ベイズの公式 - (新) 檜山正幸のキマイラ飼育記メモ編

被積分形式：

$P(dx, dy) = {\displaystyle \int_{x'\in X} P_{!Y|X}(dy \mid x')P_{!X}(dx\cap dx')} \\ P(dx, B) = {\displaystyle \int_{x'\in X} P_{!Y|X}(B \mid x')P_{!X}(dx\cap dx')} \\ P(A, B) = {\displaystyle \int_{x'\in X} P_{!Y|X}(B \mid x')P_{!X}(A\cap dx')} \\ P(A, B) = {\displaystyle \int_{x'\in A\subseteq X} P_{!Y|X}(B \mid x')P_{!X}(dx')} \\$

二変数は、ほんとは直積：

$P(dx \times dy) = {\displaystyle \int_{x'\in X} P_{!Y|X}(dy \mid x')P_{!X}(dx\cap dx')} \\ P(dx \times B) = {\displaystyle \int_{x'\in X} P_{!Y|X}(B \mid x')P_{!X}(dx\cap dx')} \\ P(A \times B) = {\displaystyle \int_{x'\in X} P_{!Y|X}(B \mid x')P_{!X}(A\cap dx')} \\ P(A \times B) = {\displaystyle \int_{x'\in A\subseteq X} P_{!Y|X}(B \mid x')P_{!X}(dx')} \\$

離散の場合：

$P(\delta x, \delta y) = {\displaystyle \sum_{x'\in X} P_{!Y|X}(\delta y \mid x')P_{!X}(\delta x\cap \delta x')} \\ P(\delta x, B) = {\displaystyle \sum_{x'\in X} P_{!Y|X}(B \mid x')P_{!X}(\delta x\cap \delta x')} \\ P(A, B) = {\displaystyle \sum_{x'\in X} P_{!Y|X}(B \mid x')P_{!X}(A\cap \delta x')} \\ P(A, B) = {\displaystyle \sum_{x'\in A\subseteq X} P_{!Y|X}(B \mid x')P_{!X}(\delta x')} \\$

デルタは基本事象：

$P(\{x\}, \{y\}) = {\displaystyle \sum_{x'\in X} P_{!Y|X}(\{y\} \mid x')P_{!X}(\{x\}\cap \{x'\})} \\ P(\{x\}, B) = {\displaystyle \sum_{x'\in X} P_{!Y|X}(B \mid x')P_{!X}(\{x\}\cap \{x'\})} \\ P(A, B) = {\displaystyle \sum_{x'\in X} P_{!Y|X}(B \mid x')P_{!X}(A\cap \{x'\})} \\ P(A, B) = {\displaystyle \sum_{x'\in A\subseteq X} P_{!Y|X}(B \mid x')P_{!X}(\{x'\})} \\$