Geometric embeddings have recently received attention for their natural ability to represent transitive asymmetric relations via containment. Box embeddings, where objects are represented by n-dimensional hyperrectangles, are a particularly promising example of such an embedding as they are closed under intersection and their volume can be calculated easily, allowing them to naturally represent calibrated probability distributions. The benefits of geometric embeddings also introduce a problem of local identifiability, however, where whole neighborhoods of parameters result in equivalent loss which impedes learning. Prior work addressed some of these issues by using an approximation to Gaussian convolution over the box parameters, however, this intersection operation also increases the sparsity of the gradient. In this work, we model the box parameters with min and max Gumbel distributions, which were chosen such that space is still closed under the operation of the intersection. The calculation of the expected intersection volume involves all parameters, and we demonstrate experimentally that this drastically improves the ability of such models to learn.

The tragedy happened to the AirFrance 447 more than 10 years ago, in 2009. The flight took off in Rio de Janeiro and was planned to land in Paris. It suddenly disappeared in the middle of the Atlantic ocean without any warning. Immediately, rescuers reached the zone and what they found were just some wreckage and corpse. All 228 people onboard died in the crash.

Offered by Duke University. In this Specialization, you will learn to analyze and visualize data in R and create reproducible data analysis reports, demonstrate a conceptual understanding of the unified nature of statistical inference, perform frequentist and Bayesian statistical inference and modeling to understand natural phenomena and make data-based decisions, communicate statistical results correctly, effectively, and in context without relying on statistical jargon, critique data-based claims and evaluated data-based decisions, and wrangle and visualize data with R packages for data analysis. You will produce a portfolio of data analysis projects from the Specialization that demonstrates mastery of statistical data analysis from exploratory analysis to inference to modeling, suitable for applying for statistical analysis or data scientist positions.

We study a decentralized cooperative multi-agent multi-armed bandit problem with $K$ arms and $N$ agents connected over a network. In our model, each arm's reward distribution is same for all agents, and rewards are drawn independently across agents and over time steps. In each round, agents choose an arm to play and subsequently send a message to their neighbors. The goal is to minimize cumulative regret averaged over the entire network. We propose a decentralized Bayesian multi-armed bandit framework that extends single-agent Bayesian bandit algorithms to the decentralized setting. Specifically, we study an information assimilation algorithm that can be combined with existing Bayesian algorithms, and using this, we propose a decentralized Thompson Sampling algorithm and decentralized Bayes-UCB algorithm. We analyze the decentralized Thompson Sampling algorithm under Bernoulli rewards and establish a problem-dependent upper bound on the cumulative regret. We show that regret incurred scales logarithmically over the time horizon with constants that match those of an optimal centralized agent with access to all observations across the network. Our analysis also characterizes the cumulative regret in terms of the network structure. Through extensive numerical studies, we show that our extensions of Thompson Sampling and Bayes-UCB incur lesser cumulative regret than the state-of-art algorithms inspired by the Upper Confidence Bound algorithm. We implement our proposed decentralized Thompson Sampling under gossip protocol, and over time-varying networks, where each communication link has a fixed probability of failure.

Stochastic inverse problems (SIP) address the behavior of a set of objects of the same kind but with variable properties, such as a population of cells. Using a population of mechanistic models from a single parametric family, SIP explains population variability by transferring real-world observations into the latent space of model parameters. Previous research in SIP focused on solving the parameter inference problem for a single population using Markov chain Monte Carlo methods. Here we extend SIP to address multiple related populations simultaneously. Specifically, we simulate control and treatment populations in experimental protocols by discovering two related latent spaces of model parameters. Instead of taking a Bayesian approach, our two-population SIP is reformulated as the constrained-optimization problem of finding distributions of model parameters. To minimize the divergence between distributions of experimental observations and model outputs, we developed novel deep learning models based on generative adversarial networks (GANs) which have the structure of our underlying constrained-optimization problem. The flexibility of GANs allowed us to build computationally scalable solutions and tackle complex model input parameter inference scenarios, which appear routinely in physics, biophysics, economics and other areas, and which can not be handled with existing methods. Specifically, we demonstrate two scenarios of parameter inference over a control population and a treatment population whose treatment either selectively affects only a subset of model parameters with some uncertainty or has a deterministic effect on all model parameters.

Unlike in the traditional statistical modeling for which a user typically hand-specify a prior, Neural Processes (NPs) implicitly define a broad class of stochastic processes with neural networks. Given a data stream, NP learns a stochastic process that best describes the data. While this "data-driven" way of learning stochastic processes has proven to handle various types of data, NPs still rely on an assumption that uncertainty in stochastic processes is modeled by a single latent variable, which potentially limits the flexibility. To this end, we propose the Boostrapping Neural Process (BNP), a novel extension of the NP family using the bootstrap. The bootstrap is a classical data-driven technique for estimating uncertainty, which allows BNP to learn the stochasticity in NPs without assuming a particular form. We demonstrate the efficacy of BNP on various types of data and its robustness in the presence of model-data mismatch.

Convolutional Neural Networks (CNNs) have proven to be state-of-the-art models for supervised computer vision tasks, such as image classification. However, large labeled data sets are generally needed for the training and validation of such models. In many domains, unlabeled data is available but labeling is expensive, for instance when specific expert knowledge is required. Active Learning (AL) is one approach to mitigate the problem of limited labeled data. Through selecting the most informative and representative data instances for labeling, AL can contribute to more efficient learning of the model. Recent AL methods for CNNs propose different solutions for the selection of instances to be labeled. However, they do not perform consistently well and are often computationally expensive. In this paper, we propose a novel AL algorithm that efficiently learns from unlabeled data by capturing high prediction uncertainty. By replacing the softmax standard output of a CNN with the parameters of a Dirichlet density, the model learns to identify data instances that contribute efficiently to improving model performance during training. We demonstrate in several experiments with publicly available data that our method consistently outperforms other state-of-the-art AL approaches. It can be easily implemented and does not require extensive computational resources for training. Additionally, we are able to show the benefits of the approach on a real-world medical use case in the field of automated detection of visual signals for pneumonia on chest radiographs.

One of the core assumptions in causal discovery is the faithfulness assumption---i.e. assuming that independencies found in the data are due to separations in the true causal graph. This assumption can, however, be violated in many ways, including xor connections, deterministic functions or cancelling paths. In this work, we propose a weaker assumption that we call 2-adjacency faithfulness. In contrast to adjacency faithfulness, which assumes that there is no conditional independence between each pair of variables that are connected in the causal graph, we only require no conditional independence between a node and a subset of its Markov blanket that can contain up to two nodes. Equivalently, we adapt orientation faithfulness to this setting. We further propose a sound orientation rule for causal discovery that applies under weaker assumptions. As a proof of concept, we derive a modified Grow and Shrink algorithm that recovers the Markov blanket of a target node and prove its correctness under strictly weaker assumptions than the standard faithfulness assumption.

We present a novel approach to Bayesian inference and general Bayesian computation that is defined through a recursive partitioning of the sample space. It does not rely on gradients, nor require any problem-specific tuning, and is asymptotically exact for any density function with a bounded domain. The output is an approximation to the whole density function including the normalization constant, via partitions organized in efficient data structures. This allows for evidence estimation, as well as approximate posteriors that allow for fast sampling and fast evaluations of the density. It shows competitive performance to recent state-of-the-art methods on synthetic and real-world problem examples including parameter inference for gravitational-wave physics.

Bayesian quadrature (BQ) is a method for solving numerical integration problems in a Bayesian manner, which allows users to quantify their uncertainty about the solution. The standard approach to BQ is based on a Gaussian process (GP) approximation of the integrand. As a result, BQ is inherently limited to cases where GP approximations can be done in an efficient manner, thus often prohibiting very high-dimensional or non-smooth target functions. This paper proposes to tackle this issue with a new Bayesian numerical integration algorithm based on Bayesian Additive Regression Trees (BART) priors, which we call BART-Int. BART priors are easy to tune and well-suited for discontinuous functions. We demonstrate that they also lend themselves naturally to a sequential design setting and that explicit convergence rates can be obtained in a variety of settings. The advantages and disadvantages of this new methodology are highlighted on a set of benchmark tests including the Genz functions, and on a Bayesian survey design problem.