PGM_Foundations

Chain Rule and Bayesian Rule

From the definition of the conditional distribution, we see that

$$P(\alpha_1 \cap ... \cap \alpha_k)=P(\alpha_1)P(\alpha_2 \vert \alpha_1)...P(\alpha_k \vert \alpha_1 \cap ...\cap \alpha_{k-1})~~(Chain~Rule)$$

$$P(\alpha \vert \beta)=\frac{P(\beta \vert \alpha)P(\alpha)}{P(\beta)}~~(Bayesian~Rule)$$

A more general conditional version of Bayes’ rule, where all our probabilities are conditioned on some background event $\gamma$, also holds $$P(\alpha \vert \beta \cap \gamma)=\frac{P(\beta \vert \alpha \cap \gamma)P(\alpha \cap \gamma)}{P(\beta \cap \gamma)}$$