bayes_theorem
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| bayes_theorem [2025/09/21 16:59] – hkimscil | bayes_theorem [2025/09/21 22:35] (current) – hkimscil | ||
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| <WRAP left> | <WRAP left> | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | & & P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} | + | P(A \mid B) & = & \dfrac{P(A \cap B)}{P(B)} |
| - | & & P(B \mid A) = \dfrac{P(B \cap A)}{P(A)} | + | P(B \mid A) & = & \dfrac{P(B \cap A)}{P(A)} |
| - | & & \text{heance } | + | \text{heance } |
| - | & & P(A \cap B) = P(A \mid B) * P(B) \;\; \text{ and | + | P(A \cap B) & = & P(A \mid B) * P(B) \;\; \text{ and |
| - | & & P(B \cap A) = P(B \mid A) * P(A) \;\;\;\;\;\;\; | + | P(B \cap A) & = & P(B \mid A) * P(A) \qquad\qquad\qquad\qquad\qquad\qquad\qquad (1) \\ |
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| - | P(B) & & = P(A \cap B) + P(\neg A \cap B) \nonumber \\ | + | P(B) & = & P(A \cap B) + P(\neg A \cap B) \nonumber \\ |
| - | & & = P(B \cap A) + P(B \cap \neg A) \nonumber \\ | + | & = & P(B \cap A) + P(B \cap \neg A) \nonumber \\ |
| - | & & = P(B \mid A) * P(A) + P(B \mid \neg A) * P(\neg A) \\ | + | & = & P(B \mid A) * P(A) + P(B \mid \neg A) * P(\neg A) \qquad\qquad (2) \\ |
| + | \\ | ||
| + | \\ | ||
| + | P(A \mid B) & = & \dfrac{P(A \cap B)}{P(B)} \nonumber | ||
| + | & = & \dfrac{P(B \cap A)}{P(B)} \;\;\; \text{ from (1) and (2) } \nonumber | ||
| + | & = & \dfrac {(1)} {(2)} \nonumber \\ | ||
| + | & = & \dfrac {P(B \mid A) * P(A)} {P(B \mid A) * P(A) + P(B \mid \neg A) * P(\neg A)} | ||
| \end{eqnarray*} | \end{eqnarray*} | ||
bayes_theorem.1758441566.txt.gz · Last modified: by hkimscil