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--- bayes_theorem [2025/09/20 20:40] – hkimscil
+++ bayes_theorem [2025/09/21 22:35] (current) – hkimscil
@@ Line 1: / Line 1: @@
 ====== Bayes' Theorem ======
-\begin{eqnarray}
+<WRAP left>
-\text{c.f.,} \nonumber \\
+\begin{eqnarray*}
-& & P(A|B) = \dfrac{P(A \cap B)}{P(B)}  \nonumber \\
+P(A \mid B) & = & \dfrac{P(A \cap B)}{P(B)}  \nonumber \\
-& & P(B|A) = \dfrac{P(B \cap A)}{P(A)}  \nonumber \\
+P(B \mid A) & = & \dfrac{P(B \cap A)}{P(A)}  \nonumber \\
-& &\text{heance }   \nonumber \\
+\text{heance }   \nonumber \\
-& & P(A \cap B) = P(A \mid B) * P(B) \text{and   }  \nonumber \\
+P(A \cap B) & = & P(A \mid B) * P(B) \;\; \text{ and   }  \nonumber \\
-& & 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) \\
  \nonumber \\
  \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 \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) \\
- \nonumber \\
+\\
-\text{by the way  } \nonumber   \\
+\\
-P(A|B) & = & \dfrac{P(A \cap B)}{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)} \\
+& = & \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*}
+</WRAP>
+<WRAP clear />
+<WRAP box left>
+\begin{eqnarray}
+& & P(A \mid B) = \dfrac{P(A \cap B)}{P(B)}  \nonumber \\
+& & P(B \mid A) = \dfrac{P(B \cap A)}{P(A)}  \nonumber \\
+& & \text{heance }   \nonumber \\
+& & P(A \cap B) = P(A \mid B) * P(B) \;\; \text{ and   }  \nonumber \\
+& & P(B \cap A) = P(B \mid A) * P(A)  \\
+ \nonumber \\
+ \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 \mid A) * P(A) + P(B \mid \neg A) * P(\neg A) \\
+ \nonumber \\
+ \nonumber \\
+& & \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>
+<WRAP clear />
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