For the record, the popular interpretation of "Popperian falsificationism" is not what Karl Popper actually believed. (According to Wikipedia, he did not even like the word "falsificationism" and preferred "critical rationalism" instead.) What most people know as "Popperian falsificationism" is a simplification optimized for memetic power, and it is quite simple to disprove. Then we can play motte and bailey with it: the motte being the set of books Karl Popper actually wrote, and the bailey being the argument of a clever internet wannabe meta-scientist about how this or that isn't scientific because it does not follow some narrow definition of falsifiability.

I have not read Popper's book, therefore I am only commenting here on the traditional internet usage of "Popperian falsificationism".

The good part is noticing that beliefs should pay rent in anticipated consequences. A theory that explains everything, predicts nothing. In the "Popperian" version, beliefs pay rent by saying which states of the world are impossible. As long as they are right, you keep them. When they get wrong once, you mercilessly kick them out.

An obvious problem: How does this work with probabilistic beliefs? Suppose we flip a fair coin, and one person believes there is a 50% chance of head/tails, and other person believes it is 99% head and 1% tails. How exactly is each of these hypotheses falsifiable? How many times exactly do I have to flip the coin and what results exactly do I need to get in order to declare each of the hypotheses as falsified? Or are they both unfalsifiable, and therefore both equally unscientific, neither of them better than the other?

That is, "Popperianism" feels a bit like Bayesianism for mathematically challenged people. Its probability theory only contains three values: yes, maybe, no. Assigning "yes" to any scientific hypothesis is a taboo (Bayesians agree), therefore we are left with "maybe" and "no", the latter for falsified hypotheses, the former for everything else. And we need to set the rules of the social game so that the "maybe" of science does not become completely worthless (i.e. equivalent to any other "maybe").

This is confusing again. Suppose you have two competing hypotheses, such as "there is a finite number of primes" and "there is an infinite number of primes". To be considered scientific, either of them must be falsifiable in principle, but of course neither can be proved. Wait, what?! How exactly would you falsify one of them without automatically proving the other?

I suppose the answer by Popper might be a combination of the following:

• mathematics is a special case, because it is not about the real world -- that is, whenever we apply math to the real world, we have two problems: whether the math itself is correct, and whether we chose the right model for the real world, and the concept of "falsifiability" only applies to the latter;
• there is always a chance that we left out something -- for example, it might turn out that the concept of "primes" or "infinity" is somehow ill-defined (self-contradictory or arbitrary or whatever), therefore one hypothesis being wrong does not necessarily imply the other being right.

Yet another problem is that scientific hypotheses actually get disproved all the time. Like, I am pretty sure there were at least dozen popular-science articles about experimental refutation of theory of relativity upvoted to the front page of Hacker News. The proper reaction is to ignore the news, and wait a few days until someone provides an explanation of why the experiment was set up wrong, or the numbers were calculated incorrectly. That is business as usual for a scientist, but would pose a philosophical problem for a "Popperian": how do you justify believing in the scientific result during the time interval between the experiment and its refutation were published? How long is the interval allowed to be: a day? a month? a century?

The underlying problem is that experimental outcomes are actually not clearly separated from hypotheses. Like, you get the raw data ("the machine X beeped today at 14:09"), but you need to combine them with some assuptions in order to get the conclusion ("therefore, the signal travelled faster than light, and the theory of relativity is wrong"). So the end result is that "data + some assumptions" disagree with "other assumptions". There as assumptions on both sides; either of them could be wrong; there is no such thing as pure falsification.

Sorry, I got carried away...

I'm not sure Bayes' Rule dictates anything beyond its plain mathematical content, which isn't terribly controversial:

$P\left(A|B\right)=\frac{P\left(B|A\right)\cdot P\left(A\right)}{P\left(B\right)}$
 

When people speak of Bayesian inference, they are talking about a mode of reasoning that uses Bayes' Rule a lot, but it's mainly motivated by a different "ontology" of probability. 

As to whether Bayesian inference and Popperian falsificationism are in conflict - I'd imagine that depends very much on the subject of investigation (does it involve a need to make immediate decisions based on limited information?) and the temperaments of the human beings trying to reach a consensus. 

Considered as an epistemology, I don't think you're missing anything.

To reconstruct Popperian falsification from Bayes, see that if you observe something that some hypothesis gave probability ~0 ("impossible"), that hypothesis is almost certainly false - it's been "falsified" by the evidence. With a large enough hypothesis space you can recover Bayes from Popper - that's Solomonoff Induction - but you'd never want to in practice.

For more about science - as institution, culture, discipline, human activity, etc. - and ideal Bayesian rationality, see the Science and Rationality sequence. I was going to single out particular essays, but honestly the whole sequence is probably relevant!

New poster. I love this topic. My own of view of the shortcoming of Bayesianism is as follows (speaking as a former die-hard Bayesian):

1. The world (multiverse) is deterministic.
2. Probability therefore does not describe an actual feature of the world. Probabilities only make sense as a statistical statements.
3. Making a statistical statement requires identifying a group of events or phenomena that are sufficiently similar that grouping makes sense. (Grouping disparate unique events makes the statistical statement meaningless, since we would have no reason to think subsequent events behave in the same way.)
4. Events and phenomena like balls in urns, medical tests for diseases with large sample sizes, even some human events like sports games, have sufficient regularity that grouping makes sense and statistical statements are meaningful.
5. Propositions about explanatory theories (are there infinite primes, is Newtonian physics “correct”) do not have sufficient regularity - a statistical statement based on any group of known propositions logically yields no predictive value about unknown propositions. (Other than where you have a good explanatory theory linking them.)
6. If probability statements about the correctness of an unknown explanatory proposition are therefore meaningless, priors and Bayesian updates are similarly meaningless. Example: Newton. Before Einstein, one’s prior for the correctness of Newton would have been high. Just as one’s prior on Einstein being correct right now is presumably high. But both are meaningless, since it is not the case that there is a proportion of universes in which they are true and a proportion in which they are false.
7. Counterpoint: why do prediction markets seem to work? No idea! Still wrestling with this. Would love to hear your thoughts.

This is a bit of an old post, but I felt I might be able to add to the discussion.  Keep in mind this is my own informal take on a rigorous philosophical topic, and I am by no means a professional.  My bias leans towards critical rationalism (Popperianism), but I'll try to be fair.  

I think you are correct in identifying induction as the fundamental tension between the two epistemologies.  Bayesian epistemology (as distinct from Bayes' Theorem) utilizes Solomonoff induction, whereas Popper is highly critical of inductive and probabilistic reasoning.  For Popper, induction isn't reasoning at all, if we view reasoning as a method by which we generate knowledge.  I'm sometimes tempted to call it "pseudo-inductive reasoning," which is just the fancy way of saying "guessing."  

To understand Popper's views, it's important to understand that knowledge for Popper has nothing to do with belief, credence, or confidence; those are all subjective states of one's mind.  Perhaps we can psychologically manipulate these mental states (perhaps upon realizing the Bayesian calculus), but those mental states don't constitute knowledge.  

If we view the scientific endeavor in three pieces, it might become more clear: 1) We propose ideas, hypotheses, and conjectures 2) We scrutinize these hypotheses and eliminate the falsified ones, and 3) We adopt those hypotheses which have been highly scrutinized as our new scientific theories, acting as if they are true until they are usurped.  For Popper, only (2) is the knowledge-generation process, while Bayesians greatly concern themselves with (1) and (3) (they don't neglect (2), but they don't see it as identically the essence of knowledge-generation).  

Popper gave a simple example of how knowledge can be contained inside books, and so is not intrinsically tied to our beliefs or actions.  Rather, (in my understanding) knowledge is the record of the progress of falsification, and the hypotheses at the top of the current leaderboard.  How we generated the hypotheses and what we do with our theories once we have them are problems for psychology and rationality, not epistemology.