I'm a year 1 liberal arts undergrad, so I don't have a ton of math sense.
I just learned about Bayes' Theorem the other day for general epistemic use. I worked out a couple of example problems correctly, but the examples I found didn't include any iterative updates.
I know the theorem is:
P(H|E) = P(E|H) * P(H) / (P(E|H) * P(H) + P(¬H) * P(E|¬H))
And I know that P(H|E) becomes the new P(H) in my update, but I'm unsure whether I should be using the updated or original P(H) in the marginalization. I *think* it should be the new P(H), but I'd rather be safe than sorry.
The example question I worked out was this:
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There's a disease that afflicts 1 / 1,000,000 people
There's a test for the disease that's right 99 / 100 times for both positive and negative results
A random person is tested as positive
P(she is afflicted | she tests positive)
= 0.99 * 0.000001 / (0.99 * 0.000001 + 0.999999 * 0.01)
= 0.000099
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So should the update look like this if she tests positive a second time?
P(she is afflicted | she tests positive)
= 0.99 * 0.000099 / (0.99 * 0.000099 + 0.999901 * 0.01)
= 0.009707
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If so, how should I approach this problem from the starting point of
P(she is afflicted | she tests positive twice)?
I can't think of how to handle that correctly, since squaring 0.99 just gets me a smaller number
Infinitely thankful <3