17.6 Appendix g to applied probability: properties of conditional

Definition . The pair $\{X,Y\}$ is conditionally independent, given Z , denoted $\{X,Y\}\mathrm{ci}|Z$ iff

An arbitrary class $\{X_t:t\in T\}$ of random vectors is conditionally independent, give Z , iff such a product rule holds for each finite subclass or two or more members of the class.

Remark . The expression “for all Borel sets $M,N$ ,” here and elsewhere, implies the sets are on the appropriate codomains. Also, the expressions below “for all Borel functions g ,” etc., imply that the functions are real-valued, such that the indicated expectations are finite.

The following are equivalent . Each is necessary and sufficient that $\{X,Y\}\mathrm{ci}|Z$ .

• (CI1) $E\left[I_M\left(X\right)I_N\left(Y\right)\right|Z]=E\left[I_M\left(X\right)\right|Z]E\left[I_N\left(Y\right)\right|Z]\mathrm{a}.\mathrm{s}.\text{for}\text{all}\text{Borel}\text{sets}M,N$
• (CI2) $E\left[I_M\left(X\right)\right|Z,Y]=E\left[I_M\left(X\right)\right|Z]\mathrm{a}.\mathrm{s}.\text{for}\text{all}\text{Borel}\text{sets}M$
• (CI3) $E\left[I_M\left(X\right)I_Q\left(Z\right)\right|Z,Y]=E\left[I_M\left(X\right)I_Q\left(Z\right)\right|Z]\mathrm{a}.\mathrm{s}.\text{for}\text{all}\text{Borel}\text{sets}M,Q$
• (CI4) $E\left[I_M\left(X\right)I_Q\left(Z\right)\right|Y]=E\left\{E\left[I_M\left(X\right)I_Q\left(Z\right)\right|Z]\right|Y\}\mathrm{a}.\mathrm{s}.\text{for}\text{all}\text{Borel}\text{sets}M,Q$
  
  ****
• (CI5) $E\left[g\right(X,Z\left)h\right(Y,Z\left)\right|Z]=E[g(X,Z)\left|Z\right]E\left[h\right(Y,Z\left)\right|Z]\mathrm{a}.\mathrm{s}.\text{for}\text{all}\text{Borel}\text{functions}g,h$
• (CI6) $E\left[g\right(X,Z\left)\right|Z,Y]=E[g(X,Z)\left|Z\right]\mathrm{a}.\mathrm{s}.\text{for}\text{all}\text{Borel}\text{functions}g$
• (CI7) For any Borel function g , there exists a Borel function e _g such that
  $$E\left[g(X,Z)\right|Z,Y]=e_g\left(Z\right)\mathrm{a}.\mathrm{s}.$$
• (CI8) $E\left[g\right(X,Z\left)\right|Y]=E\{E\left[g\right(X,Z\left)\right|Z\left]\right|Y\}\mathrm{a}.\mathrm{s}.\text{for}\text{all}\text{Borel}\text{functions}g$
  
  ****
• (CI9) $\{U,V\}\mathrm{ci}|Z$ , where $U=g(X,Z)$ and $V=h(Y,Z)$ , for any Borel functions $g,h$ .

Additional properties of conditional independence

• (CI10) $\{X,Y\}\mathrm{ci}|Z$ implies $\{X,Y\}\mathrm{ci}\left|\right(Z,U)$ , $\{X,Y\}\mathrm{ci}\left|\right(Z,V)$ , and $\{X,Y\}\mathrm{ci}\left|\right(Z,U,V)$ , where $U=h\left(X\right)$ and $V=k\left(Y\right)$ , with $h,k$ Borel.
• (CI11) $\{X,Z\}\mathrm{ci}|Y$ and $\{X,W\}\mathrm{ci}\left|\right(Y,Z)$ iff $\{X,(Z,W\left)\right\}\mathrm{ci}|Y$ .
• (CI12) $\{X,Z\}\mathrm{ci}|Y$ and $\left\{\right(X,Y),W\}\mathrm{ci}|Z$ implies $\{X,(Z,W\left)\right\}\mathrm{ci}|Y$ .
• (CI13) $\{X,Y\}$ is independent and $\{X,Z\}\mathrm{ci}|Y$ iff $\{X,(Y,Z\left)\right\}$ is independent.
• (CI14) $\{X,Y\}\mathrm{ci}|Z$ implies $E\left[g(X,Y)\right|Y=u,Z=v]=E\left[g(X,u)\right|Z=v]\mathrm{a}.\mathrm{s}.\left[P_{YZ}\right]$
• (CI15) $\{X,Y\}\mathrm{ci}|Z$ implies
  1. $E\left[g(X,Z)h(Y,Z)\right]=E\left\{E\left[g(X,Z)\right|Z]E\left[h(Y,Z)\right|Z]\right\}=E\left[e_1\left(Z\right)e_2\left(Z\right)\right]$
  2. $E\left[g\left(Y\right)\right|X\in M]P(X\in M)=E\left\{E\left[I_M\left(X\right)\right|Z]E\left[g\left(Y\right)\right|Z]\right\}$
• (CI16) $\left\{\right(X,Y),Z\}\mathrm{ci}|W$ iff $E\left[I_M\left(X\right)I_N\left(Y\right)I_Q\left(Z\right)\right|W]=E\left[I_M\left(X\right)I_N\left(Y\right)\right|W]E\left[I_Q\left(Z\right)\right|W]\mathrm{a}.\mathrm{s}.$      for all Borel sets $M,N,Q$