I have the following scenario of Bayes updating with which I struggle quite a bit.
Imagine we are interested in the probability that a given person has a disease $D$. We perform two different tests $T_1,T_2$ on a blood sample of this person.
Im interested in something like the probability that a person has the disease given both tests return positive.
$P(D|T_1 =1, T_2 = 1) = \frac{P(T_1 =1, T_2 = 1|D) * P(D)}{P(T_1 =1, T_2 = 1)}$
Importantly, both tests are not independent (as they both depend on the blood sample), but given some blood-based measure $x$, we assume conditional independence.
My first question is: How do I correctly incorporate the conditional independence assumption into the equation? Is it simply:
$P([D|T_1 =1, T_2 = 1]|x) = \frac{[P(T_1 =1, T_2 = 1|D]|x) * P(D|x)}{P([T_1 =1, T_2 = 1]|x)}$
?
Furthermore, I have a scenario in mind in which I can reasonably guess how the tests translate $x$ into a probability of having a disease e.g., $p_1(x) = 0.2 + x, p_2(x) = 0.3 + x$, but Im also not sure how to properly incorporate this information into the equation.
I appreciate any tips how to proceed, thank you!
Best N