Let C1,C2,⋯,CMC1,C2,⋯,CM be a partition of the sample space SS, and AA and BB be two events. Suppose we know that AA and BB are conditionally independent given CiCi, for all i∈{1,2,⋯,M}i∈{1,2,⋯,M}; BB is independent of all CiCi's. Prove that AA and BB are independent.

 Problem 

Let C1,C2,⋯,CM$C_1,C_2,\cdots ,C_M$ be a partition of the sample space S$S$, and A$A$ and B$B$ be two events. Suppose we know that

• A$A$ and B$B$ are conditionally independent given Ci$C_i$, for all i∈{1,2,⋯,M}$i\in \{1,2,\cdots ,M\}$;
• B$B$ is independent of all Ci$C_i$'s.

Prove that A$A$ and B$B$ are independent.

• Solution

  • Since the Ci$C_i$'s form a partition of the sample space, we can apply the law of total probability for A∩B$A\cap B$:
    
    

    P(A∩B)$P(A\cap B)$	=∑Mi=1P(A∩B|Ci)P(Ci)$=\sum _{i=1}^{M}P(A\cap B|C_i)P(C_i)$
    	=∑Mi=1P(A|Ci)P(B|Ci)P(Ci)$=\sum _{i=1}^{M}P(A|C_i)P(B|C_i)P(C_i)$	(A and B are conditionally independent)$$
    	=∑Mi=1P(A|Ci)P(B)P(Ci)$=\sum _{i=1}^{M}P(A|C_i)P(B)P(C_i)$	(B is independent of all Ci's)
    	=P(B)∑Mi=1P(A|Ci)P(Ci)$=P(B)\sum _{i=1}^{M}P(A|C_i)P(C_i)$
    	=P(B)P(A)$=P(B)P(A)$	(law of total probability).