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
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∝
φ
12
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,
x
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φ
13
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φ
25
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φ
345
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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)
p x ( x ) ∝ φ 1 2 ( x 1 , x 2 ) φ 1 3 ( x 1 , x 3 ) φ 2 5 ( x 2 , x 5 ) φ 3 4 5 ( x 3 , x 4 , x 5 )
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
x
(
x
)
>
0
p_{\mathsf{x}}(\mathbf{x})>0
p x ( x ) > 0 satisfies the graph separation property of undirected graphical models if and only if it can be represented in the factorized form
p
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x
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∝
∏
A
∈
C
ψ
x
A
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x
A
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p_{\mathsf{x}}(\mathbf{x}) \propto \prod_{\mathcal{A} \in \mathcal{C}} \psi_{\mathbf{x}_{\mathcal{A}}}\left(\mathbf{x}_{\mathcal{A}}\right)
p x ( x ) ∝ A ∈ C ∏ ψ x A ( x A )
conditional independence :
x
A
1
⊥
x
A
2
∣
x
A
3
\mathbf{x}_{\mathcal{A}_{1}} \perp \mathbf{x}_{\mathcal{A}_{2}} | \mathbf{x}_{\mathcal{A}_{3}}
x A 1 ⊥ x A 2 ∣ x A 3
2. Directed graphical models(Bayesian network)
节点表示随机变量,有向边表示条件关系
p
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,
…
,
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=
p
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p
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×
1
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∣
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⋯
p
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…
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−
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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)
p x 1 , … , x n = p x 1 ( x 1 ) p x 2 ∣ × 1 ( x 2 ∣ x 1 ) ⋯ p x n ∣ x 1 , … , x n − 1 ( x n ∣ x 1 , … , x n − 1 )
Directed acyclic graphs (DAG )
Fully-connected DAG
conditional independence :
x
A
1
⊥
x
A
2
∣
x
A
3
\mathbf{x}_{\mathcal{A}_{1}} \perp \mathbf{x}_{\mathcal{A}_{2}} | \mathbf{x}_{\mathcal{A}_{3}}
x A 1 ⊥ x A 2 ∣ x A 3
Bayes ball algorithm
primary shade:
A
3
\mathcal{A_3}
A 3 中的节点
secondary shade: primary shade 的节点,以及 secondary shade 的父节点
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3. Factor graph
有 variable nodes 和 factor nodes,是 bipartitie graph
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p_{\mathbf{x}}(\mathbf{x}) \propto \prod_{j} f_{j}\left(\mathbf{x}_{f_{j}}\right)
p x ( x ) ∝ j ∏ f j ( x f j )
因子图比 directed graph 和 undirected graph 的表示能力更强,比如
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23
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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 ) = Z 1 ϕ 1 2 ( x 1 , x 2 ) ϕ 1 3 ( x 1 , x 3 ) ϕ 2 3 ( x 2 , x 3 )
因子图可以与 DAG 相互转化(根据
x
1
,
.
.
.
,
x
n
x_1,...,x_n
x 1 , . . . , x n 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
C
I
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D
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CI(D)
C I ( D ) :the set of conditional independencies satisfied by
D
D
D
C
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CI(G)
C I ( G ) : the set of all conditional independencies implied by
G
G
G
I-map :
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⊂
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I
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D
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C I(\mathcal{G}) \subset C I(D)
C I ( G ) ⊂ C I ( D )
D-map : :
C
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⊃
C
I
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C I(\mathcal{G}) \supset C I(D)
C I ( G ) ⊃ C I ( D )
P-map :
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=
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C I(\mathcal{G}) = C I(D)
C I ( G ) = C I ( 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
C
I
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G
)
CI(G)
C I ( 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
其他内容请看: 统计推断(一) Hypothesis Test 统计推断(二) Estimation Problem 统计推断(三) Exponential Family 统计推断(四) Information Geometry 统计推断(五) EM algorithm 统计推断(六) Modeling 统计推断(七) Typical Sequence 统计推断(八) Model Selection 统计推断(九) Graphical models 统计推断(十) Elimination algorithm 统计推断(十一) Sum-product algorithm