If I have a frequency distribution describing the frequency of normal patients whose heart presents with a degree of a symptom (let's say 50 % of them present with a "3" 20% presenting with a "2" and 30% presenting with a "4".
How would I use this data test whether a new patient is healthy or unhealthy? A relative frequency would say that if a patient presents with a "3" then there is a 50% chance they are healthy, isn't it the case that they are more likely to be healthy if they sit in the middle of the distribution?
Also, how would you then go on to combine this probability with 24 other tests on the same patient to acquire an overall probability?
Many thanks
I'm assuming by "normal patients", you mean healthy.
To use p(3|healthy) to get p(healthy|3), you could invoke Bayes' Rule:
p(A|B) = p(B|A)p(A)/p(B)
in which p( ) denotes probability, and | denotes conditionality, such that "A|B" means "A given B", or "A conditional on B".
This would necessitate knowing the overall proportion of healthy patients and the overall proportion of patients presenting with a "3". Putting it all together gives
p(healthy|3) = p(3|healthy)p(healthy)/p(3)
In the simplest case, you could use Bayes' rule with B in the form above denoting the intersection of all test results (a given result in test 1 AND a given result in test 2, etc), but it would depend on what data you have available.
Letting X, Y, and Z denote given results from three tests gives
p(healthy|(X&Y&Z)) = p((X&Y&Z)|healthy)p(healthy)/p(X&Y&Z)
This can be simplified to the following IF AND ONLY IF results X, Y, and Z are independent, as are X|healthy, Y|healthy, and Z|healthy
p(healthy|(X&Y&Z)) = p(X|healthy)p(Y|healthy)p(Z|healthy)p(healthy)/(p(X)p(Y)p(Z))