Category:Bonferroni Inequalities
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This category contains pages concerning Bonferroni Inequalities:
The Bonferroni inequalities are a generalization of Boole's Inequality:
Let $E = \set {E_1, E_2, \ldots, E_n}$ be a set of $n$ events.
Let $\overline E = \set {\overline {E_1}, \overline {E_2}, \ldots, \overline {E_n} }$ be the set of complementary events to each of $\set {E_1, E_2, \ldots, E_n}$ respectively.
Let:
| \(\ds S_0\) | \(:=\) | \(\ds 1\) | ||||||||||||
| \(\ds S_1\) | \(:=\) | \(\ds \sum_{1 \mathop = 1}^n \map \Pr {E_i}\) | ||||||||||||
| \(\ds S_2\) | \(:=\) | \(\ds \sum_{1 \mathop \le i \mathop < j \mathop \le n} \map \Pr {E_i \cap E_j}\) | ||||||||||||
| \(\ds \) | \(\cdots\) | \(\ds \) | ||||||||||||
| \(\ds S_k\) | \(:=\) | \(\ds \sum_{1 \mathop \le i_1 \mathop < \cdots \mathop < i_k \mathop \le n} \map \Pr {E_{i_1} \cap E_{i_2} \cap \cdots \cap E_{i_k} }\) | ||||||||||||
| \(\ds \) | \(\cdots\) | \(\ds \) | ||||||||||||
| \(\ds S_n\) | \(:=\) | \(\ds \map \Pr {\bigcap E}\) |
where:
- $\map \Pr {E_i}$ denotes the probability of $E_i$
- $\bigcap E$ denotes the intersection of $E$.
When $n$ is odd:
| \(\ds \map \Pr {\bigcup_{i \mathop = 1}^n E_i}\) | \(=\) | \(\ds \sum_{j \mathop = 1}^n \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{j \mathop = 1}^{n - 2} \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{j \mathop = 1}^{n - 4} \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \cdots\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{j \mathop = 1}^1 \paren {-1}^{j - 1} S_j\) |
and:
| \(\ds \map \Pr {\bigcup_{i \mathop = 1}^n E_i}\) | \(=\) | \(\ds \sum_{j \mathop = 1}^n \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\ge\) | \(\ds \sum_{j \mathop = 1}^{n - 1} \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\ge\) | \(\ds \sum_{j \mathop = 1}^{n - 3} \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\ge\) | \(\ds \cdots\) | ||||||||||||
| \(\ds \) | \(\ge\) | \(\ds \sum_{j \mathop = 1}^2 \paren {-1}^{j - 1} S_j\) |
When $n$ is even:
| \(\ds \map \Pr {\bigcup_{i \mathop = 1}^n E_i}\) | \(=\) | \(\ds \sum_{j \mathop = 1}^n \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{j \mathop = 1}^{n - 1} \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{j \mathop = 1}^{n - 3} \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \cdots\) | ||||||||||||
| \(\ds \) | \(\le\) | \(\ds \sum_{j \mathop = 1}^1 \paren {-1}^{j - 1} S_j\) |
and:
| \(\ds \map \Pr {\bigcup_{i \mathop = 1}^n E_i}\) | \(=\) | \(\ds \sum_{j \mathop = 1}^n \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\ge\) | \(\ds \sum_{j \mathop = 1}^{n - 2} \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\ge\) | \(\ds \sum_{j \mathop = 1}^{n - 4} \paren {-1}^{j - 1} S_j\) | ||||||||||||
| \(\ds \) | \(\ge\) | \(\ds \cdots\) | ||||||||||||
| \(\ds \) | \(\ge\) | \(\ds \sum_{j \mathop = 1}^2 \paren {-1}^{j - 1} S_j\) |
In particular:
- $\map \Pr {\bigcap E} > 1 - \ds \sum_{i \mathop = 1}^n \map \Pr {\overline E_i}$
which is a statement of Boole's Inequality.
Pages in category "Bonferroni Inequalities"
This category contains only the following page.