David Evans , aka ‘rocket scientist‘, shared his progression from a believer in anthropogenic global warming (AGW) to skeptic in response to new evidence (longer article here). Bayesian updating is a way of modeling rational changes of mind. I want see if DE is a rational, thinking person. Dr. Jim Peacock, Chief Scientist of Australia, immediate past President of the Australian Academy of Science and a Fellow of the Royal Society of London thinks not. Using the evidence presented in his article, and a very simple Bayesian updating system, we can model the changing probabilities of AGW in DE’s mind. Bayesian updating can get very complicated, but I am going to show a very simple way of simulating it.

The only rigorous relationship I am going to show is Bayes’ Theorem named after the British cleric Thomas Bayes in “An Essay Toward Solving a Problem in the Doctrine of Chances” (Bayes 1764). Bayes’ Theorem relates the “direct” probability of a hypothesis given evidence P(H|E), with the “inverse” probability of the evidence given the hypothesis, P(E|H), the evidence P(E) and the prior P(H).

P(H|E) = P(E|H)P(H)/P(E)

As DE tells it, prior to his conversion:

*The evidence was not conclusive, but why wait until we were certain when it appeared we needed to act quickly? Soon government and the scientific community were working together and lots of science research jobs were created. We scientists had political support, the ear of government, big budgets, and we felt fairly important and useful (well, I did anyway). It was great. We were working to save the planet.*

P(H) is called the prior probability of the hypothesis. DE’s motivation for working on global warming, as stated in the first paragraph above, was basically self interest. In other words he was in some sense economically rational, and he was uncommitted on AGW. The Bayesan term for this is an uninformed prior, expressed as a 50/50 chance of AGW vs. not AGW. We write this as a matrix (0.5,0.5) where the first place is the probability of AGW and the second is the probability of not AGW. DE goes on:

*But since 1999 new evidence has seriously weakened the case that carbon emissions are the main cause of global warming, and by 2007 the evidence was pretty conclusive that carbon played only a minor role and was not the main cause of the recent global warming. As Lord Keynes famously said, “When the facts change, I change my mind. What do you do, sir?”*

P(E|H) is the probability of evidence given the hypothesis. His first piece of evidence is:

1. The greenhouse signature is missing. We have been looking and measuring for years, and cannot find it.

The probability of seeing a greenhouse signature given AGW must be greater 0.5. But given all the noise in the climate system the probability of seeing a signature is not 1 either. Lets say that P(E1|H), the probability of seeing a greenhouse signature given AGW is 0.75, and write the evidence that we have not seen AGW as the matrix (0.25,0.75).

Now we could try to estimate the P(E1) the probability of a greenhouse signature and use Bayes Theorem to calculate P(H|E1) the probability of AGW given the evidence that no signature has been found, but we would have to estimate P(E1), and I am not sure how to do that. Besides there is a simpler way. Because probabilities sum to one, we can simulate the updating of probability of AGW due to the evidence of lack of greenhouse signature by multiplying the matrices and then normalizing, i.e.

(0.5,0.5).(0.25,0.75) = (0.25,0.75)

Thus the evidence reduces the original probability of AGW of 0.5 to 0.25. The same can be done for the other pieces of evidence.

*2. [T]heory suggests that carbon emissions should raise temperatures (though by how much is hotly disputed) but there are no observations by anyone that implicate carbon emissions as a significant cause of the recent global warming.*

P(E2|H) The probability that carbon dioxide could be the cause of any given warming depends on how many other causes of warming there are. Lets say that historically, carbon dioxide has caused 50% of warmings, or causes 50% of the warmth of warming. Then P(E2|H) is (0.5,0.5). This says that the theory of global warming due to greenhouse gases (AGW) provides no effective evidence to support AGW as a cause. If CO2 was the cause of warmings 90% of the time say, then (0.9,0.1) would produce considerable support from this evidence when multiplied with the prior.

*3. The satellites that measure the world’s temperature all say that the warming trend ended in 2001, and that the temperature has dropped about 0.6C in the past year (to the temperature of 1980).*

Here we can estimate P(E3|H) with some Monte Carlo simulations to find the frequency of decadal temperature increases and decreases when the globe is warming. Say that temperatures over a ten year period are increasing 75% of the time when the earth is warming. Then P(E3|H) => (0.25,0.75).

*4. The new ice cores show that in the past six global warmings over the past half a million years, the temperature rises occurred on average 800 years before the accompanying rise in atmospheric carbon.*

While CO2 lagging temperature does not disprove AGW completely, it does tend to provide some support for the alternative, so my guess is that the probability of lagging CO2 when AGW is true is less than 0.5 and so (0.25,0.75).

Now we use a variation of a Bayesian chain rule to calculate the final probability of the hypothesis that global warming is caused by greenhouse gases given the four pieces of evidence. Below we multiply the prior matrix by the four evidence matrices and normalize the result.

P(H).P(E1|H).P(E1|H).P(E1|H).P(E1|H)

=> (0.5,0.5).(0.25,0.75).(0.5,0.5).(0.25,0.75).(0.25,0.75)

=> (0.0039, 0.1055)/0.1094

=> (0.036,0.964)

That is, the probability of AGW given the evidences E1-4 is 0.036 or 3.6%. This is low enough to lead to rejection of a hypothesis at the standard significance level of 95%. In this very rough model, the changing of David Evan’s mind is a rational outcome of updated probabilities in response to new evidence, leading to a formalized decision to dismiss AGW when it reached a high enough threshold of improbability. I think Dr. Jim Peacock, Chief Scientist of Australia, owes David Evans — ‘rocket scientist’ — an apology.