[Draft] Holy Bayesian Multiverse, Batman!

by b1shop2 min read3rd Feb 201124 comments

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Personal Blog

I couldn't find the math for the quantum suicide and immortality thought experiment, so I'm placing it here for posterity. If one actually ran the experiment, Bayes' theorem would tell us how to update our belief in the multi-world interpretation (MWI) of quantum mechanics. I conclude by arguing that we don't need to run the experiment.

Prereqs: Understand the purpose of Bayes Theorem, possess at least rudimentary knowledge of the competing quantum worldviews, and have a nostalgic appreciation for Adam West.

The Fiendish Setup:

Suppose that, after catching Batman snooping in the shadows of his evil lair, Joker ties the caped crusader into a quantum, negative binomial death machine that, every ten seconds, measures the spin value of a fresh proton. Fifty percent of the time, the result will trigger a Bat-killing Rube Goldberg machine. The other 50 percent of the time, the quantum death machine will play a suspenseful stock sound effect and search for a new proton.

From Batman's Point of View

Suppose that, ten seconds after Joker turns on the machine, a dramatic, suspenseful sound fills Joker's hideout.

The Bat has survived, and there are two possible explanations:

  1. There is only one world, and he was lucky enough to find himself in a world where the machine didn't kill him.
  2. There are many worlds, and, for obvious reasons, he could only find himself in one of the worlds where he didn't die.

What does Bayes' say about how he should update his belief in the many-worlds theorem? If we partition all QM interpretations into either the single- or many- world camp, Batman's subjective P(MW) will increase to P(MW|S) according to this formula.

Let C be the event that the Copenhagen interpretation is correct and S be the event that Batman experiences himself surviving after the first flip of the quantum coin.

 

 

Click here to see a graph of P(MW|S) graphed over Batman's original estimate of P(MW). When successive iterations is zero, it's a straight line. If Batman survives through many successive iterations, he can only claim one theory with a straight face -- many worlds.

The limit of P(MW|Sn) goes to one as n goes to infinity.

From Joker's Point of View

Unfortunately, there's no such guaranty for the experiment's observers.

If MWI is true, then, in some uncommon universes, the Joker will see Batman breathe many sighs of relief, but, in the overarching majority, Joker will emerge triumphant.

If there is only one world, then Batman might emerge unscathed after the Rube Goldberg machine eventually runs out of batteries, but it's more likely the universe will collapse into one without the Caped Crusader. Joker has no idea which he's observing.

For observers, P(MW) and P(S) are independent. That means P(MW n S) = P(MW) * P(S). Doing slightly more math than needed:

So the observer's estimate of P(MW) is unchanged by the fate of the branchonaut.

Proving Many-Worlds

Surviving a quantum death trap is a convincing argument in favor of the many-worlds hypothesis, but you'd have to be pretty risk-averse to seek it out. First of all, your survival would be completely unpersuasive to observers. Secondly, it'd leave a great many Gothams without Batman (if MWI is true) and a high chance Gotham won't have Batman (if MWI isn't true).

However, if miniscule quantum effects can snowball into cosmological consequences, then we don't need to run the rube goldberg machine. Let S be the event our human race came to being in our universe, and P(MW) be your estimate of the accuracy in MWI before considering this argument and P(MW|S) be your estimate of the accuracy in MWI after considering this argument.

Cosmology is an analagous quantum death trap, and the entire human race witnessed it from the inside.

If you haven't thought about this argument before, you should update your belief in the many-worlds interpretation radically upwards.

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Let C be the event that the Copenhagen interpretation is correct

The complement of "Many Worlds is true" is not "the Copenhagen interpretation is correct". I'm not at all confident "the Copenhagen interpretation is correct" is even a logical possibility.

Surviving a quantum death trap is a convincing argument in favor of the many-worlds hypothesis, but you'd have to be pretty risk-averse to seek it out.

I think you mean risk non-averse. Since if many worlds is wrong, you fail.

Cosmology is an analagous quantum death trap, and the entire human race witnessed it from the inside.

If you haven't thought about this argument before, you should update your belief in the many-worlds interpretation radically upwards.

This is an unconvincing argument. First, you don't know the prior probability of humanity surviving. Second, we don't know how many observers to expect in the universe. The relevant class isn't humans but "life intelligent enough to discuss anthropic considerations and MWI" - that class might be much larger.

The relevant class isn't humans but "life intelligent enough to discuss anthropic considerations and MWI" - that class might be much larger.

I'm not sure that class includes humans, actually.

It certainly doesn't include all humans. I think I wrote somewhere on LW that the anthropic Doomsday problem can be solved if the future is ruled by central powers which enforce a prohibition on anthropic talk.

Surviving a quantum death trap is a convincing argument in favor of the many-worlds hypothesis, but you'd have to be pretty risk-averse to seek it out.

I think you mean risk non-averse. Since if many worlds is wrong, you fail.

If many-worlds is right you still fail - at least if you define "self" with any sort of continuity, when you die in many-worlds you die in real life.

Cosmology is an analagous quantum death trap, and the entire human race witnessed it from the inside.

If you haven't thought about this argument before, you should update your belief in the many-worlds interpretation radically upwards.

This is an unconvincing argument. First, you don't know the prior probability of humanity surviving. Second, we don't know how many observers to expect in the universe. The relevant class isn't humans but "life intelligent enough to discuss anthropic considerations and MWI" - that class might be much larger.

Mostly agreed. Besides, the same mathematical structure exists in the Copenhagen interpretation too, though it's labeled differently; quantum mechanics is quantum mechanics. It is however evidence for the claim that conscious people can be represented by a wavefunction that contains other possibilities, even if those other possibilities are overwhelmingly probable..

But if you only die in some of the branches, then you'll experience yourself living.

If QI is true, then it's when you die in ALL of the many worlds that you no longer experience anything.

But if you only die in some of the branches, then you'll experience yourself living.

You're presupposing that there's a fact of the matter about what "you" [the person you are now] will experience at a future moment. But the person you are now does not exist at any future moment. In fact, future moments merely contain people who are very similar to you - they remember everything you remember and a little bit more.

Therefore, all we can say is that in a branch where you die, there is no-one who remembers being you, but in a branch where you survive, there is a person who remembers being you. There is no such thing as a 'you' which is identical at different moments, no 'thread of identity' that connects you with your future and past selves, and which magically 'chooses' a branch where 'you' survive. In other words, there is no 'transtemporal identity'.

Fundamentally, what you're trying to achieve in a 'quantum immortality' experiment is to experience a fantastically unlikely event. But when you rephrase this in terms that don't presuppose transtemporal identity, all you're saying is that you want there to be a person somewhere, in some Everett branch, who experiences something fantastically unlikely. Therefore, it makes no difference whether those who fail to experience something fantastically unlikely are killed or left alone.

The real question here is simply "If you experienced something fantastically unlikely, would you take this to be evidence that MWI is true and Copenhagen is false?"

(It's an awkward question because the Copenhagen interpretation is incoherent. We ought to ask the question above about a single-universe interpretation that actually makes sense, like Bohm's interpretation or the GRW theory. But for now let's just pretend that Copenhagen does make sense.)

The real question here is simply "If you experienced something fantastically unlikely, would you take this to be evidence that MWI is true and Copenhagen is false?"

And if so, does experiencing something humdrum constitute evidence that MWI is false? Surely not.

It seems to me that either (a) experiencing anything at all, regardless of how likely or unlikely it is, gives an equal amount of "anthropic evidence" in favour of MWI, whatever that means; or else (b) There is no sense whatsoever in which observations as opposed to a priori reasoning can favour MWI over Copenhagen (pretending the latter is coherent).

I don't think (a) makes sense, because surely any theory whatsoever comes equipped with an anthropic "dimmer switch" that can be "brightened" or "dimmed" arbitrarily. For instance, to 'brighten' Copenhagen we could just stipulate that there are N parallel non-interacting Copenhagen universes rather than 1. We could even let N be infinite.

So that just leaves (b).

And if so, does experiencing something humdrum constitute evidence that MWI is false?

Could constitute observational evidence for something strange, something to follow in thinking about the future, but not in thinking about counterfactuals where quantum mechanics works.

But if you only die in some of the branches, then you'll experience yourself living.

No. If you die in some of the branches, in those branches you'll experience dying, and then you won't experience anything any more.

You're right about the first part. I'll fix it when I get to a real computer.

Since we observed humanity surviving, I think it's alright to use a more specific reference class. If the anthropic principle holds for life in general, then it also holds for just JoshuaZ, right?

No matter what p(S) is, p(S|MW) is larger.

Yes, but I don't know how much larger. So claiming that one should update "radically upwards" is questionable without a lot more of an idea how much and what sorts of life we should expect.

If you make the case that p(S) should include any life, then I can concede the adverb perhaps shouldn't be so dramatic. Are we in agreement that it would be dramatic if we only included humanity exactly like us? If so, I'd like a second opinion from others if a specific anthropic principle is allowed, because it seems reasonable to me.

. Are we in agreement that it would be dramatic if we only included humanity exactly like us? I

Yes, because there are times in the history of humanity where we got close to being wiped out (the Toba event being possibly the best documented).

I'd like a second opinion from others if a specific anthropic principle is allowed, because it seems reasonable to me.

I'd be very curious if anyone else agrees with you there. I don't understand the basis for such a specific reference class, since every intelligent species that thinks about MWI will be in the same position.

Here is something I find confusing. If you believe you will have "continuity of consciousness" after being cryopreserved and revived, and you also believe in many-worlds, should you accept the Quantum Immortality hypothesis?

I never really thought of that before.

Don't update too drastically. There's a nontrivial probability that there exists some other theory you haven't thought of, that would also assign high conditional credence to you observing yourself to exist.

Is there an analogous experiment for Tegmark's multiverse?

You set up an experiment so that you survive only if some outcome, anticipated by your highly improbable theory of physics, is true.

Then you wake up in a world which is with high probability governed by your theory.

The analogous experiment for Tegmark's multiverse is called Permutation City.

It's not at all clear to me how it is that, if the many worlds theory is correct, I will experience myself not dying. Assuming the many worlds theory is accurate, in some worlds the versions of "me" that are present there will cease to exist and not feel anything anymore, and in others the versions of "me" will survive. The many worlds interpretation doesn't tell me anything about which set of universe I (singular) am in.

The probability of Batman finding that he survives after the first coinflip is not 50% or 100% in either interpretation. It's more like one in several trillion. If you add "Given that you're batman, and just did the experiment", it's clearly 100% in any interpretation.

There's another, bigger problem. Compare quantum immortality to if he had actual immortality, and just never died. You make it sound like the probabilities will come out the same. They will not. The probability of being batman after flipping the coin is lower in the many worlds version is lower, since the probability amplitude is lower. For example, if he lived 20 years before the coin flip, and, if he survives, 20 years after, in the many worlds interpretation, he'd be twice as likely to be in the "before" section, whereas if it's just guaranteed not to kill him, they'd be equal.

In addition, if you add a huge number of bystanders, one is less likely to be batman in the many worlds version, so it works out that one is about as likely to be batman before in either version, but more likely to be him after in the actual immortality version.

Since you participate in hypothetical MWIs where Batman lives or dies, to test this hypothesis you pretty much need to kill yourself. If you find yourself surviving all your attempts to kill yourself, you have great evidence for your particular version of MWI immortality.

of course if you are wrong, you are dead and gone. Bummer.

If I understand correctly, under MW you anticipate the experience of surviving with probability 1, and under C with probability 0.5. I don't think that's justified.

In both cases the probability should be either conditional on "being there to experience anything" (and equal 1), OR unconditional (equal the "external" probability of survival, 0.5). This is something in between. You take the external probability in C, but condition on the surviving branches in MW.