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

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Definition

The pair $$\{X, Y\}$$ is conditionally independent, givenZ, denoted $$\{X, Y\}$$ 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 $$M, N$$

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 $$\{X, Y\}$$ 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 $$M, N$$
(CI2): $$E[I_M (X)|Z, Y] = E[I_M(X)|Z]$$ 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 $$M, Q$$
(CI4): $$E[I_M (X) I_Q(Z)|Y] = E\{E[I_M(X) I_Q (Z)|Z]|Y\}$$ a.s. for all Borel sets $$M, Q$$

****
(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 $$g$$, $$h$$
(CI6): $$E[g(X, Z)|Z, Y] = E[g(X, Z)|Z]$$ a.s. for all Borel function $$g$$
(CI7): For any Borel function $$g$$, there exists a Borel function $$e_g$$ such that

$$E[g(X, Z)|Z,Y] = e_g(Z)$$ a.s.

(CI8): $$E[g(X,Z)|Y] = E\{E[g(X, Z)|Z]|Y\}$$ a.s. for all Borel functions $$g$$

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

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

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