Global warming, Bayes' Law, and Russell's Teapot

Suppose I wanted to demonstrate that there is not a small teapot floating between Earth and Mars (Russell's Teapot). I might formulate a null hypothesis, there is a teapot, and my alternative hypothesis, there is no teapot, and then get to work gathering data, perhaps by building a really big telescope to look for it. Russell's point, however, is that finding a teapot is so difficult that it would take a whole lot of data to refute the null hypothesis. Russell argued that this implied that we should reverse the null hypothesis to a lack of a teapot, putting the burden on teapot-believers to generate evidence in support of the teapot.

Shifting to a Bayesian approach, I would instead start by examining my prior beliefs about the likelihood that there is a small teapot floating through space. While I wouldn't assign a prior of 0, because I don't believe in ever assigning any event a prior probability of 0 (or 1), my understanding of the solar system and of teapots would lead me to assign a very low prior probability. Given the structure of the problem, observational data that did not show a teapot wouldn't change my probability assessment much, because of the low likelihood of finding a teapot, should one exist, while a picture of a teapot would increase my probability assessment quite a lot, although not to 1, because of the possibility of a false negative (like, for example, a faked photo).

So, how does this relate to global warming? First, a few claims for which there is approximately no reasoned disagreement:

• Water vapor, carbon dioxide, and other gases in the atmosphere increase the temperature of Earth by absorbing more outgoing radiation (generated by Earth) than incoming radiation (generated by the Sun). In the absence of an atmosphere, Earth would have a temperature about 30 degrees C below its current temperature. This is basic thermodynamics, and while most people (including myself) don't understand it well, as far as I know, nobody who does understand it disputes these basic claims.
• All else equal, higher carbon dioxide levels produce more greenhouse warming. Again, pretty simple thermodynamics.
• All else equal, warmer air contains more water vapor that colder air. Basic atmospheric chemistry.
• Human activity has increased the concentration of carbon dioxide in the atmosphere from less than 300 ppm in the (geologically recent) pre-industrial past to more than 390 ppm today.

The debate over the human contribution to global warming is about whether the observational evidence supports or refutes the projection that human activity has and will continue to increase global temperatures.

Viewed from the standard frequentist framework, the null hypothesis is that human activity has not contributed to global warming, and the alternative is that human activity has contributed to global warming (a one-sided test). Given the noisiness of the climate, and of our observations of it, refuting the null hypothesis is difficult, and we could easily find ourselves in a situation in which Florida is under water, Kansas is a desert, and the Arctic Ocean is navigable without ever having a enough evidence to refute the null hypothesis.

On the other hand, if we view this from a Bayesian perspective, we get a very different answer. We have a theory about the relationship between human activity, atmospheric carbon dioxide, and global temperature. We should start with a pretty strong prior belief that human activity will contribute to global warming. The observational evidence will then allow us to refine our predictions about this relationship, but the noise in the evidence will not lead us to conclude that nothing is happening. What's more, in a Bayesian framework, our understanding of the basic mechanisms shifts the burden of proof: given strong prior beliefs, it would take strong evidence of a lack of a relationship to change our beliefs, while noisy evidence will not change our beliefs much.