Word2vec (Mikolov et al., 2013b) has proven to be successful in natural language processing by capturing the semantic relationships between different words. Built on top of single-word embeddings, paragraph vectors (Le and Mikolov, 2014) find fixed-length representations for pieces of text with arbitrary lengths, such as documents, paragraphs, and sentences. In this work, we propose a novel interpretation for neural-network-based paragraph vectors by developing an unsupervised generative model whose maximum likelihood solution corresponds to traditional paragraph vectors. This probabilistic formulation allows us to go beyond point estimates of parameters and to perform Bayesian posterior inference. We find that the entropy of paragraph vectors decreases with the length of documents, and that information about posterior uncertainty improves performance in supervised learning tasks such as sentiment analysis and paraphrase detection.

Various applications involve assigning discrete label values to a collection of objects based on some pairwise noisy data. Due to the discrete---and hence nonconvex---structure of the problem, computing the optimal assignment (e.g.~maximum likelihood assignment) becomes intractable at first sight. This paper makes progress towards efficient computation by focusing on a concrete joint alignment problem---that is, the problem of recovering $n$ discrete variables $x_i \in \{1,\cdots, m\}$, $1\leq i\leq n$ given noisy observations of their modulo differences $\{x_i - x_j~\mathsf{mod}~m\}$. We propose a low-complexity and model-free procedure, which operates in a lifted space by representing distinct label values in orthogonal directions, and which attempts to optimize quadratic functions over hypercubes. Starting with a first guess computed via a spectral method, the algorithm successively refines the iterates via projected power iterations. We prove that for a broad class of statistical models, the proposed projected power method makes no error---and hence converges to the maximum likelihood estimate---in a suitable regime. Numerical experiments have been carried out on both synthetic and real data to demonstrate the practicality of our algorithm. We expect this algorithmic framework to be effective for a broad range of discrete assignment problems.

We present new excess risk bounds for general unbounded loss functions including log loss and squared loss, where the distribution of the losses may be heavy-tailed. The bounds hold for general estimators, but they are optimized when applied to $\eta$-generalized Bayesian, MDL, and ERM estimators. When applied with log loss, the bounds imply convergence rates for generalized Bayesian inference under misspecification in terms of a generalization of the Hellinger metric as long as the learning rate $\eta$ is set correctly. For general loss functions, our bounds rely on two separate conditions: the $v$-GRIP (generalized reversed information projection) conditions, which control the lower tail of the excess loss; and the newly introduced witness condition, which controls the upper tail. The parameter $v$ in the $v$-GRIP conditions determines the achievable rate and is akin to the exponent in the well-known Tsybakov margin condition and the Bernstein condition for bounded losses, which the $v$-GRIP conditions generalize; favorable $v$ in combination with small model complexity leads to $\tilde{O}(1/n)$ rates. The witness condition allows us to connect the excess risk to an 'annealed' version thereof, by which we generalize several previous results connecting Hellinger and R\'enyi divergence to KL divergence.

Graphical models provide powerful tools to uncover complicated patterns in multivariate data and are commonly used in Bayesian statistics and machine learning. In this paper, we introduce an R package BDgraph which performs Bayesian structure learning for general undirected graphical models with either continuous or discrete variables. The package efficiently implements recent improvements in the Bayesian literature. To speed up computations, the computationally intensive tasks have been implemented in C++ and interfaced with R. In addition, the package contains several functions for simulation and visualization, as well as two multivariate datasets taken from the literature and are used to describe the package capabilities. The paper includes a brief overview of the statistical methods which have been implemented in the package. The main body of the paper explains how to use the package. Furthermore, we illustrate the package's functionality in both real and artificial examples, as well as in an extensive simulation study.

A variety of machine learning tasks---e.g., matrix factorization, topic modelling, and feature allocation---can be viewed as learning the parameters of a probability distribution over bipartite graphs. Recently, a new class of models for networks, the sparse exchangeable graphs, have been introduced to resolve some important pathologies of traditional approaches to statistical network modelling; most notably, the inability to model sparsity (in the asymptotic sense). The present paper explains some practical insights arising from this work. We first show how to check if sparsity is relevant for modelling a given (fixed size) dataset by using network subsampling to identify a simple signature of sparsity. We discuss the implications of the (sparse) exchangeable subsampling theory for test-train dataset splitting; we argue common approaches can lead to biased results, and we propose a principled alternative. Finally, we study sparse exchangeable Poisson matrix factorization as a worked example. In particular, we show how to adapt mean field variational inference to the sparse exchangeable setting, allowing us to scale inference to huge datasets.

In this paper, we propose a new algorithm for learning general latent-variable probabilistic graphical models using the techniques of predictive state representation, instrumental variable regression, and reproducing-kernel Hilbert space embeddings of distributions. Under this new learning framework, we first convert latent-variable graphical models into corresponding latent-variable junction trees, and then reduce the hard parameter learning problem into a pipeline of supervised learning problems, whose results will then be used to perform predictive belief propagation over the latent junction tree during the actual inference procedure. We then give proofs of our algorithm's correctness, and demonstrate its good performance in experiments on one synthetic dataset and two real-world tasks from computational biology and computer vision -- classifying DNA splice junctions and recognizing human actions in videos.

Computing accurate estimates of the Fourier transform of analog signals from discrete data points is important in many fields of science and engineering. The conventional approach of performing the discrete Fourier transform of the data implicitly assumes periodicity and bandlimitedness of the signal. In this paper, we use Gaussian process regression to estimate the Fourier transform (or any other integral transform) without making these assumptions. This is possible because the posterior expectation of Gaussian process regression maps a finite set of samples to a function defined on the whole real line, expressed as a linear combination of covariance functions. We estimate the covariance function from the data using an appropriately designed gradient ascent method that constrains the solution to a linear combination of tractable kernel functions. This procedure results in a posterior expectation of the analog signal whose Fourier transform can be obtained analytically by exploiting linearity. Our simulations show that the new method leads to sharper and more precise estimation of the spectral density both in noise-free and noise-corrupted signals. We further validate the method in two real-world applications: the analysis of the yearly fluctuation in atmospheric CO2 level and the analysis of the spectral content of brain signals.

We study the Nonparametric Maximum Likelihood Estimator (NPMLE) for estimating Gaussian location mixture densities in $d$-dimensions from independent observations. Unlike usual likelihood-based methods for fitting mixtures, NPMLEs are based on convex optimization. We prove finite sample results on the Hellinger accuracy of every NPMLE. Our results imply, in particular, that every NPMLE achieves near parametric risk (up to logarithmic multiplicative factors) when the true density is a discrete Gaussian mixture without any prior information on the number of mixture components. NPMLEs can naturally be used to yield empirical Bayes estimates of the Oracle Bayes estimator in the Gaussian denoising problem. We prove bounds for the accuracy of the empirical Bayes estimate as an approximation to the Oracle Bayes estimator. Here our results imply that the empirical Bayes estimator performs at nearly the optimal level (up to logarithmic multiplicative factors) for denoising in clustering situations without any prior knowledge of the number of clusters.

Transfer operators such as the Perron-Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings in the machine learning community. Moreover, numerical methods to compute empirical estimates of these embeddings are akin to data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. In fact, most of the existing methods can be derived from our framework, providing a unifying view on the approximation of transfer operators. One main benefit of the presented kernel-based approaches is that these methods can be applied to any domain where a similarity measure given by a kernel is available. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis.

We're approaching the end of this series on empirical Bayesian methods, and have touched on many statistical approaches for analyzing binomial (success / total) data, all with the goal of estimating the "true" batting average of each player. There's one question we haven't answered, though: do these methods actually work? Even if we assume each player has a "true" batting average as our model suggests, we don't know it, so we can't see if our methods estimated it accurately. For example, we think that empirical Bayes shrinkage gets closer to the true probabilities than raw batting averages do, but we can't actually measure the mean-squared error. This means we can't test our methods, or examine when they work well and when they don't.