Probability Measures and Bayes' Rule
- Probability Theory
A probability measure \(\mathbb P\) assigns a number in \([0,1]\) to every event in \(\mathcal F\). The notes begin with \(\mathbb P(\Omega)=1\) and countable additivity for disjoint events:
$$\mathbb P\left(\bigcup_{n=1}^{\infty}A_n\right)=\sum_{n=1}^{\infty}\mathbb P(A_n).$$
From those axioms follow the familiar rules for complements, inclusion, and finite unions. Conditioning then changes the measure after new information arrives. For \(\mathbb P(B)>0\),
$$\mathbb P(A\mid B)=\frac{\mathbb P(A\cap B)}{\mathbb P(B)}.$$
The total-probability and Bayes formulas update a partition of the sample space when an observation occurs. The notes emphasize that Bayes' rule is a bookkeeping identity for the same joint event, not a new probability model. Independence is the special case in which \(\mathbb P(A\cap B)=\mathbb P(A)\mathbb P(B)\).