Looking for some guidance for using Baysian probability for consecutive independent tests.
Using the traditional example of disease testing efficacy (say the 1% of the population are infected and the test is 95% accurate), I would like to understand the approach should a person chose to get a second test.
- What is the probability of having the disease given both tests are positive?
- What is the probability of having the disease if they get a different test and that test is only 90% accurate?
My intuition says that the first test result is not part of the input (prior) to the second test but I can't get it right in my head. For the second test, should I partition the sample space based on the probability of being infected based on the first test? =:-/
Thanks
The assumption should be that the two test results are conditionally independent when given the disease state of the same person. Otherwise, you cannot solve the problem. So if we let the two test results be Bernoulli variables $T_1,T_2$, and disease state be $D$, then: $$\mathsf P(T_1{\,=\,}s,T_2{\,=\,}t\mid D{\,=\,}d)~=~\mathsf P(T_1{\,=\,}s\mid D{\,=\,}d)\,\mathsf P(T_2{\,=\,}t\mid D{\,=\,}d)$$
The rest is just the usual applications of Bayes' Theory and substitution for provided values.
That is $\mathsf P(D{\,=\,}1)=0.01$ and for each $k\in\{1,2\}$: $~\mathsf P(T_k{\,=\,}1\mid D{\,=\,}1)=0.95, \mathsf P(T_k{\,=\,}0\mid D{\,=\,}0)=0.95$.
$\mathsf P(D{\,=\,}1\mid T_1{\,=\,}1,T_2{\,=\,}1)$
$\mathsf P(D{\,=\,}1\mid (T_1{\,=\,}1,T_2{\,=\,}0)\cup(T_1{\,=\,}0,T_2{\,=\,}1))$