Open thread, Nov. 10 - Nov. 16, 2014

MrMind

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If it's worth saying, but not worth its own post (even in Discussion), then it goes here.

Notes for future OT posters:

1. Please add the 'open_thread' tag.

2. Check if there is an active Open Thread before posting a new one. (Immediately before; refresh the list-of-threads page before posting.)

3. Open Threads should be posted in Discussion, and not Main.

4. Open Threads should start on Monday, and end on Sunday.

A friend of mine recently succumbed to using the base rate fallacy in a line of argumentation. I tried to explain that it was a base rate fallacy, but he just replied that the base rate is actually pretty high. The argument was him basically saying something equivalent to "If I had a disease that had a 1 in a million chance of survival and I survived it, it's not because I was the 1 in a million, it's because it was due to god's intervention". So I tried to point out that either his (subjective) base rate is wrong or his (subjective) conditional probability is wrong. Here's the math that I used, let me know if I did anything wrong:

Let's assume that the prior probability for aliens is 99%. The probability of surviving the disease given that aliens cured it is 100%. And of course, the probability of surviving the disease at all is 1 out of a million, or 0.0001%.

Pr(Aliens | Survived) = Pr(Survived | Aliens) x Pr(Aliens) / Pr(Survived)
Pr(Aliens | Survived) = 100% x 99% / 0.0001%
Pr(Aliens | Survived) = 1.00 * .99 / .000001
Pr(Aliens | Survived) = .99 / .000001
Pr(Aliens | Survived) = 990,000 or 99,000,000%

There's a 99,000,000% chance that aliens exist!! But... this is probability theory, and here probabilities can only add up to 100%. Meaning that if we end up with some result that is over 100% or under 0% something in our numbers is wrong.

The Total Probability Theorem is the denominator for Bayes Theorem. In this aliens instance, that is the probability of surviving the disease without alien intervention, which is 1 out of a million. The Total Probability Theorem, meaning 1 out of a million in this case, is also equal to Pr(Survived | Aliens) x Pr(Aliens) + Pr(Survived | Some Other Cause) x Pr(Some Other Cause):

1 in a million = Pr(Survived | Aliens) x Pr(Aliens) + Pr(Survived | Some Other Cause) x Pr(Some Other Cause)
0.0001% = 100% x 99% + ??? x 1%
0.0001% = 99% + 1%*???

If we want to find ???, in this case it would be Pr(Survived | Some Other Cause), we need to solve for ??? just like we would in any basic algebra course to find x. In this case, our formula is 0.000001 = 0.99 + 0.01x.

If we solve for x, it is -98.999, or -9899%. Meaning that Pr(Survived | Some Other Cause) is -9899%. Again, a number that is outside the range of allowable probabilistic values. This means that there is something wrong with our input. Either the 1 in a million is wrong, the base rate of alien existence being 99% is wrong, or the 100% conditional probability that you would survive your 1 in a million disease due to alien intervention is wrong. The 1 in a million is already set, so either the base rate or conditional probabilities are wrong. And this is why that sort of "I could only have beaten the odds on this disease due to aliens" (or magic, or alternative medicine, or homeopathy, or Chthulu, or...) reasoning is wrong.

Again, remember the base rate. And you can't cheat by trying to jack up the base rate or you'll skew some other data unintentionally. Probability is like mass; it has to be conserved.

The argument was him basically saying something equivalent to "If I had a disease that had a 1 in a million chance of survival and I survived it, it's not because I was the 1 in a million, it's because it was due to god's intervention". So I tried to point out that either his (subjective) base rate is wrong or his (subjective) conditional probability is wrong.

There no reason why God in principle should be unable to choose which of the people of the mass of one million survives. If you don't have a model of how the one in a million gets cured ... (read more)

2MrMind12y

This. Since P(S) = P(S|A)P(A) + P(S|-A)P(-A), and P(S|A)P(A) is already .99, then P(S) cannot be .000001. Those two assertions are contradicting each other: you cannot coherently believe a composite event (suriving an illness) less than you believe each factor (surviving the illness with the aid of magic). If you believe that God will cure everyone who gets the disease (P(S|A) = 1) and God is already a certainty (P(A) = .99), then why so few people survive the illness? One possibility is that it's P(S|-A) that is one in a million (surviving without God is extremely rare). In this case: P(A|S) = P(S|A) P(A) / P(S) --> P(A|S) = P(S|A) P(A) / (P(S|A) P(A) + P(S|-A) P(-A)) --> P(A|S) =1 .99 / (1 .99 + .000001 * .01) --> P(A|S) = .99 / (.99 + .00000001) --> P(A|S) = .99 / .99000001 --> P(A|S) = .9999999... If you already believe that curing aliens are a certainty, then for sure surviving an illness that has only a millionth possibility otherwise, will bring up your belief up to almost a certainty. Another possible interpretation, that keeps P(S) = .000001, is that P(S|A) is not the certainty. Possibly God will not cure everyone who gets the disease, but only those who deserves it, and this explains why so few survive. In this case: P(S) = x .99 + y .01 = .000001 --> P(A|S) = x * .99 / .000001 a number that depends on how many people God considers worthy of surviving.

2Osho12y

I think your denominator in your original equation is missing a second term. That is why you get a non-probability for your answer. See here: http://foxholeatheism.com/wp-content/uploads/2011/12/Bayes.jpg