We provide the first polynomial-sample and polynomial-time algorithm for learning tree-structured Ising models. In particular, we show that $n$-variable tree-structured Ising models can be learned computationally-efficiently to within total variation distance~$\epsilon$ from an optimal $O(n \log n/\epsilon^2)$ samples, where $O(.)$ hides an absolute constant which does not depend on the model being learned -- neither its tree nor the magnitude of its edge strengths, on which we place no assumptions. Our guarantees hold, in fact, for the celebrated Chow-Liu [1968] algorithm, using the plug-in estimator for mutual information. While this (or any other) algorithm may fail to identify the structure of the underlying model correctly from a finite sample, we show that it will still learn a tree-structured model that is close to the true one in TV distance, a guarantee called "proper learning." Prior to our work there were no known sample- and time-efficient algorithms for learning (properly or non-properly) arbitrary tree-structured graphical models. In particular, our guarantees cannot be derived from known results for the Chow-Liu algorithm and the ensuing literature on learning graphical models, including a recent renaissance of algorithms on this learning challenge, which only yield asymptotic consistency results, or sample-inefficient and/or time-inefficient algorithms, unless further assumptions are placed on the graphical model, such as bounds on the "strengths" of the model's edges. While we establish guarantees for a widely known and simple algorithm, the analysis that this algorithm succeeds is quite complex, requiring a hierarchical classification of the edges into layers with different reconstruction guarantees, depending on their strength, combined with delicate uses of the subadditivity of the squared Hellinger distance over graphical models to control the error accumulation.

Multivariate Hawkes processes are commonly used to model streaming networked event data in a wide variety of applications. However, it remains a challenge to extract reliable inference from complex datasets with uncertainty quantification. Aiming towards this, we develop a statistical inference framework to learn causal relationships between nodes from networked data, where the underlying directed graph implies Granger causality. We provide uncertainty quantification for the maximum likelihood estimate of the network multivariate Hawkes process by providing a non-asymptotic confidence set. The main technique is based on the concentration inequalities of continuous-time martingales. We compare our method to the previously-derived asymptotic Hawkes process confidence interval, and demonstrate the strengths of our method in an application to neuronal connectivity reconstruction.

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.

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.

Estimating epistemic uncertainty of models used in low-latency applications and Out-Of-Distribution samples detection is a challenge due to the computationally demanding nature of uncertainty estimation techniques. Estimating model uncertainty using approximation techniques like Monte Carlo Dropout (MCD), DropConnect (MCDC) requires a large number of forward passes through the network, rendering them inapt for low-latency applications. We propose Select-DC which uses a subset of layers in a neural network to model epistemic uncertainty with MCDC. Through our experiments, we show a significant reduction in the GFLOPS required to model uncertainty, compared to Monte Carlo DropConnect, with marginal trade-off in performance. We perform a suite of experiments on CIFAR 10, CIFAR 100, and SVHN datasets with ResNet and VGG models. We further show how applying DropConnect to various layers in the network with different drop probabilities affects the networks performance and the entropy of the predictive distribution.