“Bayes’ Theorem”的概念、定义、翻译、参考文献-科学参考

Bayes’ Theorem

Mathematics

Let A₁, A₂,…, A_k be mutually exclusive events whose union is the whole sample space of an experiment and let B be an event with Pr(B) ≠ 0. Then Pr(A_i|B) equals

$\frac{Pr (B | A_{i}) Pr (A_{i})}{Pr (B | A_{1}) Pr (A_{1}) + \dots + Pr (B | A_{k}) Pr (A_{k})} .$

For example, let A₁ be the event of tossing a double-headed coin and A₂ the event of tossing a normal coin. Suppose that one of the coins is chosen at random so that $Pr (A_{1}) = \frac{1}{2}$ and $Pr (A_{2}) = \frac{1}{2}$ . Let B be the event of obtaining ‘heads’. Then $Pr (B | A_{1}) = 1$ and $Pr (B | A_{2}) = \frac{1}{2}$ . So

$\begin{array}{c} Pr (A_{1} | B) \\ = \frac{Pr (B | A_{1}) Pr (A_{1})}{Pr (B | A_{1}) Pr (A_{1}) + Pr (B | A_{2}) Pr (A_{2})} \\ = \frac{1 \times \frac{1}{2}}{1 \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2}} = \frac{2}{3} . \end{array}$

This says that, given that ‘heads’ was obtained, the probability that it was the double-headed coin that was tossed is 2/3.

Here, Pr(A_i) is a prior probability and Pr(A_i | B) is a posterior probability.

单词	Bayes’ Theorem
释义	Bayes’ Theorem Mathematics Let A₁, A₂,…, A_k be mutually exclusive events whose union is the whole sample space of an experiment and let B be an event with Pr(B) ≠ 0. Then Pr(A_i\|B) equals $\frac{Pr (B \| A_{i}) Pr (A_{i})}{Pr (B \| A_{1}) Pr (A_{1}) + \dots + Pr (B \| A_{k}) Pr (A_{k})} .$ For example, let A₁ be the event of tossing a double-headed coin and A₂ the event of tossing a normal coin. Suppose that one of the coins is chosen at random so that $Pr (A_{1}) = \frac{1}{2}$ and $Pr (A_{2}) = \frac{1}{2}$ . Let B be the event of obtaining ‘heads’. Then $Pr (B \| A_{1}) = 1$ and $Pr (B \| A_{2}) = \frac{1}{2}$ . So $\begin{array}{c} Pr (A_{1} \| B) \\ = \frac{Pr (B \| A_{1}) Pr (A_{1})}{Pr (B \| A_{1}) Pr (A_{1}) + Pr (B \| A_{2}) Pr (A_{2})} \\ = \frac{1 \times \frac{1}{2}}{1 \times \frac{1}{2} + \frac{1}{2} \times \frac{1}{2}} = \frac{2}{3} . \end{array}$ This says that, given that ‘heads’ was obtained, the probability that it was the double-headed coin that was tossed is 2/3. Here, Pr(A_i) is a prior probability and Pr(A_i \| B) is a posterior probability.
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