Can someone explain Quantum Bayesianism to me?

My point is that whenever you have a large enough set of identically prepared particles hitting a screen one-by-one which detects their 2-D postion what you will get is something roughly corresponding to a cross-section of the square-modulus of their wavefunction. That’s a basic prediction of quantum mechanics and the laws of probability and is interpretation-independent.

What’s remarkable in this case is that the cross-section has the same shape as when the electrons are in the hydrogen atom.

Well, the wave function (more accurately, the square of its absolute value) gives the probability of, for instance, finding a particle at a certain point in space. Making many experiments, involving many detections, will yield a distribution of particle detection events that is exactly this squared-modulus distribution. But all you ever see are just the individual particle events, which then ‘build up’ a picture of the wave function.
But the same thing you can do, for example, for a classical probability distribution, say a liouville distribution on phase space, or more prosaically, the distribution of darts thrown on a target. But that doesn’t imply that these distributions are physically ‘real’ in any sense (nor does it, of course, contradict it).

Is it really much different than saying a larger scale object is “real” because we can detect a pattern of photons being bounced off it? All methods of perception and detection are indirect extrapolations.

I wouldn’t think so: the object the photons bounced off of has an independent existence, above and beyond those photons (at least, on a straightforwardly realist reading; but then again, what other reading is there on which you can consider something to be ‘real’?); with a probability distribution, once you account for whatever underlies it (the dart throws, or particle motions in a gas etc.) has no further independent existence. At least, that’s the case in a classical world; when it comes to the quantum, things are different (in fact, it can be proven that the quantum state is not a probability distribution in the sense that there are events that underlie it like the dart throws underlie the distribution of dart throws).

Not sure how relevant this is, but to underscore the strangeness - here is a recent Science article discussing how 2 photons can be entangled w/o ever existing at the same time (via entanglement swapping).

I think entanglement gets a bit less strange if you just view it as correlation in a theory in which the sum of any two states of a physical system again is a state of that system (that is, quantum mechanics). What I mean by correlation is something like the following: let’s say I have a red marble, and a green marble, and two boxes to put them in, box A and box B.

Now, the setup of the system entails the following: whenever the red marble is in box A, the green marble is in box B; or conversely, whenever the green marble is in box A, the red marble is in box B. The marble colors are thus (anti-)correlated: once you know the color of one marble, you know the color of the other. There are thus two possible states for the system of the boxes and marbles to be in, which we can represent in the following somewhat curious way: |green[sub]A[/sub], red[sub]B[/sub]> or |red[sub]A[/sub], green[sub]B[/sub]>. Here, the indices simply denote the box, and the labels the color of the marble (the notation is a bit redundant: often, the index is omitted, and just the placement is used to indicate the ‘box’: first slot is ‘box A’, second slot is ‘box B’; but I’ll keep the indices for clarity).

Thus far the story in the classical world. But in quantum mechanics, as I said, things are a bit different: for any set of valid states, their sum is again a valid state. That is, if we just consider a single box for the moment, say box A, additionally to the states |green[sub]A[/sub]> and |red[sub]A[/sub]>, states of the form a|green[sub]A[/sub]> + b|red[sub]A[/sub]>, with certain constants a and b, are again valid system states. In the context of an experiment, i.e. if we open the box, the numbers a and b determine the probabilities of finding the marble inside to be either color; it will be green with probability |a|[sup]2[/sup], and red with probability |b|[sup]2[/sup]. Thus, if a = b = 1/√2, we will find either color with 50% probability. Note that this is not the case in the classical world: me wearing socks is a valid state, as is me wearing no socks. But me wearing socks and not wearing socks doesn’t make sense.

Now, the same considerations apply to systems of more than one box: as |green[sub]A[/sub], red[sub]B[/sub]> and |red[sub]A[/sub], green[sub]B[/sub]> are valid system states, so too is
|S[sub]AB[/sub]> = 1/√2(|green[sub]A[/sub], red[sub]B[/sub]> + |red[sub]A[/sub], green[sub]B[/sub]>)

What does it now mean for the system to be in this weird kind of state? Well, operationally, it means the following: if we open box A, and find the marble inside to be green, which will happen with a probability of 50%, then we will know instantly that the marble in box B is red; analogously, if we find the box A marble to be red (again with a probability of 50%), we know that the box B marble must be green.

This is what we observe in experiments on entangled quantum systems. Thus, entanglement is ultimately the superposition of correlated states—the mystery, then, comes ultimately from the superposition principle. From this point of view, entanglement ‘across time’ is maybe not so strange—certainly, events at two points in time can be classically correlated, even though they don’t co-occur. You could consider a set of four boxes (A, B, C, D), each set of two—A,B and C,D—correlated as above. You look into box A, note the color of the marble inside, then cast it to the flames.

Then, you fix the content of box C (which is only created at this point in time) such that it contains a marble of the same color as the one in B. Because of the correlation between boxes C and D, now box D must contain a marble whose color is opposite to that of A; thus, we have boxes A and D exhibiting the same correlation as the original A and B did, i.e. upon looking into box D, you will always find the color red if in the now destroyed you had found green, and green if you had found red. That is, they behave as if they were in either the state |green[sub]A[/sub], red[sub]D[/sub]> or |red[sub]A[/sub], green[sub]D[/sub]>. Add again superposition to the picture, and you can make them behave as if they were in the state |S[sub]AD[/sub]> as above.

And all of that is not, classically, weird. Where the weirdness comes in is with Bell’s inequality and the EPR conundrum. Unfortunately, that’s not applicable to a situation as simple as your colored marbles, but there are situations only slightly more complicated where it is applicable.

I wouldn’t say that. Superposition is very weird, classically. It’s also, ultimately, where Bell inequality violations come from: definite states in one basis are superpositions in another, such that the correlations exist across multiple bases, where the basis vectors are eigenstates of noncommuting observables; this ultimately means that you can’t find a single joint probability distribution whose marginals reproduce all pairwise correlations. But the existence of such a distribution is a necessary condition for Bell inequalities to hold.

Sure, one can’t have true superpositions classically, but one can have things that look sort of like them. Classically, I can’t say that the marble in my box is a superposition of red and green, but I can classically say that I don’t know its color, and that it has a 50% chance each of red or green. In a sense, the core of the EPR conundrum is the rigorous demonstration that this is not the same thing as a superposition. But you need a more complicated example to make this demonstration.