What we really said that we had a situation that prior
a test is a certain sensitivity and a certain specificity.
When you receive say a positive test result, what you do is you take your prior,
you multiply in the probability of this test result.
Given C, and you multiply in the probability of the test result given not C.
So this is your branch for the consideration that you have cancer.
This is your branch for the consideration you have no cancer.
When you're done with this, you arrive at a number that now combines the cancer
hypothesis with the test result.
Both for the cancer hypothesis and the not cancer hypothesis.
Now what you do, you add those up.
And they normally don't add up to one.
You get a certain quantity,
which happens to be the total probability that the test is what it was.
This case positive.
And all you do next is divide or
normalize this thing over here by the sum over here.
And the same on the right side.
The divider is the same for
both cases because this is your cancer range, your non cancer range.
But this guy doesn't rely on the cancer variable anymore.
What you now get out is the desired posterior probability, and
those add up to one if you did everything correct as shown over here.
This is your algorithm for Bayes Rule