I would like to know how the attached exercise arrived at the equation. It is a 3-way Bayes net where (C) Cancer is the parent of (T1) Test 1 and of (T2) Test 2. I'm trying to calculate P(T2=+|T1=+). In the answer the instructor arrived at the following = P(T2=+|T1=+,C).P(C|T1=+) + P(T2=+|T1=+,⌝C).P(⌝C|T1=+) After this I'm fine. It has something to do with Theory of Total Probability and conditioned on T1=+
It is the Law of Total Probability.$$\mathsf P(X{=}x\mid Y{=}y)~=~\sum_z\mathsf P(X{=}x\mid Y{=}y,Z{=}z)\cdotp\mathsf P(Z{=}z\mid Y{=}y)$$
$${{\mathsf P(T_1{=}\top\mid T_2{=}\top)}~=~{{\mathsf P(T_1{=}\top\mid T_2{=}\top,C)\cdotp\mathsf P(C\mid T_2{=}\top)}+{\mathsf P(T_1{=}\top\mid T_2{=}\top,\lnot C)\cdotp\mathsf P(\lnot C\mid T_2{=}\top)}}}$$
Now, the point is that the two tests should be conditionally independent given the status of cancer. Thus: $${{\mathsf P(T_1{=}\top\mid T_2{=}\top)}~=~{{\mathsf P(T_1{=}\top\mid C)\cdotp\mathsf P(C\mid T_2{=}\top)}+{\mathsf P(T_1{=}\top\mid \lnot C)\cdotp\mathsf P(\lnot C\mid T_2{=}\top)}}}$$