8-Uncertainty

Conditional probability

Definition of conditional probability:

$$P(a|b)=\frac{P(a\bigwedge b)}{P(b)}$$

Product rule gives an alternative formulation:

$$P(a\bigwedge b)=P(a|b)P(b)=P(b|a)P(a)$$

Chain rule is derived by successive application of product rule:

$$P(X_1,...,X_n)=P(X_1,...,X_{n-1})P(X_n|X_1,...,X_{n-1})=...=\prod _{i=1}^{n}P(X_i|X_1,...,X_{i-1})$$

Independence

A and B are independent iff

$P(A|B)=P(A)$ or $P(B|A)=P(B)$ or $P(A,B)=P(A)P(B)$

Conditional independence

$$P(Toothache,Catch|Cavity)=P(Toothache|Cavity)P(Catch|Cavity)$$

Conditional independence is our most basic and robust form of knowledge about uncertain environments.

Bayes’ Rule

$$P(a|b)=\frac{P(b|a)P(a)}{P(b)}$$

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