bayes_theorem
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| bayes_theorem [2025/09/19 19:56] – hkimscil | bayes_theorem [2025/09/21 22:35] (current) – hkimscil | ||
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| ====== Bayes' Theorem ====== | ====== Bayes' Theorem ====== | ||
| + | <WRAP left> | ||
| \begin{eqnarray*} | \begin{eqnarray*} | ||
| - | \text{c.f.,} \\ | + | P(A \mid B) & = & \dfrac{P(A \cap B)}{P(B)} |
| - | & & P(A|B) = \dfrac{P(A \cap B)}{P(B)} \\ | + | P(B \mid A) & = & \dfrac{P(B \cap A)}{P(A)} |
| - | & & P(B|A) = \dfrac{P(B \cap A)}{P(A)} \\ | + | \text{heance } |
| - | \neg{A} \\ | + | P(A \cap B) & = & P(A \mid B) * P(B) \;\; \text{ and |
| - | \sim{A} \\ | + | P(B \cap A) & = & P(B \mid A) * P(A) \qquad\qquad\qquad\qquad\qquad\qquad\qquad (1) \\ |
| - | \thicksim{A} \\ | + | |
| + | | ||
| + | P(B) & = & P(A \cap B) + P(\neg A \cap B) | ||
| + | & = & P(B \cap A) + P(B \cap \neg A) \nonumber \\ | ||
| + | & = & 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)} \qquad\qquad (3) \\ | ||
| + | |||
| \end{eqnarray*} | \end{eqnarray*} | ||
| + | </ | ||
| + | <WRAP clear /> | ||
| + | |||
| + | |||
| + | |||
| + | <WRAP box left> | ||
| + | \begin{eqnarray} | ||
| + | & & P(A \mid B) = \dfrac{P(A \cap B)}{P(B)} | ||
| + | & & P(B \mid A) = \dfrac{P(B \cap A)}{P(A)} | ||
| + | & & \text{heance } | ||
| + | & & P(A \cap B) = P(A \mid B) * P(B) \;\; \text{ and | ||
| + | & & P(B \cap A) = P(B \mid A) * P(A) \\ | ||
| + | | ||
| + | | ||
| + | & & P(B) = P(A \cap B) + P(\neg A \cap B) \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) \\ | ||
| + | | ||
| + | | ||
| + | & & \text{suppose that we not know } P(B) \nonumber \\ | ||
| + | & & 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} | ||
| + | </ | ||
| + | <WRAP clear /> | ||
| + | |||
| + | {{youtube> | ||
bayes_theorem.1758279410.txt.gz · Last modified: by hkimscil