I really don't understand the debate. Bayesian reasoning IS the reasoning that scientists use. It is the method underlying the evolution of scientific theory. Popperian falsification is just some theory, more a prescriptive than descriptive rule. It's a pie in the sky which doesn't explain how the body of scientific knowledge evolves in time.
In practice, evidence is gathered to support or falsify a given scientific premise. Newtonian mechanics was TRUE until proven otherwise. And today's theories are more or less true based on their ability to explain reality (i.e., the same thing as positive evidence in a probabilistic sense) and not be disproved (i.e., have negative evidence against them). In reality, there are limits to our understanding and the scientist with any real sense of humility should agree with Box when he said that all models are false but some are useful.
Daniel, I think what you say about an implicit agreement that the current theory could be revised in light of new contradictory evidence, this is exactly Bayesian, a form of Bayesian model selection, where it may be that no theory or model is ever thrown out completely, just assigned a very low probability. Many evolutionary arguments are just a form of Bayesian update, conditioning on new evidence.
The idea that Bayesian decision theory being descriptive of the scientific process is very beautifully detailed in classics like Pearl's book, Causality, in a way that a blog or magazine article cannot so easily convey. In a different vein, for a very readable explanation of how "truth" changes, even in mathematics, the most pure of sciences, have a look at Imre Lakatos' book, Proofs and Refutations. In this book, Lakatos makes it clear that even mathematicians can use a Bayesian update of mathematical "evidence" for or against a given hypothesis, and that old "proofs" even by the greatest of mathematicians often have holes poked in them in time.
Now pure application of Bayes' rule may just merely give the probability that a theory/model is true. In reality, we probably do have some utility/loss function that gives us a decision rule as to whether we wish to use or discard a given theory. This loss function approach will actually allow us to use "false" theories such as Newtonian mechanics, when there is some utility to it, even though the evidence against them is immense.
What Eliezer is saying in the blog and what is said in the NS article is basically descriptive, imho, let's call a spade a spade...science is already Bayesian. Those of you who cannot really accept it and think this opens up science to the possibility of witchcraft are filled with a great idealism in how science is currently conducted behind closed doors. Either that, or like the Church fathers who silenced Galileo, you're awfully scared that the opposite of your dogma, witchcraft, might have an element of truth in it. Being honest about Bayesianism means we have to consider all the alternatives.
But, to be reassuring, I don't think we've seen a terrible amount of positive evidence for witchcraft lately....
New Scientist on changing the definition of science, ungated here:
I'm a good deal less of a lonely iconoclast than I seem. Maybe it's just the way I talk.
The points of departure between myself and mainstream let's-reformulate-Science-as-Bayesianism is that:
(1) I'm not in academia and can censor myself a lot less when it comes to saying "extreme" things that others might well already be thinking.
(2) I think that just teaching probability theory won't be nearly enough. We'll have to synthesize lessons from multiple sciences like cognitive biases and social psychology, forming a new coherent Art of Bayescraft, before we are actually going to do any better in the real world than modern science. Science tolerates errors, Bayescraft does not. Nobel laureate Robert Aumann, who first proved that Bayesians with the same priors cannot agree to disagree, is a believing Orthodox Jew. Probability theory alone won't do the trick, when it comes to really teaching scientists. This is my primary point of departure, and it is not something I've seen suggested elsewhere.
(3) I think it is possible to do better in the real world. In the extreme case, a Bayesian superintelligence could use enormously less sensory information than a human scientist to come to correct conclusions. First time you ever see an apple fall down, you observe the position goes as the square of time, invent calculus, generalize Newton's Laws... and see that Newton's Laws involve action at a distance, look for alternative explanations with increased locality, invent relativistic covariance around a hypothetical speed limit, and consider that General Relativity might be worth testing. Humans do not process evidence efficiently—our minds are so noisy that it requires orders of magnitude more extra evidence to set us back on track after we derail. Our collective, academia, is even slower.