Anyway, using Bayes' formula is not the same as philosophical Bayesianism. Bayes' formula is simply a theorem of mathematics that follows from the axioms of probability theory. It's not controversial.

The controversy comes from the applicability of Bayesian philosophy. In theory and ideally, it seems nice, but there are some problems with it.

One well known problem is how to specifying your priors. A bigger issue that is often forgotten is that you can't calculate the conditional probability P(explanation | observation), without computing

P(observation) = SUM_i [ P(observation | explanation_i) P(explanation_i) ]

for the denominator of the Bayes formula.

Now, this means that you have to consider all possible explanations. But systematically scanning through all possible hypotheses is really difficult. Usually it takes rare moments of random genious ideas to come up with a hypothesis that works well. You could say this is a "global optimization" problem in hypothesis space.

Now, if you restrict the search for a particular class of hypotheses (a limited-flexibility model), then you can use computational methods, like Monte Carlo methods, to approximate this sum (integral) over a set of hypotheses. But you can never be sure if your modeling restrictions were valid or maybe the best explanation is outside "the box".

When I hear Internet people (bloggers, commenters) saying they use Bayesianism in their everyday life it's almost always about pulling numbers out of their asses and then feeling great about how smart and rational they are. Everyday life is just too messy, there are many possibilities that are hard to capture in a model.

Addendum: I also really hate how Lesswrongians hijacked and overmystify the term "Bayesian" (and even "rationalist") as if it were some panacea/silver bullet that magically turns you into some superhero. It's like a "one weird trick" BS just for smarter people.