None of these explanations are intuitive. Here is a visual explanation in two short paragraphs. Imagine a Venn diagram of two partly overlapping circles. The probability space is your sheet of paper on which the diagram is drawn. The area of the yellow circle with respect to the entire sheet is the probability of event Y. Same measure of the cyan circle is the probability of event C. The overlapping area is green (yellow and cyan pigments, when mixed, result in green), and corresponds to both events occurring.
To simplify calculations, we don't scale circle areas by the area of the entire sheet (which incidentally turns our probabilities into frequencies). This way, P(C) and P(Y) are simply areas of the corresponding circles. Now, P(Y|C) is the probability of event Y occurring, conditional on event C occurring also. This is visualized as simply as taking the green area (because both events have to happen) and dividing it by the area of the entire circle C (because we are interested in conditional probability on C); the resulting ratio is P(Y|C). Now it should be clear as sky that P(Y|C) * P(C) = green area = P(C|Y) * P(Y), which is another way to write Bayes Theorem.
This explains the equation, but it doesn't explain the "philosophy." Being Bayesian is in direct conflict with the "scientific method." The scientific method says: make a theory, then test it against observation. To a Bayesian, first comes the observation, and they pick the theory that best explains it.
If T = theory, and O = observation
P(T|O) = P(O|T) * P(T) / P(O)
tells you: given my observation what is the probability of my theory, P(T|O). And the equation gives you how this quantity depends on how your theory predicts the observation, P(O|T).
Not true. Maximum likelihood picks the hypothesis/theory that best "explains" the data, hence you run into issues with overfitting, etc. A Bayesian starts with a prior over a set of hypotheses (the P(T) term), and then uses the data to update their confidence in the various hypotheses. Assuming you have a sane prior, you end up with a simple theory with a record of making decent predictions (i.e. the kind of thing you look for in science, cf. Occam's razor, etc).
I don't see any conflict with the scientific method (at least in a fundamental sense). Scientists aren't oracles - hypotheses need to come from somewhere. Some call it intuition - I would call it an implicit application of Bayesian reasoning, where the data is experience, and the prior is governed by genetic constraints of the brain. From this you obtain a set of intuitive hypotheses to be (further) tested (i.e. those with large posterior P(T|O)).
Testing a set of hypotheses then just involves collecting observations that differentiate them (i.e. where they make conflicting predictions). This can still be considered an application of Bayes' rule, but usually one tries to collect enough data that it's quite obvious which is consistently making the best predictions (in other words has posterior close to 1), in which case it becomes a theory.
I never disagreed with the equation (of course it's correct). My point is that the prior always comes first, even in UII. You're not simply picking the hypothesis that best explains the data (assuming by best explains you mean has the greatest likelihood P(O|T)), otherwise you just end up with the hypothesis containing a lookup table of all previous data. You need to take into account your confidence in the hypothesis before the data arrived (e.g. based on the complexity/size of programs expressing that hypothesis for UII).
Ah. Your use of not true made it look like an outright dismissal of his whole statement. As for the order of when to pick the prior, I think what is more important is that the data not influence your choice of prior. If you were some oracular machine you could see the data and generate hypothesis and priors for them independent of the data and still not fall for the problem you state.
And then there is the problem of how do you form sensible hypotheses without at least knowing the shape of the data first. The form of these hypotheses are themselves a restriction on the possible space. I think that is what the GGP was getting at.
Well, the OP was about a theorem, so I explained the theorem... Bayesian theorem formalizes the process through which human intuition works and with which we pick theories. It is hugely ironic that people not familiar with the Bayesian theorem often consider it unintuitive, since the theorem itself is the basic process of human intuition formalized.
Now, the whole point of the classic scientific method, developed before the scientific mainstream even knew about Bayesian theorem, is to "fight" human intuition since it (probably rightfully so) considers human intuition as something that is only useful to make a guess, but as not really useful (and one that often gets in the way) when one needs to confirm a theory. To summarize, the scientific method was developed to "fight" Bayesianism before Bayesianism existed. Hence people writing long articles and philosophizing about what should be a trivial issue.
But Baye's theorem is not obvious to humans. The intuition is P(A|B) = P(B|A). Base rate fallacy etc. etc.
That said, Bayesianism is not the same thing as Baye's Theorem. You can be a frequentist and still apply Bayes Theorems or the like. Bayes theorem is straight forward once you train your mind on it.
Bayesianism is much more than that and is not a trivial issue. But you are also correct in that human philosophical intuition on probability is Bayesian. With a very broken application. Few people that claim to be Bayesians are actually doing pure Bayesian probability.
Hand wringing over priors and model structure is what Bayesianism is all about. Philosophically, the viewpoints of bayesianism and how to best pick priors are interesting. Also interesting is how Quantum Mechanics fits neatly into the bayesian perspective.
I have a feeling that Bayes is unintuitive specifically because people pick extreme or otherwise bad examples to explain it. As you mention, the way we calculate odds is oddly (pardon the pun) similar to Bayes.
Here is an African Safari example. The probability of an individual hunter being attacked by a lion P(L|H) (assuming any lion who tracks down a hunter will also attack him) equals to the conditional probability of the hunter tracking down a lion P(H|L) (which is a measure of how good the hunter is at finding lions assuming there are lions) times the general probability of encountering a lion in the bush P(L) divided by the general probability of encountering another hunter like himself in the bush P(H). Hence why you can get away with hunting lions alone only if you're very bad at finding lions or if there are no lions.
> The scientific method says: make a theory, then test it against observation.
That's... not the scientific method I learned. The one I learned starts with "Observe a phenomenon". Of course, then you develop a hypothesis and formalize the observation process so that conclusions can be more reliably drawn from it (an experiment), but you're still observing something first.
To simplify calculations, we don't scale circle areas by the area of the entire sheet (which incidentally turns our probabilities into frequencies). This way, P(C) and P(Y) are simply areas of the corresponding circles. Now, P(Y|C) is the probability of event Y occurring, conditional on event C occurring also. This is visualized as simply as taking the green area (because both events have to happen) and dividing it by the area of the entire circle C (because we are interested in conditional probability on C); the resulting ratio is P(Y|C). Now it should be clear as sky that P(Y|C) * P(C) = green area = P(C|Y) * P(Y), which is another way to write Bayes Theorem.