Bayes' formula can yield P>1

[Coffee Jesus]

I'm trying to apply Bayes' formula to hypothesis testing. O is the observation. T is the hypothesis being true.
P(T|O) = P(O|T) ⋅ P(T) / P(O)
The equation works sometimes, and sometimes it yields a result greater than 1 when the result should be less than 1.

Problem 1. The equation works.
Four six-sided dice are picked randomly. One is a special die that has two 1 sides.
What is the probability that you rolled the special die (T) given that you rolled a 1 (O)?

P(T|O) = P(O|T) ⋅ P(T) / P(O)
(1/3) ⋅ (1/4) / (5/24) = 2/5

Correct. There are five equally likely outcomes that involve rolling a 1. Two of those outcomes involve rolling the special die.


Problem 2. The equation fails.
The hypothesis has a 4/10 probability of being true. P(T) = 4/10
The probability of the observation is 1/100. P(O) = 1/100
The probability of the observation is 3/100 if the hypothesis is true. P(O|T)
What's the probability of the hypothesis being true if we make the observation?

(3/100) ⋅ (4/10) / (1/100) = 6/5 = 1.2

Wrong. P(T|O) < 1 because there are still some possibilities in which the observation is made despite the claim being false.

[Categories+Sheaves]

Oh, cool. That's actually a pretty cute mistake--or feat of misdirection, not sure which. But still cool!

The problem comes up because the spread of values you assigned is inconsistent. IT should be straightforward to see that P(O and T) = P(O|T)*P(T) = (3/100)*(4/10) = 12/1000
But P(O) = 1/100 = 10/1000 < 12/1000, which means we've already made a contradiction. Having O happen is a strictly weaker condition than having O happen and having T be true, so P(O) should never be less than P(O and T). But that's a consequence of the values you assigned, so there can't exist a probability space with those properties in the first place. Since we're only supposed to use Bayes' theorem on well-defined probability measures, your example 2 isn't a problem for the theorem

You understand? Ja? Did I do good at making sense?

[Coffee Jesus]

Ja?

Almighty Ja!

http://www.youtube.com/watch?v=QYQ4uV8NDJo