Download or read book Learning Directed Graphical Models with Latent Variables written by Basil N. Saeed and published by . This book was released on 2020 with total page 56 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the problem of learning directed graphical models with latent variables, represented by directed maximal ancestral graphs, from a conditional independence oracle. We show that given a set of separation statements from some directed maximal ancestral graph G* = (V*,E*), we can map posets with ground set V* to minimal IMAPs of G* such that the sparsest of these minimal IMAPs is Markov equivalent to G*. We give a diagrammatic interpretation of these minimal IMAPs in terms of the Hasse diagram of the poset of posets. Namely, the Hasse diagram of these minimal IMAPs corresponds to the Hasse diagram of the poset of posets after identifying posets that map to the same minimal IMAP. We show that moving between these minimal IMAPs using legitimate mark changes corresponds to covering relations in the poset obtained after identification. Finally, we conjecture that a greedy search to minimize sparsity over this contracted space by moving between minimal IMAPs using legitimate mark changes converges to G*.
Download or read book Learning in Graphical Models written by M.I. Jordan and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 658 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the past decade, a number of different research communities within the computational sciences have studied learning in networks, starting from a number of different points of view. There has been substantial progress in these different communities and surprising convergence has developed between the formalisms. The awareness of this convergence and the growing interest of researchers in understanding the essential unity of the subject underlies the current volume. Two research communities which have used graphical or network formalisms to particular advantage are the belief network community and the neural network community. Belief networks arose within computer science and statistics and were developed with an emphasis on prior knowledge and exact probabilistic calculations. Neural networks arose within electrical engineering, physics and neuroscience and have emphasised pattern recognition and systems modelling problems. This volume draws together researchers from these two communities and presents both kinds of networks as instances of a general unified graphical formalism. The book focuses on probabilistic methods for learning and inference in graphical models, algorithm analysis and design, theory and applications. Exact methods, sampling methods and variational methods are discussed in detail. Audience: A wide cross-section of computationally oriented researchers, including computer scientists, statisticians, electrical engineers, physicists and neuroscientists.
Download or read book Learning and Inference in Latent Variable Graphical Models written by Wei Ping and published by . This book was released on 2016 with total page 167 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probabilistic graphical models such as Markov random fields provide a powerful framework and tools for machine learning, especially for structured output learning. Latent variables naturally exist in many applications of these models; they may arise from partially labeled data, or be introduced to enrich model flexibility. However, the presence of latent variables presents challenges for learning and inference.For example, the standard approach of using maximum a posteriori (MAP) prediction is complicated by the fact that, in latent variable models (LVMs), we typically want to first marginalize out the latent variables, leading to an inference task called marginal MAP. Unfortunately, marginal MAP prediction can be NP-hard even on relatively simple models such as trees, and few methods have been developed in the literature. This thesis presents a class of variational bounds for marginal MAP that generalizes the popular dual-decomposition method for MAP inference, and enables an efficient block coordinate descent algorithm to solve the corresponding optimization. Similarly, when learning LVMs for structured prediction, it is critically important to maintain the effect of uncertainty over latent variables by marginalization. We propose the marginal structured SVM, which uses marginal MAP inference to properly handle that uncertainty inside the framework of max-margin learning.We then turn our attention to an important subclass of latent variable models, restricted Boltzmann machines (RBMs). RBMs are two-layer latent variable models that are widely used to capture complex distributions of observed data, including as building block for deep probabilistic models. One practical problem in RBMs is model selection: we need to determine the hidden (latent) layer size before performing learning. We propose an infinite RBM model and apply the Frank-Wolfe algorithm to solve the resulting learning problem. The resulting algorithm can be interpreted as inserting a hidden variable into a RBM model at each iteration, to easily and efficiently perform model selection during learning. We also study the role of approximate inference in RBMs and conditional RBMs. In particular, there is a common assumption that belief propagation methods do not work well on RBM-based models, especially for learning. In contrast, we demonstrate that for conditional models and structured prediction, learning RBM-based models with belief propagation and its variants can provide much better results than the state-of-the-art contrastive divergence methods.
Download or read book Structure Learning in Graphical Modeling written by Mathias Drton and published by . This book was released on 2017 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: A graphical model is a statistical model that is associated with a graph whose nodes correspond to variables of interest. The edges of the graph reflect allowed conditional dependencies among the variables. Graphical models have computationally convenient factorization properties and have long been a valuable tool for tractable modeling of multivariate distributions. More recently, applications such as reconstructing gene regulatory networks from gene expression data have driven major advances in structure learning, that is, estimating the graph underlying a model. We review some of these advances and discuss methods such as the graphical lasso and neighborhood selection for undirected graphical models (or Markov random fields) and the PC algorithm and score-based search methods for directed graphical models (or Bayesian networks). We further review extensions that account for effects of latent variables and heterogeneous data sources.
Download or read book Global Variational Learning for Graphical Models with Latent Variables written by Ahmed M. Abdelatty and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Probabilistic Graphical Models have been used intensively for developing Machine Learning applications including Computer Vision, Natural Language processing, Collaborative Filtering, and Bioinformatics. Moreover, Graphical Models with latent variables are very powerful tools for modeling uncertainty, since latent variables can be used to represent unobserved factors, and they also can be used to model the correlations between the observed variables. However, global learning of Latent Variable Models (LVMs) is NP-hard in general, and the state-of-the-art algorithm for learning them such as Expectation Maximization algorithm can get stuck in local optimum. In this thesis, we address the problem of global variational learning for LVMs. More precisely, we propose a convex variational approximation for Maximum Likelihood Learning and apply Frank-Wolfe algorithm to solve it. We also investigate the use of the Global Optimization Algorithm (GOP) for Bayesian Learning, and we demonstrate that it converges to the global optimum.
Download or read book Latent Variable Models and Factor Analysis written by David J. Bartholomew and published by Wiley. This book was released on 1999-08-10 with total page 214 pages. Available in PDF, EPUB and Kindle. Book excerpt: Hitherto latent variable modelling has hovered on the fringes of the statistical mainstream but if the purpose of statistics is to deal with real problems, there is every reason for it to move closer to centre stage. In the social sciences especially, latent variables are common and if they are to be handled in a truly scientific manner, statistical theory must be developed to include them. This book aims to show how that should be done. This second edition is a complete re-working of the book of the same name which appeared in the Griffin’s Statistical Monographs in 1987. Since then there has been a surge of interest in latent variable methods which has necessitated a radical revision of the material but the prime object of the book remains the same. It provides a unified and coherent treatment of the field from a statistical perspective. This is achieved by setting up a sufficiently general framework to enable the derivation of the commonly used models. The subsequent analysis is then done wholly within the realm of probability calculus and the theory of statistical inference. Numerical examples are provided as well as the software to carry them out ( where this is not otherwise available). Additional data sets are provided in some cases so that the reader can aquire a wider experience of analysis and interpretation.
Download or read book Handbook of Graphical Models written by Marloes Maathuis and published by CRC Press. This book was released on 2018-11-12 with total page 536 pages. Available in PDF, EPUB and Kindle. Book excerpt: A graphical model is a statistical model that is represented by a graph. The factorization properties underlying graphical models facilitate tractable computation with multivariate distributions, making the models a valuable tool with a plethora of applications. Furthermore, directed graphical models allow intuitive causal interpretations and have become a cornerstone for causal inference. While there exist a number of excellent books on graphical models, the field has grown so much that individual authors can hardly cover its entire scope. Moreover, the field is interdisciplinary by nature. Through chapters by leading researchers from different areas, this handbook provides a broad and accessible overview of the state of the art. Key features: * Contributions by leading researchers from a range of disciplines * Structured in five parts, covering foundations, computational aspects, statistical inference, causal inference, and applications * Balanced coverage of concepts, theory, methods, examples, and applications * Chapters can be read mostly independently, while cross-references highlight connections The handbook is targeted at a wide audience, including graduate students, applied researchers, and experts in graphical models.
Download or read book Latent Clustered Causal Models written by Annie Yun and published by . This book was released on 2021 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the problem of learning directed graphical models in the presence of latent variables. We define latent clustered causal models as a particular restriction on directed graphical models with latent variables and corresponding clusters of observed nodes, characterized by edges between only observed and latent variables. We discuss this model's particular applicability towards genomics applications and examine its relationship to prior causal structure recovery work. We show identifiability results on this model and design a consistent three-stage algorithm that discovers clusters of observed nodes, a partial ordering over clusters, and finally, the entire structure over both observed and latent nodes. We also evaluate our method on synthetic datasets and demonstrate its performance in low sample-size regimes.
Download or read book Selecting Models from Data written by P. Cheeseman and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 475 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is a selection of papers presented at the Fourth International Workshop on Artificial Intelligence and Statistics held in January 1993. These biennial workshops have succeeded in bringing together researchers from Artificial Intelligence and from Statistics to discuss problems of mutual interest. The exchange has broadened research in both fields and has strongly encour aged interdisciplinary work. The theme ofthe 1993 AI and Statistics workshop was: "Selecting Models from Data". The papers in this volume attest to the diversity of approaches to model selection and to the ubiquity of the problem. Both statistics and artificial intelligence have independently developed approaches to model selection and the corresponding algorithms to implement them. But as these papers make clear, there is a high degree of overlap between the different approaches. In particular, there is agreement that the fundamental problem is the avoidence of "overfitting"-Le., where a model fits the given data very closely, but is a poor predictor for new data; in other words, the model has partly fitted the "noise" in the original data.
Download or read book Advances in Probabilistic Graphical Models written by Peter Lucas and published by Springer. This book was released on 2007-06-12 with total page 386 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book brings together important topics of current research in probabilistic graphical modeling, learning from data and probabilistic inference. Coverage includes such topics as the characterization of conditional independence, the learning of graphical models with latent variables, and extensions to the influence diagram formalism as well as important application fields, such as the control of vehicles, bioinformatics and medicine.
Download or read book Probabilistic Graphical Models written by Daphne Koller and published by MIT Press. This book was released on 2009-07-31 with total page 1270 pages. Available in PDF, EPUB and Kindle. Book excerpt: A general framework for constructing and using probabilistic models of complex systems that would enable a computer to use available information for making decisions. Most tasks require a person or an automated system to reason—to reach conclusions based on available information. The framework of probabilistic graphical models, presented in this book, provides a general approach for this task. The approach is model-based, allowing interpretable models to be constructed and then manipulated by reasoning algorithms. These models can also be learned automatically from data, allowing the approach to be used in cases where manually constructing a model is difficult or even impossible. Because uncertainty is an inescapable aspect of most real-world applications, the book focuses on probabilistic models, which make the uncertainty explicit and provide models that are more faithful to reality. Probabilistic Graphical Models discusses a variety of models, spanning Bayesian networks, undirected Markov networks, discrete and continuous models, and extensions to deal with dynamical systems and relational data. For each class of models, the text describes the three fundamental cornerstones: representation, inference, and learning, presenting both basic concepts and advanced techniques. Finally, the book considers the use of the proposed framework for causal reasoning and decision making under uncertainty. The main text in each chapter provides the detailed technical development of the key ideas. Most chapters also include boxes with additional material: skill boxes, which describe techniques; case study boxes, which discuss empirical cases related to the approach described in the text, including applications in computer vision, robotics, natural language understanding, and computational biology; and concept boxes, which present significant concepts drawn from the material in the chapter. Instructors (and readers) can group chapters in various combinations, from core topics to more technically advanced material, to suit their particular needs.
Download or read book Probabilistic Graphical Models for Computer Vision written by Qiang Ji and published by Academic Press. This book was released on 2019-12-12 with total page 322 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probabilistic Graphical Models for Computer Vision introduces probabilistic graphical models (PGMs) for computer vision problems and teaches how to develop the PGM model from training data. This book discusses PGMs and their significance in the context of solving computer vision problems, giving the basic concepts, definitions and properties. It also provides a comprehensive introduction to well-established theories for different types of PGMs, including both directed and undirected PGMs, such as Bayesian Networks, Markov Networks and their variants. Discusses PGM theories and techniques with computer vision examples Focuses on well-established PGM theories that are accompanied by corresponding pseudocode for computer vision Includes an extensive list of references, online resources and a list of publicly available and commercial software Covers computer vision tasks, including feature extraction and image segmentation, object and facial recognition, human activity recognition, object tracking and 3D reconstruction
Download or read book Provable Algorithms for Learning and Variational Inference in Undirected Graphical Models written by Frederic Koehler and published by . This book was released on 2021 with total page 263 pages. Available in PDF, EPUB and Kindle. Book excerpt: Graphical models are a general-purpose tool for modeling complex distributions in a way which facilitates probabilistic reasoning, with numerous applications across machine learning and the sciences. This thesis deals with algorithmic and statistical problems of learning a high-dimensional graphical model from samples, and related problems of performing inference on a known model, both areas of research which have been the subject of continued interest over the years. Our main contributions are the first computationally efficient algorithms for provably (1) learning a (possibly ill-conditioned) walk-summable Gaussian Graphical Model from samples, (2) learning a Restricted Boltzmann Machine (or other latent variable Ising model) from data, and (3) performing naive mean-field variational inference on an Ising model in the optimal density regime. These different problems illustrate a set of key principles, such as the diverse algorithmic applications of "pinning" variables in graphical models. We also show in some cases that these results are nearly optimal due to matching computational/cryptographic hardness results
Download or read book Large scale Directed Graphical Models Learning written by Gunwoong Park and published by . This book was released on 2016 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Directed graphical models are a powerful statistical method to compactly describe directional or causal relationships among the set of variables in large-scale data. However, a number of statistical and computational challenges arise that make learning directed graphical models often impossible for large-scale data. These issues include: (1) model identifiability; (2) computational guarantee; (3) sample size guarantee; and (4) combining interventional experiments with observational data. In this thesis, we focus on learning directed graphical models by addressing the above four issues. In Chapter 3, we discuss learning Poisson DAG models for modeling large-scale multivariate count data problems where each node is a Poisson random variable conditioning on its parents. We address the question of (1) model identifiability and learning algorithms with (2) computational complexity and (3) sample complexity. We prove that Poisson DAG models are fully identifiable from observational data using the notion of overdispersion, and present a polynomial-time algorithm that learns the Poisson DAG model under suitable regularity conditions. Chapter 4 focuses on learning a broader class of DAG models in large-scale settings. We address the issue of (1) model identifiability and learning algorithms with (2) computational complexity and (3) sample complexity. We introduce a new class of identifiable DAG models which include many interesting classes of distributions such as Poisson, Binomial, Geometric, Exponential, Gamma, and many more, and prove that this class of DAG models is fully identifiable using the idea of overdispersion. Furthermore, we develop statistically consistent and computationally tractable learning algorithms for the new class of identifiable DAG models in high-dimensional settings. Our algorithms exploits the sparsity of the graphs and overdispersion property. Chapter 5 concerns learning general DAG models using a combination of observational and interventional (or experimental) data. Prior work has focused on algorithms using Markov equivalence class (MEC) for the DAG and then using do-calculus rules based on interventions to learn the additional directions. However it has been shown that existing passive and active learning strategies that rely on accurate recovery of the MEC do not scale well to large-scale graphs because recovering MEC for DAG models are not successful large-scale graphs. Hence, we prove (1) model identifiability using the notion of the moralized graphs, and develop passive and active learning algorithms (4) combining interventional experiments with observational data. Lastly in Chapter 6, we concern learning directed cyclic graphical (DCG) models. We focus on (1) model identifiability for directed graphical models with feedback. We provide two new identifiability assumptions with respect to sparsity of a graph and the number of d-separation rules, and compare these new identifiability assumptions to the widely-held faithfulness and minimality assumptions. Furthermore we develop search algorithms for small-scale DCG models based on our new identifiability assumptions.
Download or read book Probabilistic Graphical Models written by Ying Liu (Ph. D.) and published by . This book was released on 2014 with total page 173 pages. Available in PDF, EPUB and Kindle. Book excerpt: In undirected graphical models, each node represents a random variable while the set of edges specifies the conditional independencies of the underlying distribution. When the random variables are jointly Gaussian, the models are called Gaussian graphical models (GGMs) or Gauss Markov random fields. In this thesis, we address several important problems in the study of GGMs. The first problem is to perform inference or sampling when the graph structure and model parameters are given. For inference in graphs with cycles, loopy belief propagation (LBP) is a purely distributed algorithm, but it gives inaccurate variance estimates in general and often diverges or has slow convergence. Previously, the hybrid feedback message passing (FMP) algorithm was developed to enhance the convergence and accuracy, where a special protocol is used among the nodes in a pseudo-FVS (an FVS, or feedback vertex set, is a set of nodes whose removal breaks all cycles) while standard LBP is run on the subgraph excluding the pseudo-FVS. In this thesis, we develop recursive FMP, a purely distributed extension of FMP where all nodes use the same integrated message-passing protocol. In addition, we introduce the subgraph perturbation sampling algorithm, which makes use of any pre-existing tractable inference algorithm for a subgraph by perturbing this algorithm so as to yield asymptotically exact samples for the intended distribution. We study the stationary version where a single fixed subgraph is used in all iterations, as well as the non-stationary version where tractable subgraphs are adaptively selected. The second problem is to perform model learning, i.e. to recover the underlying structure and model parameters from observations when the model is unknown. Families of graphical models that have both large modeling capacity and efficient inference algorithms are extremely useful. With the development of new inference algorithms for many new applications, it is important to study the families of models that are most suitable for these inference algorithms while having strong expressive power in the new applications. In particular, we study the family of GGMs with small FVSs and propose structure learning algorithms for two cases: 1) All nodes are observed, which is useful in modeling social or flight networks where the FVS nodes often correspond to a small number of high-degree nodes, or hubs, while the rest of the networks is modeled by a tree. 2) The FVS nodes are latent variables, where structure learning is equivalent to decomposing an inverse covariance matrix (exactly or approximately) into the sum of a tree-structured matrix and a low-rank matrix. We perform experiments using synthetic data as well as real data of flight delays to demonstrate the modeling capacity with FVSs of various sizes.
Download or read book Bayesian Statistics 7 written by J. M. Bernardo and published by Oxford University Press. This book was released on 2003-07-03 with total page 1114 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the 7th Valencia International Meeting on Bayesian Statistics. This conference is held every four years and provides the main forum for researchers in the area of Bayesian statistics to come together to present and discuss frontier developments in the field.
Download or read book Large scale Directed Graphical Models Learning written by and published by . This book was released on 2016 with total page 164 pages. Available in PDF, EPUB and Kindle. Book excerpt: Directed graphical models are a powerful statistical method to compactly describe directional or causal relationships among the set of variables in large-scale data. However, a number of statistical and computational challenges arise that make learning directed graphical models often impossible for large-scale data. These issues include: (1) model identifiability; (2) computational guarantee; (3) sample size guarantee; and (4) combining interventional experiments with observational data. In this thesis, we focus on learning directed graphical models by addressing the above four issues. In Chapter 3, we discuss learning Poisson DAG models for modeling large-scale multivariate count data problems where each node is a Poisson random variable conditioning on its parents. We address the question of (1) model identifiability and learning algorithms with (2) computational complexity and (3) sample complexity. We prove that Poisson DAG models are fully identifiable from observational data using the notion of overdispersion, and present a polynomial-time algorithm that learns the Poisson DAG model under suitable regularity conditions. Chapter 4 focuses on learning a broader class of DAG models in large-scale settings. We address the issue of (1) model identifiability and learning algorithms with (2) computational complexity and (3) sample complexity. We introduce a new class of identifiable DAG models which include many interesting classes of distributions such as Poisson, Binomial, Geometric, Exponential, Gamma, and many more, and prove that this class of DAG models is fully identifiable using the idea of overdispersion. Furthermore, we develop statistically consistent and computationally tractable learning algorithms for the new class of identifiable DAG models in high-dimensional settings. Our algorithms exploits the sparsity of the graphs and overdispersion property. Chapter 5 concerns learning general DAG models using a combination of observational and interventional (or experimental) data. Prior work has focused on algorithms using Markov equivalence class (MEC) for the DAG and then using do-calculus rules based on interventions to learn the additional directions. However it has been shown that existing passive and active learning strategies that rely on accurate recovery of the MEC do not scale well to large-scale graphs because recovering MEC for DAG models are not successful large-scale graphs. Hence, we prove (1) model identifiability using the notion of the moralized graphs, and develop passive and active learning algorithms (4) combining interventional experiments with observational data. Lastly in Chapter 6, we concern learning directed cyclic graphical (DCG) models. We focus on (1) model identifiability for directed graphical models with feedback. We provide two new identifiability assumptions with respect to sparsity of a graph and the number of d-separation rules, and compare these new identifiability assumptions to the widely-held faithfulness and minimality assumptions. Furthermore we develop search algorithms for small-scale DCG models based on our new identifiability assumptions.
