I applaud the authors for putting their argument in a bayesian form. At the very least, it forces them to make their argument explicit. Also, in theory it lets one describe disagreement in terms of values of numbers (priors and conditional probabilities). However, I suspect that this is rarely going to happen in historical disagreements and the qualitative structure will capture most of the disagreement. (In principal, anyone should be able to supply numbers someone else's structure, but I'm skeptical that it is a useful comparison.)
Disclaimer: I have not read this book. I'm posting it in the expectation that others may enjoy it as much as I'm sure I would if I had time to read it myself.
This looks interesting as an extended worked example of Bayesian reasoning (the "scientific approach" of the title).
Edited to add:
There are many signs in the above block of text that this book is not up to Lesswrong standards. As gwern suggests, reading it should be done with an adversarial attitude.
I propose some more useful goals than finding someone for whom we can cheer loudly as a properly qualified member of our tribe: find worked examples that let you practice your art; find structured activities that will actually lead you to practice your art; try to critically assess arguments that use the tools we think powerful, then discuss your criticism on a forum like Lesswrong where your errors are likely to be discovered and your insights are likely to be rewarded (with tasty karma).