1. Nonparametric statistical models 2. Gaussian processes 3. Empirical processes 4. Function spaces and approximation theory 5. Linear nonparametric estimators 6. The minimax paradigm 7. Likelihood-based procedures 8. Adaptive inference.

Mathematical foundations of infinite-dimensional statistical models

Although considerable research has concentrated on online campaigning, it is still unclear how politicians use different social media platforms in political communication. Focusing on the German fe...

https://www.tandfonline.com/doi/pdf/10.1080/10584609.2017.1334728

Election Campaigning on Social Media: Politicians, Audiences, and the Mediation of Political Communication on Facebook and Twitter

We survey latent variable models for solving data-analysis problems. A latent variable model is a probabilistic model that encodes hidden patterns in the data. We uncover these patterns from their conditional distribution and use them to summarize data and form predictions. Latent variable models are important in many fields, including computational biology, natural language processing, and social network analysis. Our perspective is that models are developed iteratively: We build a model, use it to analyze data, assess how it succeeds and fails, revise it, and repeat. We describe how new research has transformed these essential activities. First, we describe probabilistic graphical models, a language for formulating latent variable models. Second, we describe mean field variational inference, a generic algorithm for approximating conditional distributions. Third, we describe how to use our analyses to solve problems: exploring the data, forming predictions, and pointing us in the direction of improved mo...

Build, Compute, Critique, Repeat: Data Analysis with Latent Variable Models

We introduce HD (or “Hierarchical-Deep”) models, a new compositional learning architecture that integrates deep learning models with structured hierarchical Bayesian (HB) models. Specifically, we show how we can learn a hierarchical Dirichlet process (HDP) prior over the activities of the top-level features in a deep Boltzmann machine (DBM). This compound HDP-DBM model learns to learn novel concepts from very few training example by learning low-level generic features, high-level features that capture correlations among low-level features, and a category hierarchy for sharing priors over the high-level features that are typical of different kinds of concepts. We present efficient learning and inference algorithms for the HDP-DBM model and show that it is able to learn new concepts from very few examples on CIFAR-100 object recognition, handwritten character recognition, and human motion capture datasets.

/pdf/learning-with-hierarchical-deep-models-82ltplwvcf.pdf

Learning with Hierarchical-Deep Models

Video analysis often begins with background subtraction. This problem is often approached in two steps-a background model followed by a regularisation scheme. A model of the background allows it to be distinguished on a per-pixel basis from the foreground, whilst the regularisation combines information from adjacent pixels. We present a new method based on Dirichlet process Gaussian mixture models, which are used to estimate per-pixel background distributions. It is followed by probabilistic regularisation. Using a non-parametric Bayesian method allows per-pixel mode counts to be automatically inferred, avoiding over-/under- fitting. We also develop novel model learning algorithms for continuous update of the model in a principled fashion as the scene changes. These key advantages enable us to outperform the state-of-the-art alternatives on four benchmarks.

/pdf/background-subtraction-with-dirichletprocess-mixture-models-bz3b18ok2n.pdf

Background Subtraction with DirichletProcess Mixture Models

Hierarchical modeling is a fundamental concept in Bayesian statistics. The basic idea is that parameters are endowed with distributions which may themselves introduce new parameters, and this construction recurses. In this review we discuss the role of hierarchical modeling in Bayesian nonparametrics, focusing on models in which the infinite-dimensional parameters are treated hierarchically. For example, we consider a model in which the base measure for a Dirichlet process is itself treated as a draw from another Dirichlet process. This yields a natural recursion that we refer to as a hierarchical Dirichlet process. We also discuss hierarchies based on the Pitman-Yor process and on completely random processes. We demonstrate the value of these hierarchical constructions in a wide range of practical applications, in problems in computational biology, computer vision and natural language processing.

/pdf/bayesian-nonparametrics-hierarchical-bayesian-nonparametric-4xtt1vncfz.pdf

Michael I. Jordan

Papers

Bayesian Nonparametrics: Hierarchical Bayesian nonparametric models with applications