That's mainly because Copenhagen never specified macrsoscopic ...but the idea of an unequivocal "cut" was at the back of a lot of copenhagenists minds, and it has been eaten away by various things over the years.

Just because something is called an 'interpretation' does not mean it doesn't have testable predictions. For example, macroscopic superposition discerns between CI and MWI (although CI keeps changing its definition of 'macroscopic').

I notice that I am getting confused again. Is RQM trying to say that reality via some unknown process the universe produces results to measurements, and we use wavefunctions as something like an interpolation tool to account for those observations, but different observations lead to different inferences and hence to different wavefunctions?

I know I'm late to the party, but I couldn't help but notice that this interesting question hadn't been answered (here, at least). So here it is: as far as I know, B 'splits' immediately, but this in an unphysical question.

In MWI we would have observers A and B, who could observe Aup or Adown and Bup or Bdown (and start in |Aunknown> and |Bunknown> before measuring) respectively. If we write |PAup> and |PAdown> for the wavefunctions corresponding to the particle near observer A being in the up resp. down states, and introduce similar notation for the particle near observer B, then the initial configuration is:

|Aunkown> * |Bunknown> * (|PAup> * |PBdown> - |PAdown> * |PBup>) / \sqrt(2)

Now if we let person A measure the particle the complete wavefunction changes to:

|Bunknown> * (|Aup> * |PAup> * |PBdown> - |Adown> * |PAdown> * |PBup>) / \sqrt(2)

Important is that this is a local change to the wavefunction, what happened here is merely that A measured the particle near A. Since observer A is a macroscopic object we would expect the two branches of the wavefunction above (separated by the minus sign) to be quite far apart in configuration space, so the worlds have definitely split here. But B still isn't correlated to any particular branch: from the point of A, person B is now in a superposition. In particular observer B doesn't notice anything from this splitting - as we would expect (splitting being a local process and observers A and B being far apart). This is also why I called the question as to when B splits 'unphysical' above, since it is a property known only locally at A, and in fact the answer to this question wouldn't change any of B's anticipations.

This might seem a lot like RQM, and that is because RQM happens to get the answer to this question right. The problem with RQM (at least, the problem I ran into while reading the paper) was that the author claims that measurements are ontologically fundamental, and wavefunctions are only a mathematical tool. This seems to confuse the map with the territory: if wavefunctions are only part of our maps, what are they maps of? Also if wavefunctions aren't part of the territory an explanation is needed for the observation that different observers can get the same results when measuring a system, i.e. an explanation is needed for the fact that all observations are consistent. It seems unnecessarily complicated to demand that wavefunctions aren't real, and then separately explain why all observations are consistent as they would have been if the wavefunction were real.

I think this is what Eliezer might have meant with

As far as I can tell, the only possible coherent state of affairs corresponding to RQM - the only reality in which you can embed these systems relating to each other - is MWI

RQM seems to assert precisely what MWI asserts, except that it denies the existence of objective reality, and then needs a completely new and different explanation for the consistency between measurements by different observers. I found the insults hurled at RQM by Eliezer disrespectful but, on close inspection, well-deserved. Denying reality doesn't seem like a good property for a theory of physics to have.

I don't have the referenced textbook to check the result being suggested.

You can see http://projecteuclid.org/euclid.aoms/1177704038, which proves the result.

Replace the original distribution with the sampling distribution from the stopped problem, and it's not longer a stopped problem, it's normal pulls from that sampling distribution.

How would this affect a frequentist?

I'm not sure it's more clear,I think it is not. Your "remapped" problem makes it look like it's a result of low-data-volume and not a problem of how the sampling distribution was actually constructed.

I'm giving low data because those are the simplest kinds of cases to think of. If you had lots of data with the same distribution/likelihood, it would be the same. I leave it as an exercise to find a case with lots of data and the same underlying distribution ...

I was mainly trying to convince you that nothing's actually wrong with having 33% false positive rate in contrived cases.

If you are willing to sample literally forever it seems like you'd be able to convince the Bayesian that the mean is not 0 with arbitrary Bayes factor.

No, there's a limit on that as well. See http://www.ejwagenmakers.com/2007/StoppingRuleAppendix.pdf

Instead, as pointed out by Edwards et al. (1963, p. 239): “(...) if you set out to collect data until your posterior probability for a hypothesis which unknown to you is true has been reduced to .01, then 99 times out of 100 you will never make it, no matter how many data you, or your children after you, may collect (...)”.

If you can generate arbitrarily high Bayes factors, then you can reduce your posterior to .01, which means that it can only happen 1 in 100 times. You can never have a guarantee of always getting strong evidence for a false hypothesis. If you find a case that does, it will be new to me and probably change my mind.

But it doesn't change the results if we switch and say we fool 33% of the Bayesians with Bayes factor of 3. We are still fooling them.

That doesn't concern me. I'm not going to argue for why, I'll just point out that if it is a problem, it has absolutely nothing to do with optional stopping. The exact same behavior (probability 1/3 of generating a Bayes factor of 3 in favor of a false hypothesis) shows up in the following case: a coin either always lands on heads, or lands on heads 1/3 of the time and tails 2/3 of the time. I flip the coin a single time. Let's say the coin is the second coin. There's a 33% chance of getting heads, which would produce a Bayes factor of 3 in favor of the 100%H coin.

If there's something wrong with that, it's a problem with classic Bayes, not optional stopping.

It is my thesis that every optional stopping so-called paradox can be converted into a form without optional stopping, and those will be clearer as to whether the problem is real or not.

Again, the prior doesn't matter, they are computing Bayes factors.

It matters here, because you said "So you might be able to create a rule that fools 99 out of the 100 Bayesians". The probability of getting data given a certain rule depends on which hypothesis is true, and if we're assuming the hypothesis is like the prior, then we need to know the prior to calculate those numbers.

Let's say the null is true, but the frequentist mastermind has devised some data generating process that (let's say he has infinite data at his disposal) that can produce evidence in favor of competing hypothesis at a Bayes factor of 3, 99% of the time.

That's impossible. http://doingbayesiandataanalysis.blogspot.com/2013/11/optional-stopping-in-data-collection-p.html goes through the math.

Using either Bayesian HDI with ROPE, or a Bayes factor, the false alarm rate asymptotes at a level far less than 100% (e.g., 20-25%). In other words, using Bayesian methods, the null hypothesis is accepted when it is true, even with sequential testing of every datum, perhaps 75-80% of the time.

In fact, you can show easily that this can succeed at most 33% of the time. By definition, the Bayes factor is how likely the data is given one hypothesis, divided by how likely the data is given the other. The data in the class "results in a bayes factor of 3 against the null" has a certain chance of happening given that the null is true, say p. This class of course contains many individual mutually exclusive sets of data, each with a far lower probability, but they sum to p. Now, the chance of this class of possible data sets happening given that the null is not true has an upper bound of 1. Each individual probability (which collectively sum to at most 1) must be 3 times as much as the corresponding probability in the group that sums to p. Ergo, p is upper bounded by 33%.

I think 2 is uncontroversial, other than if you have a perfect prior why do any experiment at all?

By perfect I mean well calibrated. I don't see why knowing that your priors in general are well calibrated implies that more information doesn't have positive expected utility.

The issue is that with optional stopping you bias the Bayes factor.

Only in some cases, and only with regard to someone who knows more than the Bayesian. The Bayesian himself can't predict that the factor will be biased; the expected factor should be 1. It's only someone who knows better that can predict this.

So let's think of this like a frequentist who has a laboratory full of bayesians in cages.

Before I analyse this case, can you clarify whether the hypothesis happens to be true, false, or chosen at random? Also give these Bayesians' priors, and perhaps an example of the rule you'd use.

You seem to be saying that a rule based on posteriors would not be susceptible to such hacking?

I'm saying that all inferences are still correct. So if your prior is correct/well calibrated, then your posterior is as well. If you end up with 100 studies that all found an effect for different things at a posterior of 95%, 5% of them should be wrong.

The goal of the p-hacker is to increase the probability of type 1 error.

So what I should say is that the Bayesian doesn't care about the frequency of type 1 errors. If you're going to criticise that, you can do so without regard to stopping rules. I gave an example in a different reply of hacking bayes factors, now I'll give one with hacking posteriors:

Two kinds of coins: one fair, one 10%H/90%T. There are 1 billion of the fair ones, and 1 of the other kind. You take a coin, flip it 10 times, then say which coin you think it is. The Bayesian gets the biased coin, and no matter what he flips, will conclude that the coin is fair with overwhelming probability. The frequentist gets the coin, get ~9 tails, and says "no way is this fair". There, the frequentist does better because the Bayesian's prior is bad (I said there are a billion fair ones and only one biased one, but only looked at the biased ones).

It doesn't matter if you always conclude with 95% posterior that the null is false when it is true, as long as you have 20 times as many cases that the null is actually false. Yes, this opens you up to being tricked; but if you're worried about deliberate deception, you should include a prior over that. If you're worried about publication bias when reading other studies, include a prior over that, etc.

Then you are doing a very confusing thing that isn't likely to give much insight. Frequentist inference and Bayesian inference are different and it's useful to at least understand both ideas(even if you reject frequentism).

I think I understand frequentism. My claim here was that the specific claim of "the stopping rule paradox proves that frequentism does better than Bayes" is wrong, or is no stronger than the standard objection that Bayes relies on having good priors.

So saying the "frequentist encodes a better prior" is to miss the whole point of how frequentist statistics works.

What I meant is that you can get the same results as the frequentist in the stopping rule case if you adopt a particular prior. I might not be able to show that rigorously, though.

And the point in the paper I linked has nothing to do with the prior, it's about the bayes factor, which is independent of the prior.

That paper only calculates what happens to the bayes factor when the null is true. There's nothing that implies the inference will be wrong.

There are a couple different version of the stopping rule cases. Some are disguised priors, and some don't affect calibration/inference or any Bayesian metrics.

Whether or not the Bayesian calibration is overall correct depends not just on the Bayes factor but the prior.

It depends only on the prior. I consider all these "stopping rule paradoxes" disguised cases where you give the Bayesian a bad prior, and the frequentist formula encodes a better prior.

In practice what p-hacking is about is convincing the world of an effect, so you are trying to create bias toward any data looking like a novel effect.

You still wouldn't have more chances of showing a novel effect than you thought you would when you went into the experiment, if your priors are correct. If you say "I'll stop when I have a novel effect", do this many times, and then look at all the times you found a novel effect, 95% of the time the effect should actually be true. If this is wrong, you must have bad priors.