02 Bayes Theorem
Bayes’ Theorem¶
It determines the probability of an event with uncertain knowledge.
\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]
where - \(P(A|B)\) = posterior,
- \(P(B|A)\) = likelihood,
- \(P(A)\) = prior probability - \(P(B)\) = marginal probability
General Formula¶
\[ \begin{aligned} P(A_i|B) &= \frac{P(A_i \land B)}{P(B)} \\ &= \frac{P(B | A_i) \cdot P(A_i)}{\sum\limits_{j=1}^{n} P(B|A_j) \cdot P(A_j)} \\ \end{aligned} \]
where \(A_1, A_2, \dots, A_n\) are all mutually exclusive events
Conditions¶
- Events must be disjoint (no overlapping)
- Events must be exhaustive: they combine to include all possibilities
Phrases¶
- “out of”
- “of those who”