7,129
个编辑
更改
跳到导航
跳到搜索
第1行：
第1行： − {{machine learning bar}}+ − {{More footnotes|date=May 2017}}+ − A '''graphical model''' or '''probabilistic graphical model''' ('''PGM''') or '''structured probabilistic model''' is a [[probabilistic model]] for which a [[Graph (discrete mathematics)|graph]] expresses the [[conditional dependence]] structure between [[random variable]]s. They are commonly used in [[probability theory]], [[statistics]]—particularly [[Bayesian statistics]]—and [[machine learning]].+ − A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning.+ − − '''<font color="#ff8000">图模型 Graphical Model</font>'''或'''<font color="#ff8000">概率图模型 Probabilistic Graphical Model</font>'''(PGM)或结构化概率模型是一种用图表示随机变量之间条件依赖关系的概率模型。它们通常用于概率论、统计学---- 尤其是贝叶斯统计学---- 和'''<font color="#ff8000">机器学习 Machine Learning</font>'''。 − − − − [[File:Graph model.svg|thumb|right|alt=An example of a graphical model.| An example of a graphical model. Each arrow indicates a dependency. In this example: D depends on A, B, and C; and C depends on B and D; whereas A and B are each independent.]] − − An example of a graphical model. Each arrow indicates a dependency. In this example: D depends on A, B, and C; and C depends on B and D; whereas A and B are each independent. − − 这是一个图模型的例子。每个箭头表示一个依赖关系。在这个例子中: D 依赖于 A、 B 和 C; C 依赖于 B 和 D; 而 A 和 B 相互独立。 − − − − ==Types of graphical models== − 图模型的类别 − − Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a distribution over a multi-dimensional space and a graph that is a compact or [[Factor graph|factorized]] representation of a set of independences that hold in the specific distribution. Two branches of graphical representations of distributions are commonly used, namely, [[Bayesian network]]s and [[Markov random field]]s. Both families encompass the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce.<ref name=koller09>{{cite book − − Generally, probabilistic graphical models use a graph-based representation as the foundation for encoding a distribution over a multi-dimensional space and a graph that is a compact or factorized representation of a set of independences that hold in the specific distribution. Two branches of graphical representations of distributions are commonly used, namely, Bayesian networks and Markov random fields. Both families encompass the properties of factorization and independences, but they differ in the set of independences they can encode and the factorization of the distribution that they induce.<ref name=koller09>{{cite book − − 一般来说，概率图模型中图的表示方法常常作为对多维空间上的分布进行编码的基础，而图是一组独立分布的紧凑或分解表示。常用的概率图模型大致分为两类：贝叶斯网络和马尔可夫随机场。这两种都包含因子分解和独立性的性质，但是它们在它们可以编码的一系列独立性和它们所诱导的分布的因子分解上有所不同。 09{ cite book − − |author=Koller, D. − − |author=Koller, D. − − 作者 Koller，d。 − − |author2=Friedman, N. − − |author2=Friedman, N. − − 弗里德曼，n。 − − |title=Probabilistic Graphical Models − − |title=Probabilistic Graphical Models − − | 题目概率图形模型 − − |url=http://pgm.stanford.edu/ − − |url=http://pgm.stanford.edu/ − − Http://pgm.stanford.edu/ − − |publisher=MIT Press − − |publisher=MIT Press − − 出版商: MIT 出版社 − − |location=Massachusetts − − |location=Massachusetts − − 马萨诸塞州 | 地点 − − |year=2009 − − |year=2009 − − 2009年 − − |pages=1208 − − |pages=1208 − − 第1208页 − − |isbn=978-0-262-01319-2 − − |isbn=978-0-262-01319-2 − − [国际标准图书编号978-0-262-01319-2] − − |doi= − − |doi= − − 不会吧 − − |accessdate= − − |accessdate= − − 访问日期 − − |author2-link=Nir Friedman − − |author2-link=Nir Friedman − − | author2-link Nir Friedman − − |author-link=Daphne Koller − − |author-link=Daphne Koller − − 作者链接 Daphne Koller − − |archive-url=https://web.archive.org/web/20140427083249/http://pgm.stanford.edu/ − − |archive-url=https://web.archive.org/web/20140427083249/http://pgm.stanford.edu/ − − | 档案-网址 https://web.archive.org/web/20140427083249/http://pgm.stanford.edu/ − − |archive-date=2014-04-27 − − |archive-date=2014-04-27 − − | 档案-日期2014-04-27 − − |url-status=dead − − |url-status=dead − − 状态死机 − − }}</ref> − − }}</ref> − − {} / ref − − − − ===Bayesian network=== − 贝叶斯网络 − {{main|Bayesian network}} − − − − If the network structure of the model is a [[directed acyclic graph]], the model represents a factorization of the joint [[probability]] of all random variables. More precisely, if the events are <math>X_1,\ldots,X_n</math> then the joint probability satisfies − − If the network structure of the model is a directed acyclic graph, the model represents a factorization of the joint probability of all random variables. More precisely, if the events are <math>X_1,\ldots,X_n</math> then the joint probability satisfies − − 如果模型的网络结构是有向无环图，则模型表示所有随机变量的联合概率的因子分解。更确切地说，如果事件是<math>X_1,\ldots,X_n</math>，那么联合概率满足 + + − <math>P[X_1,\ldots,X_n]=\prod_{i=1}^nP[X_i|\text{pa}(X_i)]</math> − − − − − − where <math>\text{pa}(X_i)</math> is the set of parents of node <math>X_i</math> (nodes with edges directed towards <math>X_i</math>). In other words, the [[joint distribution]] factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an [[ancestral graph]]) consists of the random variables <math>A, B, C, D</math> − − where <math>\text{pa}(X_i)</math> is the set of parents of node <math>X_i</math> (nodes with edges directed towards <math>X_i</math>). In other words, the joint distribution factors into a product of conditional distributions. For example, the graphical model in the Figure shown above (which is actually not a directed acyclic graph, but an ancestral graph) consists of the random variables <math>A, B, C, D</math> − − 其中 <math>\text{pa}(X_i)</math> 是节点 <math>X_i</math> 的父节点集(节点的边际指向 <math>X_i</math> )。换句话说，联合分布因子可以表示为条件分布的乘积。例如，上图中的图模型(实际上不是有向无环图，而是原始图)是由随机变量<math>A, B, C, D</math> 组成 − − with a joint probability density that factors as − − with a joint probability density that factors as − − 联合概率密度因子为 + − <math>P[A,B,C,D] = P[A]\cdot P[B]\cdot P[C,D|A,B]</math> − − − − − − Any two nodes are [[Conditional independence|conditionally independent]] given the values of their parents. In general, any two sets of nodes are conditionally independent given a third set if a criterion called [[d-separation|''d''-separation]] holds in the graph. Local independences and global independences are equivalent in Bayesian networks. − Any two nodes are conditionally independent given the values of their parents. In general, any two sets of nodes are conditionally independent given a third set if a criterion called d-separation holds in the graph. Local independences and global independences are equivalent in Bayesian networks.+ − 任何两个节点都是和其父节点的值是条件独立的。一般来说，如果 d- 分离准则在图中成立，那么任意两组节点和给定第三个节点集都是条件独立的。在贝叶斯网络中，局部独立性和全局独立性是等价的。 + − This type of graphical model is known as a directed graphical model, [[Bayesian network]], or belief network. Classic machine learning models like [[hidden Markov models]], [[neural networks]] and newer models such as [[variable-order Markov model]]s can be considered special cases of Bayesian networks.+ − + − This type of graphical model is known as a directed graphical model, Bayesian network, or belief network. Classic machine learning models like hidden Markov models, neural networks and newer models such as variable-order Markov models can be considered special cases of Bayesian networks.+ − + − 这种类型的图形模型被称为有向图形模型、贝氏网路或信念网络。经典的机器学习模型：隐马尔可夫模型、神经网络和更新的模型如可变阶马尔可夫模型都可以看作是贝叶斯网络的特殊情况。+ − + − + − − − 其他类别 − − − 朴素贝叶斯分类器，其中我们会使用一颗单结点树 − − 依赖网络中环的出现 − − 树增广朴素贝叶斯分类器或简称为TAN模型 − − 因子图是连接变量和因子的无向二分图。 每个因子代表与其连接的变量有关的函数。 这对于理解和实现信念传播算法很有帮助。 − − 在节点树算法中，通常会使用派系中一棵派系树或者是节点树。 − − − 链式图是既可以有向也可以无向的边，但是没有任何有向环（即，如果我们从任意一个顶点开始并依据任何箭头的方向沿该图形移动，则只要我们沿着箭头走了一步我们将无法返回到该顶点）。 有向无环图和无向图都是链式图的特例，因此可以提供一种统一和泛化贝叶斯网络和马尔可夫网络的方法。 − − </ref> − − − − / 参考 − + − |first1=Thomas |last1=Richardson |first2=Peter |last2=Spirtes − − |first1=Thomas |last1=Richardson |first2=Peter |last2=Spirtes − − 1 Thomas | last 1 Richardson | first2 Peter | last 2 Spirtes − − |title=Ancestral graph Markov models − − |title=Ancestral graph Markov models − − | 标题祖先图马尔科夫模型 − − |journal=[[Annals of Statistics]] − − |journal=Annals of Statistics − − 统计年鉴 − − |volume=30 |issue=4 |year=2002 |pages=962–1030 − − |volume=30 |issue=4 |year=2002 |pages=962–1030 − − 第三十卷，第四期2002年，第962-1030页 − − |doi=10.1214/aos/1031689015 − − |doi=10.1214/aos/1031689015 − − |doi=10.1214/aos/1031689015 − − |mr=1926166 | zbl = 1033.60008 − − |mr=1926166 | zbl = 1033.60008 − − 1926 / 166先生1033.60008 − − − − |citeseerx=10.1.1.33.4906}}</ref> − − 10.1.1.33.4906} / ref 第271行：
第49行： − + − + − The framework of the models, which provides algorithms for discovering and analyzing structure in complex distributions to describe them succinctly and extract the unstructured information, allows them to be constructed and utilized effectively.<ref name=koller09/> Applications of graphical models include [[causal inference]], [[information extraction]], [[speech recognition]], [[computer vision]], decoding of [[low-density parity-check codes]], modeling of [[gene regulatory network]]s, gene finding and diagnosis of diseases, and [[graphical models for protein structure]]. − − The framework of the models, which provides algorithms for discovering and analyzing structure in complex distributions to describe them succinctly and extract the unstructured information, allows them to be constructed and utilized effectively. Applications of graphical models include causal inference, information extraction, speech recognition, computer vision, decoding of low-density parity-check codes, modeling of gene regulatory networks, gene finding and diagnosis of diseases, and graphical models for protein structure. − − 该模型的框架为发现和分析复杂分布中的结构、简洁地描述结构和提取非结构化信息提供了算法，使模型得到有效的构建和利用。图形模型的应用包括因果推理、信息抽取、语音识别、计算机视觉、低密度奇偶校验码的解码、基因调控网络的建模、基因发现和疾病诊断，以及蛋白质结构的图形模型。 − + − + − 信念传递网络+ − − 结构方程模型 − + + − + − − − − *{{cite book − − | last = Barber − − | last = Barber − − 理发师 − − | first = David − − | first = David − − 首先是大卫 − − | title = Bayesian Reasoning and Machine Learning − − | title = Bayesian Reasoning and Machine Learning − − 贝叶斯推理和机器学习 − − | publisher = Cambridge University Press − − | publisher = Cambridge University Press − − 出版商剑桥大学出版社 − − | year = 2012 − − | year = 2012 − − 2012年 − − | isbn = 978-0-521-51814-7 − − | isbn = 978-0-521-51814-7 − − [国际标准图书编号978-0-521-51814-7] − − }} − − }} − − }} − − − − * {{cite book − − | last = Bishop − − | last = Bishop − − 最后一个主教 − − | first = Christopher M. − − | first = Christopher M. − − 首先是克里斯托弗 m。 − − | authorlink = Christopher Bishop − − | authorlink = Christopher Bishop − − 克里斯托弗 · 毕晓普 − − | title = Pattern Recognition and Machine Learning − − | title = Pattern Recognition and Machine Learning − − 模式识别与机器学习 − − | publisher = Springer − − | publisher = Springer − − 出版商斯普林格 − − | year = 2006 − − | year = 2006 − − 2006年 − − | url = https://www.springer.com/us/book/9780387310732 − − | url = https://www.springer.com/us/book/9780387310732 − − Https://www.springer.com/us/book/9780387310732 − − | isbn=978-0-387-31073-2 − − | isbn=978-0-387-31073-2 − − [国际标准图书编号978-0-387-31073-2] − − | chapter= Chapter 8. Graphical Models − − | chapter= Chapter 8. Graphical Models − − | 第八章。图形模型 − − | chapterurl=https://www.microsoft.com/en-us/research/wp-content/uploads/2016/05/Bishop-PRML-sample.pdf − − | chapterurl=https://www.microsoft.com/en-us/research/wp-content/uploads/2016/05/Bishop-PRML-sample.pdf − − Https://www.microsoft.com/en-us/research/wp-content/uploads/2016/05/bishop-prml-sample.pdf − − | pages=359–422 − − | pages=359–422 − − 第359-422页 − − | mr=2247587 − − | mr=2247587 − − 2247587先生 − − }} − − }} − − }} − − * {{cite book − − |author=Cowell, Robert G. − − |author=Cowell, Robert G. − − 作者: 罗伯特 · g。 − − |author2=Dawid, A. Philip |author3=Lauritzen, Steffen L. |author4-link=David Spiegelhalter |author4=Spiegelhalter, David J. − − |author2=Dawid, A. Philip |author3=Lauritzen, Steffen L. |author4-link=David Spiegelhalter |author4=Spiegelhalter, David J. − − 2 Dawid，a. Philip | author3 Lauritzen，Steffen l. | author4-link David Spiegelhalter | author4 Spiegelhalter，David j. − − |title=Probabilistic networks and expert systems |publisher=Springer |location=Berlin |year=1999 |pages= |isbn=978-0-387-98767-5 |doi= |accessdate= |mr=1697175 |ref=cowell |author2-link=Philip Dawid }} A more advanced and statistically oriented book − − |title=Probabilistic networks and expert systems |publisher=Springer |location=Berlin |year=1999 |pages= |isbn=978-0-387-98767-5 |doi= |accessdate= |mr=1697175 |ref=cowell |author2-link=Philip Dawid }} A more advanced and statistically oriented book − − 一本更先进和面向统计的书。1697175 + + + + − * {{Cite book+ − − |first=Judea |last=Pearl |authorlink = Judea Pearl − − |first=Judea |last=Pearl |authorlink = Judea Pearl − − 朱迪亚珍珠 − − | year = 1988 − − | year = 1988 − − 1988年 − − | title = Probabilistic Reasoning in Intelligent Systems − − | title = Probabilistic Reasoning in Intelligent Systems − − | 题目: 智能系统中的概率推理 − − | edition = 2nd revised − − | edition = 2nd revised − − 第二版修订版 − − | location = San Mateo, CA − − | location = San Mateo, CA − − | 地点: 加利福尼亚州圣马特奥 − − | publisher = [[Morgan Kaufmann]] − − | publisher = Morgan Kaufmann − − 出版商摩根 · 考夫曼 − − | mr = 0965765 − − | mr = 0965765 − − 0965765先生 − − |isbn = 978-1-55860-479-7 − − |isbn = 978-1-55860-479-7 − − [国际标准图书编号978-1-55860-479-7] − − }} A computational reasoning approach, where the relationships between graphs and probabilities were formally introduced. − − }} A computational reasoning approach, where the relationships between graphs and probabilities were formally introduced. − − 一种计算推理方法，图和概率之间的关系被正式引入。 − − − − ===Journal articles=== − − * {{Cite journal − − | author = Edoardo M. Airoldi |authorlink1=Edoardo Airoldi − − | author = Edoardo M. Airoldi |authorlink1=Edoardo Airoldi − − 1 Edoardo Airoldi − − | title = Getting Started in Probabilistic Graphical Models − − | title = Getting Started in Probabilistic Graphical Models − − 开始使用概率图模型 − − | journal = [[PLoS Computational Biology]] − − | journal = PLoS Computational Biology − − 2012年3月24日 | PLoS 计算生物学 − − | volume = 3 − − | volume = 3 − − 第三卷 − − | issue = 12 − − | issue = 12 − − 第12期 − − | pages = e252 − − | pages = e252 − − 第252页 − − | year = 2007 − − | year = 2007 − − 2007年 − − | doi = 10.1371/journal.pcbi.0030252 − − | doi = 10.1371/journal.pcbi.0030252 − − 10.1371 / journal.pcbi. 0030252 − − | pmid = 18069887 − − | pmid = 18069887 − − 18069887 − − | pmc = 2134967 − − | pmc = 2134967 − − 2134967 − − }} − − }} − − }} + − − + 第587行：
第89行： − + − + − − + − {{Statistics|analysis}}+ − − − − − − Category:Bayesian statistics − − 类别: 贝叶斯统计 − − [[Category:Graphical models| ]] − − <noinclude> − − <small>This page was moved from [[wikipedia:en:Graphical model]]. Its edit history can be viewed at [[概率图模型/edithistory]]</small></noinclude> − −
无编辑摘要
此词条暂由Yuling翻译，未经人工整理和审校，带来阅读不便，请见谅。
此词条暂由Yuling翻译，未经人工整理和审校，带来阅读不便，请见谅。
'''<font color="#ff8000">图模型 Graphical Model</font>''','''亦称<font color="#ff8000">概率图模型 Probabilistic Graphical Model（PGM）</font>'''或'''结构化概率模型 structured probabilistic model'''，是一种用图表示随机变量之间条件依赖关系的概率模型。它们通常用于概率论、统计学，尤其是[[贝叶斯统计学]]和[[机器学习]]。
[[File:Graph model.svg|thumb|right|这是一个图模型的例子。每个箭头表示一个依赖关系。在这个例子中: D 依赖于 A、 B 和 C; C 依赖于 B 和 D; 而 A 和 B 相互独立。]]
==图模型的类别==
一般来说，概率图模型中图的表示方法常常作为对多维空间上的分布进行编码的基础，而图是一组独立分布的紧凑或分解表示。常用的概率图模型大致分为两类：贝叶斯网络和马尔可夫随机场。这两种都包含因子分解和独立性的性质，但是它们在它们可以编码的一系列独立性和它们所诱导的分布的因子分解上有所不同。 <ref name=koller09>{{ cite book|author=Koller, D.|author2=Friedman, N.|title=Probabilistic Graphical Models|url=http://pgm.stanford.edu/|publisher=MIT Press|location=Massachusetts|year=2009|pages=1208|isbn=978-0-262-01319-2|archive-url=https://web.archive.org/web/20140427083249/http://pgm.stanford.edu/|archive-date=2014-04-27|url-status=dead}}</ref>
===贝叶斯网络===
如果模型的网络结构是[[有向无环图]]，则模型表示所有随机变量的联合概率的因子分解。更确切地说，如果事件是<math>X_1,\ldots,X_n</math>，那么联合概率满足：
:<math>P[X_1,\ldots,X_n]=\prod_{i=1}^nP[X_i|\text{pa}(X_i)]</math>
:<math>P[X_1,\ldots,X_n]=\prod_{i=1}^nP[X_i|\text{pa}(X_i)]</math>
其中 <math>\text{pa}(X_i)</math> 是节点 <math>X_i</math> 的父节点集（节点的边际指向 <math>X_i</math> )。换句话说，联合分布因子可以表示为条件分布的乘积。例如，上图中的图模型（实际上不是有向无环图，而是原始图）是由随机变量<math>A, B, C, D</math> 组成，联合概率密度因子为
:<math>P[A,B,C,D] = P[A]\cdot P[B]\cdot P[C,D|A,B]</math>
:<math>P[A,B,C,D] = P[A]\cdot P[B]\cdot P[C,D|A,B]</math>
任何两个节点都是和其父节点的值是条件独立的。一般来说，如果 d-分离准则在图中成立，那么任意两组节点和给定第三个节点集都是条件独立的。在贝叶斯网络中，局部独立性和全局独立性是等价的。
这种类型的图形模型被称为[[有向图模型]]、贝氏网路或信念网络。经典的机器学习模型：[[隐马尔可夫模型]]、[[神经网络]]和更新的模型如可变阶马尔可夫模型都可以看作是贝叶斯网络的特殊情况。
===其他类别===
*[[朴素贝叶斯分类器 Naive Bayes classifier]],其中我们会使用一颗单结点树
*[[依赖网络 Dependency network]]中环的出现
*树增广朴素贝叶斯分类器 Tree-augmented classifier 或简称为TAN模型
*[[因子图 factor graph]]是连接变量和因子的无向二分图。 每个因子代表与其连接的变量有关的函数。 这对于理解和实现[[信念传播算法]]很有帮助。
*在[[节点树 clique tree]] 算法中，通常会使用派系中一棵派系树或者是节点树。
*[[链式图 chain graph]] 是既可以有向也可以无向的边，但是没有任何有向环（即，如果我们从任意一个顶点开始并依据任何箭头的方向沿该图形移动，则只要我们沿着箭头走了一步我们将无法返回到该顶点）。 有向无环图和无向图都是链式图的特例，因此可以提供一种统一和泛化贝叶斯网络和马尔可夫网络的方法。<ref>{{cite journal|last=Frydenberg|first=Morten|year=1990|title=The Chain Graph Markov Property|journal=[[Scandinavian Journal of Statistics]]|volume=17|issue=4|pages=333–353|mr=1096723|jstor=4616181 }}</ref>
===Other types===
*[[Naive Bayes classifier]] where we use a tree with a single root
*[[Dependency network (graphical model)|Dependency network]] where cycles are allowed
*Tree-augmented classifier or '''TAN model'''
*A [[factor graph]] is an undirected [[bipartite graph]] connecting variables and factors. Each factor represents a function over the variables it is connected to. This is a helpful representation for understanding and implementing [[belief propagation]].
* A [[clique tree]] or junction tree is a [[tree (graph theory)|tree]] of [[clique (graph theory)|cliques]], used in the [[junction tree algorithm]].
* A [[chain graph]] is a graph which may have both directed and undirected edges, but without any directed cycles (i.e. if we start at any vertex and move along the graph respecting the directions of any arrows, we cannot return to the vertex we started from if we have passed an arrow). Both directed acyclic graphs and undirected graphs are special cases of chain graphs, which can therefore provide a way of unifying and generalizing Bayesian and Markov networks.<ref>{{cite journal|last=Frydenberg|first=Morten|year=1990|title=The Chain Graph Markov Property|journal=[[Scandinavian Journal of Statistics]]|volume=17|issue=4|pages=333–353|mr=1096723|jstor=4616181 }}
</ref>
* An [[ancestral graph]] is a further extension, having directed, bidirected and undirected edges.<ref>{{cite journal
* An [[ancestral graph]] is a further extension, having directed, bidirected and undirected edges.<ref>{{cite journal
|first1=Thomas |last1=Richardson |first2=Peter |last2=Spirtes|title=Ancestral graph Markov models|journal=Annals of Statistics|volume=30 |issue=4 |year=2002 |pages=962–1030|doi=10.1214/aos/1031689015|mr=1926166 | zbl = 1033.60008|citeseerx=10.1.1.33.4906}}</ref>
|citeseerx=10.1.1.33.4906}}</ref>
* [[Random field]] techniques
* [[Random field]] techniques
==Applications==
==应用==
该模型的框架为发现和分析复杂分布中的结构、简洁地描述结构和提取非结构化信息提供了算法，使模型得到有效的构建和利用。图形模型的应用包括因果推理、信息抽取、[[语音识别]]、[[l计算机视觉]]、低密度奇偶校验码的解码、基因调控网络的建模、基因发现和疾病诊断，以及蛋白质结构的图形模型。
==See also==
==另见==
* [[Belief propagation]]
* [[Belief propagation 信念传递网络]]
* [[Structural equation model 结构方程模型]]
* [[Structural equation model]]
==Notes==
==参考文献==
{{reflist}}
{{reflist}}
==进一步阅读==
==Further reading==
===书籍===
===Books and book chapters===
*{{cite book| last = Barber| first = David| title = Bayesian Reasoning and Machine Learning| publisher = Cambridge University Press| year = 2012| isbn = 978-0-521-51814-7}}
* {{cite book| last = Bishop| first = Christopher M.| title = Pattern Recognition and Machine Learning| publisher = Springer| year = 2006| url = https://www.springer.com/us/book/9780387310732| isbn=978-0-387-31073-2| chapter= Chapter 8. Graphical Models| chapterurl=https://www.microsoft.com/en-us/research/wp-content/uploads/2016/05/Bishop-PRML-sample.pdf| pages=359–422| mr=2247587}}
* {{cite book|author=Cowell, Robert G.|author2=Dawid, A. Philip |author3=Lauritzen, Steffen L. author4=Spiegelhalter, David J.|title=Probabilistic networks and expert systems |publisher=Springer |location=Berlin |year=1999 |pages= |isbn=978-0-387-98767-5 |doi= |accessdate= |mr=1697175 |ref=cowell |author2-link=Philip Dawid }} 一本更先进和面向统计的书。
* {{cite book |author=Jensen, Finn |title=An introduction to Bayesian networks |publisher=Springer |location=Berlin |year=1996 |pages= |isbn=978-0-387-91502-9 |doi= |accessdate=}}
* {{cite book |author=Jensen, Finn |title=An introduction to Bayesian networks |publisher=Springer |location=Berlin |year=1996 |pages= |isbn=978-0-387-91502-9 |doi= |accessdate=}}
* {{Cite book|first=Judea |last=Pearl | year = 1988| title = Probabilistic Reasoning in Intelligent Systems| edition = 2nd revised| location = San Mateo, CA| publisher = Morgan Kaufmann | mr = 0965765|isbn = 978-1-55860-479-7}}一种计算推理方法，图和概率之间的关系被正式引入。
===论文===
* {{Cite journal| author = Edoardo M. Airoldi |authorlink1=Edoardo Airoldi| title = Getting Started in Probabilistic Graphical Models| journal = PLoS Computational Biology| volume = 3| issue = 12| pages = e252| year = 2007| doi = 10.1371/journal.pcbi.0030252| pmid = 18069887| pmc = 2134967}}
*{{Cite journal | last1 = Jordan | first1 = M. I. | authorlink=Michael I. Jordan| doi = 10.1214/088342304000000026 | title = Graphical Models | journal = Statistical Science | volume = 19 | pages = 140–155| year = 2004 | pmid = | pmc = | doi-access = free }}
*{{Cite journal | last1 = Jordan | first1 = M. I. | authorlink=Michael I. Jordan| doi = 10.1214/088342304000000026 | title = Graphical Models | journal = Statistical Science | volume = 19 | pages = 140–155| year = 2004 | pmid = | pmc = | doi-access = free }}
*{{Cite journal|last=Ghahramani|first=Zoubin|date=May 2015|title=Probabilistic machine learning and artificial intelligence|journal=Nature|language=English|volume=521|issue=7553|pages= 452–459|doi=10.1038/nature14541|pmid=26017444}}
*{{Cite journal|last=Ghahramani|first=Zoubin|date=May 2015|title=Probabilistic machine learning and artificial intelligence|journal=Nature|language=English|volume=521|issue=7553|pages= 452–459|doi=10.1038/nature14541|pmid=26017444}}
===Other===
===其他===
*[http://research.microsoft.com/en-us/um/people/heckerman/tutorial.pdf Heckerman's Bayes Net Learning Tutorial]
*[http://research.microsoft.com/en-us/um/people/heckerman/tutorial.pdf Heckerman's Bayes Net Learning Tutorial]
*[http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html A Brief Introduction to Graphical Models and Bayesian Networks]
*[http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html A Brief Introduction to Graphical Models and Bayesian Networks]
*[http://www.cedar.buffalo.edu/~srihari/CSE574 Sargur Srihari's lecture slides on probabilistic graphical models]
*[http://www.cedar.buffalo.edau/~srihari/CSE574 Sargur Srihari's lecture slides on probabilistic graphical models]
==External links==
==其他链接==
* [http://sandeep-aparajit.blogspot.com/2013/06/how-does-conditional-random-field-crf.html Graphical models and Conditional Random Fields]
* [http://sandeep-aparajit.blogspot.com/2013/06/how-does-conditional-random-field-crf.html Graphical models and Conditional Random Fields]
* [https://www.cs.cmu.edu/~epxing/Class/10708/ Probabilistic Graphical Models taught by Eric Xing at CMU]
* [https://www.cs.cmu.edu/~epxing/Class/10708/ Probabilistic Graphical Models taught by Eric Xing at CMU]
[[Category:贝叶斯统计]]
[[Category:图模型]]
[[Category:Bayesian statistics]]
[[Category:待整理页面]]