Из серии Foundations and Trends in Machine Learning издательства NOWPress, 2008, -305 pp.The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instances — including the key problems of computing marginals and modes of probability distributions — are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms — among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations — can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.Introduction
Background
Graphical Models as Exponential Families
Sum-Product, Bethe–Kikuchi, and Expectation-Propagation
Mean Field Methods
Variational Methods in Parameter Estimation
Convex Relaxations and Upper Bounds
Integer Programming, Max-product, and Linear Programming Relaxations
Moment Matrices, Semidefinite Constraints, and Conic Programming Relaxation
Discussion
A Background Material
B Proofs and Auxiliary Results: Exponential Families and Duality
C Variational Principles for Multivariate Gaussians
D Clustering and Augmented Hypergraphs

E Miscellaneous Results

Из серии Foundations and Trends in Machine Learning издательства NOWPress, 2008, -305 pp.概率图形模型的形式主义为捕捉随机变量之间的复杂依赖关系和建立大规模的多变量统计模型提供了一个统一的框架。图解模型已经成为许多统计、计算和数学领域的研究重点，包括生物信息学、通信理论、统计物理学、组合优化、信号和图像处理、信息检索和统计机器学习。许多在具体实例中出现的问题--包括计算概率分布的边际和模式的关键问题--最好在一般情况下研究。通过指数族表示法，并利用指数族的累积函数和熵之间的共轭对偶性，我们开发了计算似然、边际概率和最有可能的配置等问题的一般变量表示法。我们描述了各种各样的算法--其中包括和积、集群变异方法、期望传播、均值场方法、最大积和线性编程松弛以及圆锥编程松弛--都可以用这些变异表示的精确或近似形式来理解。变分方法为马尔科夫链蒙特卡洛提供了一个补充性的替代方案，作为大规模统计模型推断的近似方法的一般来源。

背景介绍

作为指数族的图形模型

和积法、贝特-菊池法和期望-传播法

均值场方法

参数估计中的变分方法

凸式松弛和上界

整数编程、最大乘积和线性编程松弛

矩矩阵、半定式约束和圆锥编程松弛

讨论

A 背景材料

B 证明和辅助结果。指数族和二重性

C 多变量高斯的变异原理

D 聚类和增强的超图

E 杂项结果

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