I loved reading Nate Silver (he of the accurate 2012 election forecasts) new book ‘The Signal and the Noise: Why So Many Predictions Fail but Some Don’t.’ I especially liked his chapter on climate change: A Climate of Healthy Skepticism. In it he uses Bayes’ Theorem to address the probability of climate change being accurately predicted. His previous explanation, earlier in the book, of Bayes’ Theorem, which helps you adjust your belief rationally in the face of new evidence, is the first explanation of the theory I have fully grasped. Here is Nate applying it to the new evidence on climate change. The supporting detail is in the chapter of his book and the rest of his book too.
Uncertainty is an essential and nonnegotiable part of a forecast. As we have found, sometimes an honest and accurate expression of the uncertainty is what has the potential to save property and lives. In other cases, as when trading stock options or wagering on an NBA team, you may be able to place bets on your ability to forecast accurately.
However, there is another reason to quantify uncertainty carefully and explicitly. It is essential to scientific progress, especially under Bayes’Theorem.
Suppose that in 2001, you had started out with a strong prior belief in the hypothesis that industrial carbon emissions would continue to cause a temperature rise. (In my view, such a belief would have been appropriate because of our strong causal understanding of the greenhouse effect and the empirical evidence for it up to that point.) Say you had attributed the chance of the global warming hypothesis being true at 95%.
But then you observe some new evidence: over the next decade, from 2001 through 2011, global temperatures do not rise. In fact, they fall, although very slightly. Under Bayes’ Theorem, you should revise your estimate of the probability of global warming hypothesis downward: the question is by how much.
If you had come to a proper estimate of the uncertainty in the near-term temperature patterns, there is about a 15% chance that there will be no net warming over a decade even if the global warming hypothesis is true because of the variability of climate. Conversely, if temperature changes are purely random and unpredictable, the chance of a cooling decade would be 50% since and increase and a decrease in temperature are equally likely. Under Bayes’ Theorem, a no net warming decade would cause you to revise your estimate of the global warming hypothesis’s likelihood to 85% from 95%.
Here’s his math:
Initial estimate of how likely it is that global temperatures are increasing 95%: x
Probability of no net warming over 10 years if global warming hypothesis is correct: 15%: y
Probability of no net warming over 10 years if global warming hypothesis is false: 50%: z
Revised estimate of how likely it is that global warming is occurring, given no net temperature increase over 10 years:
Bayes’ Theorem gives us:
xy/xy +z(1-x) = 0.95.0.15/0.95.0.15 +.50(1-.95)= 85%
On the other hand, if you had asserted that there was just a 1% chance that temperatures would fail to increase over the decade, your theory is now in much worse shape because you are claiming that this was a more definitive test. Under Bayes’ Theorem, the probability you would attach to the global warming hypothesis has now dropped to just 28%.
When we advance more confident claims and they fail to come to fruition, this constitutes much more powerful evidence against our hypothesis.
He goes on to say some other interesting things, but my point here is not to contribute to the climate change debate, but simply to note that under Bayes’ Theorem, if your prior belief that climate change is not happening, that it is 0% probable, then no data will change your mind, as it is still 0% likely with the new data, and I guess if you are 100% certain it is happening then ditto as your standard for disproof will be higher and Bayes will predict it is still 100% whatever the new data. Personally, my own prior belief in 2 degree C warming per century was around 80% to begin with, so with the ten year hiatus, it drops to 54% in the light of the new data given Nate’s other assumptions.
So I guess my suggestion is that anyone who thinks global warming is either 0% likely or 100% likely is under Bayes’ Theorem not open to changing their prior conviction, or what I might call them, they are belief fundamentalists on this issue.
More on the Reverend Bayes (1701-1761) at: http://en.wikipedia.org/wiki/Thomas_Bayes
Creativeconflictwisdom's Blog 
A mathematician like tamino (https://tamino.wordpress.com/) would be far more able to respond. However, a couple of points as a policy analysis dealing with climate change:
1. “anyone who thinks global warming is either 0% likely or 100% likely is under Bayes’ Theorem not open to changing their prior conviction” might be true, but the calculation of the 0% and 100% depends on your assumptions. I understand that climate scientists do not consider a change in global average temperature in periods of up to fifteen years to be a statistically significant indication of the lack of global warming. We’ll see what happens next.
2. None of the skeptics have come up with any explanation for what happens to increasing heat capture with higher levels of GHGs. Some simply deny that GH effect from CO2 exists. The apparent “pause” in atmospheric warming could be seen as confounding in light of the certainty with which most scientists accept the GH affect. It now appears that much of the increased heat is going into the oceans. The oceans are a far larger heat sink than the atmosphere– http://www.skepticalscience.com/Ocean-Heat-Content-And-The-Importance-Of-The-Deep-Ocean.html and http://www.skepticalscience.com/argument.php?a=11&p=5
In addition, there are recent studies on this issue, and the climate blogs are burning with furious debate between the so-called skeptics and those in the science community who choose to engage with them. See, e.g., http://arstechnica.com/science/2013/04/oceans-continue-to-warm-especially-the-deeps/ and http://news.yahoo.com/oceans-may-explain-slowdown-climate-change-study-170410831.html I’d like to see this latest data & analysis worked into Baye’ Theorem, assuming you could get some scientists to project what is going to happen next.
@louploup2. I was using climate change to illustrate Bayes’ Theorem as I found Nate Silver’s use of it compelling. The 0% and 100% were ways of expressing evidence proof fundamentalism within the framework of Bayes’ maths and doesn’t prove anything but it aligned with my own experience of the 0% and 100% folk as being not open to new data. I fully concur with the other data you highlight would probably shift my posterior estimate back to perhaps 80%. There is also the issue of latent heat. The phase transition from ice to water, as I recall from High School physics experiments I did myself, results in a flat line in the temperature rise graph until the ice is melted if you apply uniform heat to ice. The heating is particularly concentrated at the poles so the latent heat effect is likely to be strong but I don’t have the numbers to hand. Also the heating at the North Polar regions is slowing the jet stream and meaning that weather stays put, hence the US mild winter 2011/12 and the hard winter in Europe this year.
My problem remains as ever with the skeptics, the meta issue that there is no data that would make them change their mind. They have evidence proof often ad hominem (‘liberal conspiracy’) beliefs not data driven conclusions.
I have a reverse Bayes’ Theorem that says not how much should you change your mind given X new data but what new data Y would be needed to change your Bayesian Prior from Z to W %. At least I can now express it Bayes’ terms thanks to Nate.
Thanks as ever for your comments
Yes, all your points are accurate; I’ve been following Arctic situation and jet stream work by Jennifer Francis and others. It will be very interesting to see what happens in Northern Hemisphere in next El Nino (if I’m remembering right).
And especially “the meta issue that there is no data that would make them change their mind.” When I dig into the identity of the worst offenders, I find much overlap with anti-government, conspiracy oriented reactionaries. Others are what I call infantile libertarians. Most of them conflate the science with the political solutions they hate. Some of them are quite scary in the depth of their anger and hostility on top of irrational certainty and lack of ability to think critically.
@louploup2. Looking at conservative reaction to climate change analytically. I am reading Rebecca Costa’s interesting book ‘The Watchman’s Rattle’. She suggests that societies collapse when their complexity exceeds the current state of the evolution of the human brain to handle complexity, though a smart culture can defer this or even avoid it. But in case of various historical collapses like the Mayans, the Romans, the Khmers she sees complexity of sustaining the civilization getting to a critical point at which knowledge of what is happening overwhelms the civilization’s abilities and knowledge. At that point belief systems arise often with human sacrifice or other magical thinking memes and the search for knowledge disappears in the superstitious mists. Belief trumps knowledge.
My thinking now, not hers but applying her approach. I think conservatives are distinguished in psychological tests be being far more fearful than liberals. Not lacking courage or bad people, but fearful about life, about change (that’s why they are called conservative) and have much lower tolerance of ambiguity. They, faced with the threatening and complex nature of climate science, may therefore be pre-disposed to retreat into mental citadels (Wilfred Owen’s ‘To miss the march of this retreating world
Into vain citadels that are not walled’.) and beliefs rather than knowledge. Their search for knowledge when it does happen is skewed by confirmation bias but generally they just adopt data proof beliefs about liberal climate conspiracies as this is mentally safer than reality..
In addition, on the issue of AGW and climate change impacts, even if we did not emit another pound of GHG, it appears clear that we have already committed ourselves to a different climate. Regardless of how sensitive the climate is the GHGs we have already emitted, the global climate is not likely to be Holocene-like again for a long time. I.e., the debate between “0% v 100%” is irrelevant.
This new reality is not just anecdotal; see Hansen’s compelling statistical analysis of the change in extreme weather, discussed very clearly at http://www.skepticalscience.com/hansens-new-climate-dice-loaded-misunderstood.html
I work on climate change at the local level in communities all over North America. I think it is very difficult to deny that the climate is already significantly changing, regardless of the flatness of the average temperature graph in the recent past. There are simply too many clear impacts in numerous areas: botany, entymology, hydrology, forest ecology, weather, marine ecology, etc.
@louploup2. Absolutely. My actual example for my personal take was over the 2 degrees C per century rate and we are not yet certain on the rate given the complexity; but directionally you are right on.And Nate Silver was also talking more about the general direction though he does think at least 1.5 degrees C is the likely rate.