Saturday, April 14, 2012

Another Reason Why The Base Rate Fallacy Is A Fallacy

So a couple of years ago I had someone use this line of reasoning on me:

"If I had some disease that had a 1 in a million chance of survival and I survived it, it's not because I was that one in a million, it's because God did it"

This is the Base Rate Fallacy because even if there is high conditional probability of surviving the disease given god did it, this is ignoring the prior probability of god's existence which is extremely low.

But ok. What if I didn't know that the prior probability of god's existence was low? What if I assumed it was high? If so, why is it still a fallacy? Because of the Total Probability Theorem.

Positing god in this case would skew the Total Probability to be something other than what it is by necessity of the above reasoning (i.e. it has to be 1 in a million for her "logic" to work). So the reasoning above could also be called a Total Probability Fallacy, if it wasn't already called the Base Rate Fallacy.

Let H be "god did it". E is the 1 in a million chance of surviving the disease, .0001%. P(E | H), or surviving the disease given god did it, is 100%. So then we have:

P(H | E) = P(E | H) * P(H) / P(E)
= 1.00 * "high" / .000001

Here's the thing. P(E) can also be represented by including the alternative hypothesis and its conditional probability, P(E | ~H) * P(~H). It then becomes:

P(E) = .000001 = P(E | H) * P(H) + P(E | ~H) * P(~H).
.000001 = 1.00 * "high" + P(E | ~H) * P(~H)

So what is the probability of surviving the disease given some other hypothesis? Like random chance? Isn't that the same as P(E)?

The only way to get this to work is if P(H), the probability that god was responsible, is so close to zero that it might as well be zero. That is the only way it can all work out so that .000001 = P(E | H) * P(H) + P(E | ~H) * P(H). If P(H) is "high", then the second term P(E | ~H) * P(~H), or surviving the disease given some other hypothesis, would have to be a literally impossibly low number to compensate, like a negative probability.

If P(H) was equal to P(E) -- "extraordinary claims require extraordinary evidence" -- then this would necessitate that the second term was equal to zero and not the first, meaning that it would be impossible to survive the disease without god's intervention... meaning that the actual probability is not 1 in a million like she needed it to be for her logic to work but a much, much, much lower number.

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