Indepencies

• Two event \alpha and \beta said to be independant in a joint distribution P, denoted as (A \bot B) (is symmetrix), if

  $P(\alpha | \beta) =$$P(\alpha)$ or $P(\beta | \alpha) =$$P(\beta)$ or $P(\alpha \cap \beta) =$$P(\alpha) P(\beta)$
• Two random variables X,Y are independant if for all values
• Number of entries in joint distributino grows exponentially with number of variables.
• Two events \alpha, \beta , \gamma :

  $P(\alpha | \beta \cap \gamma) =$$P(\alpha | \gamma), P(\beta | \alpha \cap \gamma) =$$P(\beta| \gamma), P(\alpha \cap \beta | \gamma) =$$P(\alpha | \gamma)P(\beta|\gamma)$
• $P(X,Y,Z) =$$P(X|Y,Z) P(Y,Z) =$$P(X|Y,Z) P(Y|Z) P(Z) =$$P(X|Z) P(Y|Z) P(Z)$