1. Undirected graphical models(Markov random fields)

• 节点表示随机变量，边表示与节点相关的势函数
  $p_{\mathbf{x}}(\mathbf{x}) \propto \varphi_{12}\left(x_{1}, x_{2}\right) \varphi_{13}\left(x_{1}, x_{3}\right) \varphi_{25}\left(x_{2}, x_{5}\right) \varphi_{345}\left(x_{3}, x_{4}, x_{5}\right)$px​(x)∝φ12​(x1​,x2​)φ13​(x1​,x3​)φ25​(x2​,x5​)φ345​(x3​,x4​,x5​)
  

• clique：全连接的节点集合

• maximal clique：不是其他 clique 的真子集

**Theorem (Hammersley-Clifford) **: A strictly positive distribution $p_{\mathsf{x}}(\mathbf{x})>0$px​(x)>0 satisfies the graph separation property of undirected graphical models if and only if it can be represented in the factorized form
$p_{\mathsf{x}}(\mathbf{x}) \propto \prod_{\mathcal{A} \in \mathcal{C}} \psi_{\mathbf{x}_{\mathcal{A}}}\left(\mathbf{x}_{\mathcal{A}}\right)$px​(x)∝A∈C∏​ψxA​​(xA​)

• conditional independence：$\mathbf{x}_{\mathcal{A}_{1}} \perp \mathbf{x}_{\mathcal{A}_{2}} | \mathbf{x}_{\mathcal{A}_{3}}$xA1​​⊥xA2​​∣xA3​​

2. Directed graphical models(Bayesian network)

• 节点表示随机变量，有向边表示条件关系
  $p_{\mathrm{x}_{1}, \ldots, \mathrm{x}_{n}}=p_{\mathrm{x}_{1}}\left(x_{1}\right) p_{\mathrm{x}_{2} | \times_{1}}\left(x_{2} | x_{1}\right) \cdots p_{\mathrm{x}_{n} | x_{1}, \ldots, x_{n-1}}\left(x_{n} | x_{1}, \ldots, x_{n-1}\right)$px1​,…,xn​​=px1​​(x1​)px2​∣×1​​(x2​∣x1​)⋯pxn​∣x1​,…,xn−1​​(xn​∣x1​,…,xn−1​)
  

• Directed acyclic graphs (DAG)

• Fully-connected DAG

• conditional independence：$\mathbf{x}_{\mathcal{A}_{1}} \perp \mathbf{x}_{\mathcal{A}_{2}} | \mathbf{x}_{\mathcal{A}_{3}}$xA1​​⊥xA2​​∣xA3​​

  

• Bayes ball algorithm

  • primary shade: $\mathcal{A_3}$A3​ 中的节点
  • secondary shade: primary shade 的节点，以及 secondary shade 的父节点

  [外链图片转存失败,源站可能有防盗链机制,建议将图片保存下来直接上传(img-8UKlSiFL-1580777349196)(C:\Users\1\AppData\Roaming\Typora\typora-user-images\1574319393095.png)]

3. Factor graph

• 有 variable nodes 和 factor nodes，是 bipartitie graph
  $p_{\mathbf{x}}(\mathbf{x}) \propto \prod_{j} f_{j}\left(\mathbf{x}_{f_{j}}\right)$px​(x)∝j∏​fj​(xfj​​)
  

• 因子图比 directed graph 和 undirected graph 的表示能力更强，比如 $p(x)=\frac{1}{Z}\phi_{12}(x_1,x_2)\phi_{13}(x_1,x_3)\phi_{23}(x_2,x_3)$p(x)=Z1​ϕ12​(x1​,x2​)ϕ13​(x1​,x3​)ϕ23​(x2​,x3​)

• 因子图可以与 DAG 相互转化(根据 $x_1,...,x_n$x1​,...,xn​ 依次根据 conditional independence 决定父节点)，DAG又可以转化为 undirected graph

4. Measuring goodness of graphical representations

• 给定分布 D 和图 G，他们之间没必要有联系
• $CI(D)$CI(D)：the set of conditional independencies satisfied by $D$D
• $CI(G)$CI(G)： the set of all conditional independencies implied by $G$G
• I-map：$C I(\mathcal{G}) \subset C I(D)$CI(G)⊂CI(D)
• D-map: ：$C I(\mathcal{G}) \supset C I(D)$CI(G)⊃CI(D)
• P-map：$C I(\mathcal{G}) = C I(D)$CI(G)=CI(D)
• minimal I-map: Aminimal I-mapisanI-mapwiththepropertythatremovinganysingle edge would cause the graph to no longer be an I-map.
  Remarks: G 中去掉一个边会使该 map 中有更多的 conditional independence，也即 $CI(G)$CI(G) 更大，更不易满足 I-map条件。I-map 可以表示分布 D，但是 D-map 不能

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