Question: the following application of bayes rule often occurs in actual...
The following application of Bayes rule often occurs in actual
medical practice. Suppose you have tested positive for a disease.
What is the probability you actually have the disease? It depends
on the sensitivity and specificity of the test, and on the
prevalence (prior probability) of the disease.
a positive test as: Test = pos
a negative test as: Test = neg
presence of disease as: Disease = true
absence of disease as: Disease = false
We know from clinical studies done on the test before FDA approval that the sensitivity and specificity of the test are:
p (Test = pos | Disease = true) = 0.95 (true positive rate, or sensitivity)
p (Test = neg | Disease = false) = 0.90 (true negative rate, or specificity)
From which we can also deduce:
p (Test = neg | Disease = true) = 0.05 (false negative rate)
p (Test = pos | Disease = false) = 0.10 (false positive rate)
We also know from public health surveys that the disease is relatively rare. The prevalence in the general population is:
p (Disease = true) = 0.01
From which we can deduce:
p (Disease = false) = 0.99
a) Use Bayes rule to calculate p (Disease = true | Test = pos), i.e. the probability you actually have the disease, given the test was positive.
b) Calculate the ratio p (Disease = true | Test = pos) / p (Disease = true). [ In Bayesian statistics, a ratio like this is interpreted as the effect of new evidence on our beliefs about a probability. In this case, we are concerned with the probability of disease, and the new evidence is the test result.]