因子图(factor graph)

本文介绍了因子图的概念,一种用于表示全局函数因子分解的双向图,并解释了如何使用sum-product算法高效地求解变量的边缘分布。因子图在概率论及机器学习中有着广泛的应用。

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因子图(factor graph)

  Factor Graph 是概率图的一种,概率图有很多种,最常见的就是Bayesian Network (贝叶斯网络)和Markov Random Fields(马尔可夫随机场)。
  在概率图中,求某个变量的边缘分布是常见的问题。这问题有很多求解方法,其中之一就是可以把Bayesian Network和Markov Random Fields 转换成Facor Graph,然后用sum-product算法求解。基于Factor Graph可以用sum-product算法可以高效的求各个变量的边缘分布。

更详细的理解

  将一个具有多变量的全局函数因子分解,得到几个局部函数的乘积,以此为基础得到的一个双向图叫做因子图。
  所谓factor graph(因子图),就是对函数因子分解的表示图,一般内含两种节点,变量节点和函数节点。我们知道,一个全局函数能够分解为多个局部函数的积,因式分解就行了,这些局部函数和对应的变量就能体现在因子图上。
  在概率论及其应用中, 因子图是一个在贝叶斯推理中得到广泛应用的模型。

sum-product算法

  在因子图中,所有顶点,要不然就是变量节点不然就是函数节点,边线表示他们之间的函数关系。在讲解朴素贝叶斯和马尔可夫的时候,我们变线上标注的符 号,也就是Psi函数表示符号,就是表示我们模型中x和y的联系函数。Psi函数在不同的环境下有着不同的含义,因此解释这种东西总是比较棘手的。在动态模型里面,或者任何其他的图概率模型,都是可以用因子图表示的,而Psi在这里,表征的通常都是概率或者条件概率。 因子图和Psi函数表示法,在machine learning的paper中是比较常用的。
  参考资料:http://www.cnblogs.com/549294286/archive/2013/06/06/3121454.html

图例

  第一个公式等价于下图:
  这里写图片描述
这里写图片描述

  下面就是隐马尔可夫模型的因子图:
这里写图片描述

We review the use of factor graphs for the modeling and solving of large-scale inference problems in robotics. Factor graphs are a family of probabilistic graphical models, other examples of which are Bayesian networks and Markov random fields, well known from the statistical modeling and machine learning literature. They provide a powerful abstraction that gives insight into particular inference problems, making it easier to think about and design solutions, and write modular software to perform the actual inference. We illustrate their use in the simultaneous localization and mapping problem and other important problems associated with deploying robots in the real world. We introduce factor graphs as an economical representation within which to formulate the different inference problems, setting the stage for the subsequent sections on practical methods to solve them.We explain the nonlinear optimization techniques for solving arbitrary nonlinear factor graphs, which requires repeatedly solving large sparse linear systems. The sparse structure of the factor graph is the key to understanding this more general algorithm, and hence also understanding (and improving) sparse factorization methods. We provide insight into the graphs underlying robotics inference, and how their sparsity is affected by the implementation choices we make, crucial for achieving highly performant algorithms. As many inference problems in robotics are incremental, we also discuss the iSAM class of algorithms that can reuse previous computations, re-interpreting incremental matrix factorization methods as operations on graphical models, introducing the Bayes tree in the process. Because in most practical situations we will have to deal with 3D rotations and other nonlinear manifolds, we also introduce the more sophisticated machinery to perform optimization on nonlinear manifolds. Finally, we provide an overview of applications of factor graphs for robot perception, showing the broad impact factor graphs had in robot perception. (只有第一章内容,我也才发现)
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