We propose a new family of combinatorial inference problems for graphical models. Unlike classical statistical inference where the main interest is point estimation or parameter testing, combinatorial inference aims at testing the global structure of the underlying graph. Examples include testing the graph connectivity, the presence of a cycle of certain size, or the maximum degree of the graph. To begin with, we develop a unified theory for the fundamental limits of a large family of combinatorial inference problems. We propose new concepts including structural packing and buffer entropies to characterize how the complexity of combinatorial graph structures impacts the corresponding minimax lower bounds. On the other hand, we propose a family of novel and practical structural testing algorithms to match the lower bounds. We provide thorough numerical results on both synthetic graphical models and brain networks to illustrate the usefulness of these proposed methods.
Graphical model inference is relatively straightforward when @math , but becomes notoriously challenging when @math . In high-dimensions, estimation procedures were studied by @cite_26 @cite_4 @cite_25 @cite_34 among others, while for variable selection procedures see @cite_7 @cite_18 @cite_32 @cite_35 @cite_34 and references therein. Recently, motivated by @cite_0 , various inferential methods for high-dimensional graphical models were suggested [e.g.] liu2013, jankova2014confidence,chen2015asymptotically,ren2015asymptotic, neykov2015aunified, gu2015local , most of which focus on testing the presence of a single edge (except @cite_27 who took the FDR approach to conduct multiple tests and @cite_9 who developed procedures of edge testing in Gaussian copula models). None of the aforementioned works address the problem of combinatorial structure testing.
@cite_35 @cite_18 @cite_26 @cite_4 @cite_7 @cite_9 @cite_32 @cite_0 @cite_27 @cite_34 @cite_25
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This paper proposes a unified framework to quantify local and global inferential uncertainty for high dimensional nonparanormal graphical models. In particular, we consider the problems of testing the presence of a single edge and constructing a uniform confidence subgraph. Due to the presence of unknown marginal transformations, we propose a pseudo likelihood based inferential approach. In sharp contrast to the existing high dimensional score test method, our method is free of tuning parameters given an initial estimator, and extends the scope of the existing likelihood based inferential framework. Furthermore, we propose a U-statistic multiplier bootstrap method to construct the confidence subgraph. We show that the constructed subgraph is contained in the true graph with probability greater than a given nominal level. Compared with existing methods for constructing confidence subgraphs, our method does not rely on Gaussian or sub-Gaussian assumptions. The theoretical properties of the proposed inferential methods are verified by thorough numerical experiments and real data analysis. Graphical models (e.g., Bayesian and constraint networks, influence diagrams, and Markov decision processes) have become a central paradigm for knowledge representation and reasoning in both artificial intelligence and computer science in general. These models are used to perform many reasoning tasks, such as scheduling, planning and learning, diagnosis and prediction, design, hardware and software verification, and bioinformatics. These problems can be stated as the formal tasks of constraint satisfaction and satisfiability, combinatorial optimization, and probabilistic inference. It is well known that the tasks are computationally hard, but research during the past three decades has yielded a variety of principles and techniques that significantly advanced the state of the art. In this book we provide comprehensive coverage of the primary exact algorithms for reasoning with such models. The main feature exploited by the algorithms is the model's graph. We present inference-based, message-passing schemes (e.g., variable-elimination) and search-based, conditioning schemes (e.g., cycle-cutset conditioning and AND OR search). Each class possesses distinguished characteristics and in particular has different time vs. space behavior. We emphasize the dependence of both schemes on few graph parameters such as the treewidth, cycle-cutset, and (the pseudo-tree) height. We believe the principles outlined here would serve well in moving forward to approximation and anytime-based schemes. The target audience of this book is researchers and students in the artificial intelligence and machine learning area, and beyond. Table of Contents: Preface Introduction What are Graphical Models Inference: Bucket Elimination for Deterministic Networks Inference: Bucket Elimination for Probabilistic Networks Tree-Clustering Schemes AND OR Search Spaces and Algorithms for Graphical Models Combining Search and Inference: Trading Space for Time Conclusion Bibliography Author's Biography Graphical models with higher order factors are an important tool for pattern recognition that has recently attracted considerable attention. Inference based on such models is challenging both from the view point of software design and optimization theory. In this article, we use the new C++ template library OpenGM to empirically compare inference algorithms on a set of synthetic and real-world graphical models with higher order factors that are used in computer vision. While inference algorithms have been studied intensively for graphical models with second order factors, an empirical comparison for higher order models has so far been missing. This article presents a first set of experiments that intends to fill this gap. The idea of using statistical inference for analyzing and understanding images has been used for at least 20 years, going back, for instance, to the work of Grenander [Gr] and Cooper [Co]. To apply these techniques, one needs, of course, a probabilistic model for some class of images or some class of structures present in images. Many models of this type have been introduced. There are stochastic models for image textures [GGGD], [ZMW], for contours in images [Mu], [GCK], for the decomposition of an image into regions [G-G], [M-S], for disparity maps, for grammatical parsing of shapes [Fu], for template matching, for speci c tasks such as face recognition [HGYGM]. The common framework for all these studies is to describe some class of images I(x; y) by means of a set of auxiliary variables fx g representing the salient structures in the images, e.g. edges, texture statistics, inferred depth values or relations, illumination features, medial axes or shape features, locations of key points such as eyes in a face, labels (as in character recognition), etc. Then i) a prior probability model for the hidden' variables p(fx g) and ii) an imagingmodel p(Ijfx g) for I, given the hidden variables, are de ned. Finally, an image is analyzed using Bayes's rule p(fx gjI) p(Ijfx g)p(fx g)