoting causal influence. A causal graph is a skeletal picture, showing causal associations and impact between entities. Common methods used for graphically representing causal scenarios are neurons, truth tables, causal Bayesian networks, cognitive maps and Petri Nets. Causality is often defined in terms of precedence (the cause precedes the effect), concurrency (often, an effect is provoked simultaneously by two or more causes), circularity (a cause provokes the effect and the effect reinforces the cause) and imprecision (the presence of the cause favors the effect, but not necessarily causes it). We will show that, even though the traditional graphical models are able to represent separately some of the properties aforementioned, they fail trying to illustrate indistinctly all of them. To approach that gap, we will introduce Fuzzy Stochastic Timed Petri Nets as a graphical tool able to represent time, co-occurrence, looping and imprecision in causal flow.

The defining challenge for causal inference from observational data is the presence of `confounders', covariates that affect both treatment assignment and the outcome. To address this challenge, practitioners collect and adjust for the covariates, hoping that they adequately correct for confounding. However, including every observed covariate in the adjustment runs the risk of including `bad controls', variables that \emph{induce} bias when they are conditioned on. The problem is that we do not always know which variables in the covariate set are safe to adjust for and which are not. To address this problem, we develop Nearly Invariant Causal Estimation (NICE). NICE uses invariant risk minimization (IRM) [Arj19] to learn a representation of the covariates that, under some assumptions, strips out bad controls but preserves sufficient information to adjust for confounding. Adjusting for the learned representation, rather than the covariates themselves, avoids the induced bias and provides valid causal inferences. NICE is appropriate in the following setting. i) We observe data from multiple environments that share a common causal mechanism for the outcome, but that differ in other ways. ii) In each environment, the collected covariates are a superset of the causal parents of the outcome, and contain sufficient information for causal identification. iii) But the covariates also may contain bad controls, and it is unknown which covariates are safe to adjust for and which ones induce bias. We evaluate NICE on both synthetic and semi-synthetic data. When the covariates contain unknown collider variables and other bad controls, NICE performs better than existing methods that adjust for all the covariates.

There is significant growth and interest in the use of synthetic data as an enabler for machine learning in environments where the release of real data is restricted due to privacy or availability constraints. Despite a large number of methods for synthetic data generation, there are comparatively few results on the statistical properties of models learnt on synthetic data, and fewer still for situations where a researcher wishes to augment real data with another party's synthesised data. We use a Bayesian paradigm to characterise the updating of model parameters when learning in these settings, demonstrating that caution should be taken when applying conventional learning algorithms without appropriate consideration of the synthetic data generating process and learning task. Recent results from general Bayesian updating support a novel and robust approach to Bayesian synthetic-learning founded on decision theory that outperforms standard approaches across repeated experiments on supervised learning and inference problems.

In many projects I carried out, companies, despite having fantastic AI business ideas, display a tendency to slowly become frustrated when they realize that they do not have enough data… However, solutions do exist! The purpose of this article is to briefly introduce you to some of them (the ones that are proven effective in my practice) rather than to list all existing solutions. The problem of data scarcity is very important since data are at the core of any AI project. The size of a dataset is often responsible for poor performances in ML projects. Most of the time, data related issues are the main reason why great AI projects cannot be accomplished.

The newly emerged machine learning (e.g. deep learning) methods have become a strong driving force to revolutionize a wide range of industries, such as smart healthcare, financial technology, and surveillance systems. Meanwhile, privacy has emerged as a big concern in this machine learning-based artificial intelligence era. It is important to note that the problem of privacy preservation in the context of machine learning is quite different from that in traditional data privacy protection, as machine learning can act as both friend and foe. Currently, the work on the preservation of privacy and machine learning (ML) is still in an infancy stage, as most existing solutions only focus on privacy problems during the machine learning process. Therefore, a comprehensive study on the privacy preservation problems and machine learning is required. This paper surveys the state of the art in privacy issues and solutions for machine learning. The survey covers three categories of interactions between privacy and machine learning: (i) private machine learning, (ii) machine learning aided privacy protection, and (iii) machine learning-based privacy attack and corresponding protection schemes. The current research progress in each category is reviewed and the key challenges are identified. Finally, based on our in-depth analysis of the area of privacy and machine learning, we point out future research directions in this field.

Handwritten Digit Recognition is an interesting machine learning problem in which we have to identify the handwritten digits through various classification algorithms. There are a number of ways and algorithms to recognize handwritten digits, including Deep Learning/CNN, SVM, Gaussian Naive Bayes, KNN, Decision Trees, Random Forests, etc. In this article, we will deploy a variety of machine learning algorithms from the Sklearn's library on our dataset to classify the digits into their categories. We will use Sklearn's load_digits dataset, which is a collection of 8x8 images (64 features)of digits. The dataset contains a total of 1797 sample points.