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### Bayes’ Theorem: Basics

- Total probability Theorem:

\[P(B)=\sum_{i=1}^{M}P(B|A_{i})P(A_{i})\]

- Bayes’ Theorem:

\[P(H|X)=\frac{P(X|H)P(H)}{P(X)}=P(X|H)\times P(H)/P(X)\]

- Let
**X**be a data sample (“*evidence*”): class label is unknown - Let H be a
*hypothesis*that X belongs to class C - Classification is to determine P(H|
**X**), (i.e.,*posteriori probability):*the probability that the hypothesis holds given the observed data sample**X** - P(H) (
*prior probability*): the initial probability - E.g.,
**X**will buy computer, regardless of age, income, … - P(
**X**): probability that sample data is observed - P(
**X**|H) (likelihood): the probability of observing the sample**X**, given that the hypothesis holds - E.g.,
**X**will buy computer, the prob. that X is 31..40, medium income

**Speaker notes:**

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