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Appendix G to Applied Probability: Properties of conditional independence, given a random vector

Module by: Paul E Pfeiffer. E-mail the author

Summary: This property lies at the root of the theory of Markov processes, which are characterized by the condition that past and future are conditionally independent, given the present. The notions of past, present, and future are somewhat elastic. One may have an extended present, a finite or infinite past or future. The mathematical patterns of past, present, and future provide important results on the "behavior" of Markov processes.

Definition. The pair {X,Y}{X,Y} is conditionally independent, givenZ, denoted {X,Y} ci |Z{X,Y} ci |Z iff

E [ I M ( X ) I N ( Y ) | Z ] = E [ I M ( X ) | Z ] E [ I N ( Y ) | Z ] a . s . for all Borel sets M , N E [ I M ( X ) I N ( Y ) | Z ] = E [ I M ( X ) | Z ] E [ I N ( Y ) | Z ] a . s . for all Borel sets M , N
(1)

An arbitrary class {Xt:tT}{Xt:tT} 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,NM,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} ci |Z{X,Y} ci |Z.

  • (CI1) : E [ I M ( X ) I N ( Y ) | Z ] = E [ I M ( X ) | Z ] E [ I N ( Y ) | Z ] a . s . for all Borel sets M , N E [ I M ( X ) I N ( Y ) | Z ] = E [ I M ( X ) | Z ] E [ I N ( Y ) | Z ] a . s . for all Borel sets M , N
  • (CI2) : E [ I M ( X ) | Z , Y ] = E [ I M ( X ) | Z ] a . s . for all Borel sets M E [ I M ( X ) | Z , Y ] = E [ I M ( X ) | Z ] a . s . for all Borel sets M
  • (CI3) : E [ I M ( X ) I Q ( Z ) | Z , Y ] = E [ I M ( X ) I Q ( Z ) | Z ] a . s . for all Borel sets M , Q E [ I M ( X ) I Q ( Z ) | Z , Y ] = E [ I M ( X ) I Q ( Z ) | Z ] a . s . for all Borel sets M , Q
  • (CI4) : E[IM(X)IQ(Z)|Y]=E{E[IM(X)IQ(Z)|Z]|Y}a.s.forallBorelsetsM,QE[IM(X)IQ(Z)|Y]=E{E[IM(X)IQ(Z)|Z]|Y}a.s.forallBorelsetsM,Q

    ****
  • (CI5) : E [ g ( X , Z ) h ( Y , Z ) | Z ] = E [ g ( X , Z ) | Z ] E [ h ( Y , Z ) | Z ] a . s . for all Borel functions g , h E [ g ( X , Z ) h ( Y , Z ) | Z ] = E [ g ( X , Z ) | Z ] E [ h ( Y , Z ) | Z ] a . s . for all Borel functions g , h
  • (CI6) : E [ g ( X , Z ) | Z , Y ] = E [ g ( X , Z ) | Z ] a . s . for all Borel functions g E [ g ( X , Z ) | Z , Y ] = E [ g ( X , Z ) | Z ] a . s . for all Borel functions g
  • (CI7) : For any Borel function g, there exists a Borel function eg such that
    E[g(X,Z)|Z,Y]=eg(Z)a.s.E[g(X,Z)|Z,Y]=eg(Z)a.s.
    (2)
  • (CI8) : E[g(X,Z)|Y]=E{E[g(X,Z)|Z]|Y}a.s.forallBorelfunctionsgE[g(X,Z)|Y]=E{E[g(X,Z)|Z]|Y}a.s.forallBorelfunctionsg

    ****
  • (CI9) : {U,V} ci |Z{U,V} ci |Z, where U=g(X,Z)U=g(X,Z) and V=h(Y,Z)V=h(Y,Z), for any Borel functions g,hg,h.

Additional properties of conditional independence

  • (CI10): {X,Y} ci |Z{X,Y} ci |Z implies {X,Y} ci |(Z,U){X,Y} ci |(Z,U), {X,Y} ci |(Z,V){X,Y} ci |(Z,V), and {X,Y} ci |(Z,U,V){X,Y} ci |(Z,U,V), where U=h(X)U=h(X) and V=k(Y)V=k(Y), with h,kh,k Borel.
  • (CI11): {X,Z} ci |Y{X,Z} ci |Y and {X,W} ci |(Y,Z){X,W} ci |(Y,Z) iff {X,(Z,W)} ci |Y{X,(Z,W)} ci |Y.
  • (CI12): {X,Z} ci |Y{X,Z} ci |Y and {(X,Y),W} ci |Z{(X,Y),W} ci |Z implies {X,(Z,W)} ci |Y{X,(Z,W)} ci |Y.
  • (CI13): {X,Y}{X,Y} is independent and {X,Z} ci |Y{X,Z} ci |Y iff {X,(Y,Z)}{X,(Y,Z)} is independent.
  • (CI14): {X,Y} ci |Z{X,Y} ci |Z implies E[g(X,Y)|Y=u,Z=v]=E[g(X,u)|Z=v]a.s.[PYZ]E[g(X,Y)|Y=u,Z=v]=E[g(X,u)|Z=v]a.s.[PYZ]
  • (CI15): {X,Y} ci |Z{X,Y} ci |Z implies
    1. E[g(X,Z)h(Y,Z)]=E{E[g(X,Z)|Z]E[h(Y,Z)|Z]}=E[e1(Z)e2(Z)]E[g(X,Z)h(Y,Z)]=E{E[g(X,Z)|Z]E[h(Y,Z)|Z]}=E[e1(Z)e2(Z)]
    2. E[g(Y)|XM]P(XM)=E{E[IM(X)|Z]E[g(Y)|Z]}E[g(Y)|XM]P(XM)=E{E[IM(X)|Z]E[g(Y)|Z]}
  • (CI16): {(X,Y),Z} ci |W{(X,Y),Z} ci |W iff E[IM(X)IN(Y)IQ(Z)|W]=E[IM(X)IN(Y)|W]E[IQ(Z)|W]a.s.E[IM(X)IN(Y)IQ(Z)|W]=E[IM(X)IN(Y)|W]E[IQ(Z)|W]a.s.      for all Borel sets M,N,QM,N,Q

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