Physicist and science writer Jon Butterworth has written a layman's introduction to Bayes' theorem, touching at one point on its use in climate change:
If you have a prior assumption that modern life is rubbish and technology is intrinsically evil, then you will place a high prior probability on Carbon Dioxide emissions dooming us all. On the other hand, if your prior bias is toward the idea that there is massive plot by huge multinational environmental corporations, academics and hippies to deprive you of the right to drive the kids to school in a humvee, you will place a much lower weight on mounting evidence of anthropogenic climate change. If your prior was roughly neutral, you will by now be pretty convinced that we have a problem with global warming. In any case, anyone paying attention as evidence mounts would eventually converge on the right answer, whatever their prior - though it may come too late to affect the outcome, of course.
The use of Bayes' theorem in climate science is so much more interesting than this: as readers at BH no doubt know, the IPCC's use of a uniform prior in ECS for its "neutral" starting point biased the posterior towards higher estimates of future warming. Secondly, do we really have "mounting evidence"? Surely what we have is comparisons of unvalidated physical models to observations.