贝叶斯

graph TD x1((x1)) --> x3((x3)) x1 --> x4((x4))

%%{init: {'theme':'dark'}}%% graph TD x1((x1)) --> x3((x3)) x1 --> x4((x4))

$$\displaystyle{ \begin{aligned}P \left( x _{ 3 } , x _{ 4 } \mid x _{ 1 } \right) & = \frac{ P \left( x _{ 1 } , x _{ 3 } , x _{ 4 } \right) }{ P \left( x _{ 1 } \right) } \\ & = \frac{ P \left( x _{ 3 } \mid x _{ 1 } \right) P \left( x _{ 4 } \mid x _{ 1 } \right) P \left( x _{ 1 } \right) }{ P \left( x _{ 1 } \right) }\end{aligned} }$$

graph RL z((z)) --> x((x)) --> y((y))

%%{init: {'theme':'dark'}}%% graph RL z((z)) --> x((x)) --> y((y))

$$\displaystyle{ \begin{aligned}P \left( y , z \mid x \right) & = \frac{ P \left( y \mid x \right) P \left( x \mid z \right) P \left( z \right) }{ P \left( x \right) } \\ & = \frac{ P \left( y \mid x \right) P \left( z \mid x \right) P \left( x \right) }{ P \left( x \right) }\end{aligned} }$$

graph TD x1((x1)) --> x4((x4)) x2((x2)) --> x4

%%{init: {'theme':'dark'}}%% graph TD x1((x1)) --> x4((x4)) x2((x2)) --> x4

$$\displaystyle{ \begin{aligned}P \left( x _{ 1 } , x _{ 2 } \right) & = \sum _{ x _{ 4 } } P \left( x _{ 1 } , x _{ 2 } , x _{ 4 } \right) \\ & = \sum _{ x _{ 4 } } P \left( x _{ 4 } \mid x _{ 1 } , x _{ 2 } \right) P \left( x _{ 1 } \right) P \left( x _{ 2 } \right) \\ & = P \left( x _{ 1 } \right) P \left( x _{ 2 } \right)\end{aligned} }$$