You can conjecture Bayes' theorem. You can also conjecture all the rest, however some things (such as induction, justificationism, foundationalism) contradict Popper's epistemology. So at least one of them has a mistake to fix. Fixing that may or may not lead to drastic changes, abandonment of the main ideas, etc

Fully agreed. In principle, if Popper's epistemology is of the second, self-modifying type, there would be nothing wrong with drastic changes. One could argue that something like that is exactly how I arrived at my current beliefs, I wasn't born a Bayesian.

I can also see some ways to make induction and foundationalism easer to swallow.

I don't know the etiquette or format of this website well or how it works. When I have comments on the book, would it make sense to start a new thread or post somewhere/somehow?

A discussion post sounds about right for this, if enough people like it you might consider moving it to the main site.

I think you are claiming that seeing a white swan is positive support for the assertion that all swans are white. (If not, please clarify).

This is precisely what I am saying.

If so, this gets into important issues. Popper disputed the idea of positive support. The criticism of the concept begins by considering: what is support? And in particular, what is the difference between "X supports Y" and "X is consistent with Y"?

The beauty of Bayes is how it answers these questions. To distinguish between the two statements we express them each in terms of probabilities.

"X is consistent with Y" is not really a Bayesian way of putting things, I can see two ways of interpreting it. One is as P(X&Y) > 0, meaning it is at least theoretically possible that both X and Y are true. The other is that P(X|Y) is reasonably large, i.e. that X is plausible if we assume Y.

"X supports Y" means P(Y|X) > P(Y), X supports Y if and only if Y becomes more plausible when we learn of X. Bayes tells us that this is equivalent to P(X|Y) > P(X), i.e. if Y would suggest that X is more likely that we might think otherwise then X is support of Y.

Suppose we make X the statement "the first swan I see today is white" and Y the statement "all swans are white". P(X|Y) is very close to 1, P(X|~Y) is less than 1 so P(X|Y) > P(X), so seeing a white swan offers support for the view that all swans are white. Very, very weak support, but support nonetheless.

(The above is not meant to be condescending, I apologise if you know all of it already).

To show they are correct. Popper's epistemology is different: ideas never have any positive support, confirmation, verification, justification, high probability, etc...

This is a very tough bullet to bite.

How do we decide which idea is better than the others? We can differentiate ideas by criticism. When we see a mistake in an idea, we criticize it (criticism = explaining a mistake/flaw). That refutes the idea. We should act on or use non-refuted ideas in preference over refuted ideas.

One thing I don't like about this is the whole 'one strike and you're out' feel of it. It's very boolean, the real world isn't usually so crisp. Even a correct theory will sometimes have some evidence pointing against it, and in policy debates almost every suggestion will have some kind of downside.

There is also the worry that there could be more than one non-refuted idea, which makes it a bit difficult to make decisions. Bayesianism, on the other hand, when combined with expected utility theory, is perfect for making decisions.