Application of Bayes Theorem

Question

I am reading Nate Silver's book "The Signal and the Noise" and have a question about Bayes Theorem. I've created my own example and am trying to wrap my mind around the conclusion.

Let's say, before any information, I think there is a 5% chance humans have caused global warming.

Then, I hear information that scientists think there is a 99% chance that humans have caused global warming.

I also know that the probability that the 99% claim is wrong is 10%.

Using the Bayes Theorem calculation, the result is a 34% chance that humans cause global warming.

Here is the calculation:

X = initial probability of humans causing global warming = 5%

Y = probability of humans causing global warming, given scientist evidence = 99%

Z = probability of humans not causing global warming, given scientist evidence = 10%

The formula presented in the book (page 247) is:

Revised probability (given the new information) = XY / (XY + Z(1-X))

Revised probability (given the new information) = 34%

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My intuition says that, after this new knowledge, the chances that humans have caused global warming is instead (10% * 1%) + (90% * 99%) or 90%.

This would be based on the fact that theres a 10% they're wrong and 90% chance they're right.

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What is wrong about my application of the theorem or understanding of the theorem that causes this mental roadblock?

Thanks.

Answer 1

I think the problem is that $Z$ and $Y$ are likely not correct. I may be getting parts of your example wrong, so let me explicate all my steps. There are two possible interpretations I came up with:

Define the following Bernoulli RVs:

• $A$: Humans cause global warming
• $B$: Scientists are correct

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First interpretation: We have the following data:

• $P(A) = 5\%$, the initial probability that humans cause GW
• $P(B) = 90\%$, the probability that scientists are correct
• $P(A\mid B) = 99\%$, the probability that humans cause GW, given that scientists are correct

We can now calculate $P(B \mid A)$ with Bayes' Theorem; the probability that scientists are correct given that humans cause GW. This isn't what we wanted.

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Alternatively: We have the following data:

• $P(A) = 5\%$, the initial probability that humans cause GW
• $P(B\mid A) = 90 \%$, the probability that scientists are correct given that humans cause GW
• $P(A \mid B) = 99 \%$, the probability that humans cause GW given that scientists are correct.

Bayes' Theorem gives us a method to calculate $P(B)$, the probability that scientists are correct.

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In neither of these interpretations, we have obtained the desired $P(A \mid B)$. This is due to the fact that what we are given (some scientists playing oracle) does not relate in any way to humans causing GW or not. This is because they only say "There is a chance of $99\%$ that humans cause GW" which doesn't relate to the RVs $A$ and $B$, but rather to:

• $C$: The RV $A$ is Bernoulli distributed with $p = 0.99$

As an example of information that would relate $A$ to $B$ (yielding $P(B \mid A)$ and making Bayes' applicable), consider:

• "The scientists' measurements occur with a probability of $99\%$ if humans cause GW."

However, this is rather strange (since it would mean that the measurements almost cannot vary if humans cause GW - this would likely mean that the probability of obtaining those measurements would be small, which is contradicted by the probability of $90\%$ that the scientists are correct).

In conclusion, your example doesn't lend itself for Bayes' Theorem unless more information is given or some information is altered.

Answer 2

I think that the problem is that you are adding all the information about GW as a single factor, while each individual journal entry added to the scientific literature advanced the case a bit further. Let's imagine that there have been 3000 published journal entries advancing the idea of anthropogenic global warming. We know from the Ioannidis article cited in the same that 2/3rds of them will be wrong, and lets imagine that each of them has at least a 75% probability of humans causing global warming given the evidence presented in the article. I am choosing 75% because by definition an article advancing a theory must exhibit a probability greater than 50% lest it would not be counted among those advancing the theory. Since 51% is little better than flipping a coin, then I can assume that the journals would be unlikely to unmistakenly publish an article with such a low probability as less than 75%. Of course they could do so mistakenly, but those mistakes are among those accounted in Ioannidis's 67%. Now your values are as such:

X = initial probability of humans causing global warming = 5%

Y = probability of humans causing global warming, given scientist evidence = 75%

Z = probability of humans not causing global warming, given scientist evidence = 67%

The formula presented in the book (page 247) is:

Revised probability (given the new information) = XY / (XY + Z(1-X))

Revised probability (given the new information) = 5.5%

And that's after the first journal entry. After the second it is up to 6.1% by the 100th positive article you're up to 99.98% likelihood. By the 3000th it may as well be 100% because Excel does not even handle that many 9’s.

This model, however, does not account for the journal articles that deny the cause of AGW in those cases we would see different math: in this case I assigned the Y value to 25% for the reverse logic of articles that support AGW... by definition they have to be <50%, and in order to not be counted in Ioannidis's 67% it must be lower still. This raises a new problem: when do the AGW refuting articles get published. For simplicity sake I spaced the negative articles every 10 and re-ran the model. It takes longer to get to 99.999%, but we still get there. We reach five nines by the next iteration, the 1710th. It drops back below again several times, but the last time is on the 1820th.

Last issue is my 75% assumption… let’s say it is 70% instead of 75%? This changes the math significantly as Bayes now takes us down to near zero instead of up to near certainty. But then 75% was my floor, not the full range of how much each journal article may advance or refute AGW. I think we can safely assume a mean advance of >75%