Sometimes Bayesianism is proved, other times it is acknowledged to have known flaws. Sometimes it may be completely compatible with Popper, other times it is dethroning Popper. It seems to me that perhaps Bayesianism is a bit underspecified. I wonder why they haven't sorted out these internal disputes.
There are disputes among the Bayesians. But you are confusing different issues. First, the presence of internal disputes about the borders of an idea is not a priori a problem with an idea that is in progress. The fact that evolutionary biologists disagree about how much neutral drift matters isn't a reason to reject evolution. (It is possible that I'm reading an unintended implication here.)
Moreover, most of what you are talking about here are not contradictions but failure to understand. That Bayesianism has flaws is a distinct claim from when someone talks about something like Cox's theorem which is the sort of result that Bayesians are talking about that you refer to as "Sometimes Bayesianism is proved"(which incidentally is a terribly unhelpful and vague way of discussing the point). The point of results like Cox's theorem is that if one very broad attempts under certain very weak assumptions to formalize epistemology you must end up with some form of Bayesianism. At the same time it is important to keep in mind that this isn't saying all that much. It doesn't for example say anything about what one's priors should be. Thus one has the classical disagreement between objective and subjective Bayesians based on what sort of priors to use (and within each of those there is further breakdown. LessWrong seems to mainly have objective Bayesians favoring some form Occam prior, although just what is not clear.) Similarly, when discussing whether or not Bayesianism is compatible with Popper depends a lot on what one means by "Bayesianism", "compatible" and "Popper". Bayesianism is certainly not compatible with a naive-Popperian approach, which is what many are talking about when they say that it is not compatible (and as you've already noted Popper himself wasn't a naive Popperian). But some people use Popper to mean the idea that given an interesting hypothesis one should search out for experiments which would be likely to falsify the hypothesis if it is false (an idea that actually predates Popper) but what one means by falsify can be a problem.
I was directed to this book (http://www-biba.inrialpes.fr/Jaynes/prob.html) in conversation here:
http://lesswrong.com/lw/3ox/bayesianism_versus_critical_rationalism/3ug7?context=1#3ug7
I was told it had a proof of Bayesian epistemology in the first two chapters. One of the things we were discussing is Popper's epistemology.
Here are those chapters:
http://www-biba.inrialpes.fr/Jaynes/cc01p.pdf
http://www-biba.inrialpes.fr/Jaynes/cc02m.pdf
I have not found any proof here that Bayesian epistemology is correct. There is not even an attempt to prove it. Various things are assumed in the first chapter. In the second chapter, some things are proven given those assumptions.
Some first chapter assumptions are incorrect or unargued. It begins with an example with a policeman, and says his conclusion is not a logical deduction because the evidence is logically consistent with his conclusion being false. I agree so far. Next it says "we will grant that it had a certain degree of validity". But I will not grant that. Popper's epistemology explains that *this is a mistake* (and Jaynes makes no attempt at all to address Popper's arguments). In any case, simply assuming his readers will grant his substantive claims is no way to argue.
The next sentences blithely assert that we all reason in this way. Jaynes' is basically presenting the issues of this kind of reasoning as his topic. This simply ignores Popper and makes no attempt to prove Jaynes' approach is correct.
Jaynes goes on to give syllogisms, which he calls "weaker" than deduction, which he acknowledges are not deductively correct. And then he just says we use that kind of reasoning all the time. That sort of assertion only appeals to the already converted. Jaynes starts with arguments which appeal to the *intuition* of his readers, not on arguments which could persuade someone who disagreed with him (that is, good rational arguments). Later when he gets into more mathematical stuff which doesn't (directly) rest on appeals to intution, it does rest on the ideas he (supposedly) established early on with his appeals to intuition.
The outline of the approach here is to quickly gloss over substantive philosophical assumptions, never provide serious arguments for them, take them as common sense, do not detail them, and then later provide arguments which are rigorous *given the assumptions glossed over earlier*. This is a mistake.
So we get, e.g., a section on Boolean Algebra which says it will state previous ideas more formally. This briefly acknowledges that the rigorous parts depend on the non-rigorous parts. Also the very important problem of carefully detailing how the mathematical objects discussed correspond to the real world things they are supposed to help us understand does not receive adequate attention.
Chapter 2 begins by saying we've now formulated our problem and the rest is just math. What I take from that is that the early assumptions won't be revisted but simply used as premises. So the rest is pointless if those early assumptions are mistaken, and Bayesian Epistemology cannot be proven in this way to anyone who doesn't grant the assumptions (such as a Popperian).
Moving on to Popper, Jaynes is ignorant of the topic and unscholarly. He writes:
http://www-biba.inrialpes.fr/Jaynes/crefsv.pdf
> Karl Popper is famous mostly through making a career out of the doctrine that theories may not be proved true, only false
This is pure fiction. Popper is a fallibilist and said (repeatedly) that theories cannot be proved false (or anything else).
It's important to criticize unscholarly books promoting myths about rival philosophers rather than addressing their actual arguments. That's a major flaw not just in a particular paragraph but in the author's way of thinking. It's especially relevant in this case since the author of the books tries to tell us about how to think.
Note that Yudkowsky made a similar unscholarly mistake, about the same rival philosopher, here:
http://yudkowsky.net/rational/bayes
> Previously, the most popular philosophy of science was probably Karl Popper's falsificationism - this is the old philosophy that the Bayesian revolution is currently dethroning. Karl Popper's idea that theories can be definitely falsified, but never definitely confirmed
Popper's philosophy is not falsificationism, it was never the most popular, and it is fallibilist: it says ideas cannot be definitely falsified. It's bad to make this kind of mistake about what a rival's basic claims are when claiming to be dethroning him. The correct method of dethroning a rival philosophy involves understanding what it does say and criticizing that.
If Bayesians wish to challenge Popper they should learn his ideas and address his arguments. For example he questioned the concept of positive support for ideas. Part of this argument involves asking the questions: 'What is support?' (This is not asking for its essential nature or a perfect definition, just to explain clearly and precisely what the support idea actually says) and 'What is the difference between "X supports Y" and "X is consistent with Y"?' If anyone has the answer, please tell me.