Probability is in the Mind

Monsterwithgirl_2

Followup toThe Mind Projection Fallacy

Yesterday I spoke of the Mind Projection Fallacy, giving the example of the alien monster who carries off a girl in a torn dress for intended ravishing—a mistake which I imputed to the artist's tendency to think that a woman's sexiness is a property of the woman herself, woman.sexiness, rather than something that exists in the mind of an observer, and probably wouldn't exist in an alien mind.

The term "Mind Projection Fallacy" was coined by the late great Bayesian Master, E. T. Jaynes, as part of his long and hard-fought battle against the accursèd frequentists.  Jaynes was of the opinion that probabilities were in the mind, not in the environment—that probabilities express ignorance, states of partial information; and if I am ignorant of a phenomenon, that is a fact about my state of mind, not a fact about the phenomenon.

I cannot do justice to this ancient war in a few words—but the classic example of the argument runs thus:

You have a coin.
The coin is biased.
You don't know which way it's biased or how much it's biased.  Someone just told you, "The coin is biased" and that's all they said.
This is all the information you have, and the only information you have.

You draw the coin forth, flip it, and slap it down.

Now—before you remove your hand and look at the result—are you willing to say that you assign a 0.5 probability to the coin having come up heads?

The frequentist says, "No.  Saying 'probability 0.5' means that the coin has an inherent propensity to come up heads as often as tails, so that if we flipped the coin infinitely many times, the ratio of heads to tails would approach 1:1.  But we know that the coin is biased, so it can have any probability of coming up heads except 0.5."

The Bayesian says, "Uncertainty exists in the map, not in the territory.  In the real world, the coin has either come up heads, or come up tails.  Any talk of 'probability' must refer to the information that I have about the coin—my state of partial ignorance and partial knowledge—not just the coin itself.  Furthermore, I have all sorts of theorems showing that if I don't treat my partial knowledge a certain way, I'll make stupid bets.  If I've got to plan, I'll plan for a 50/50 state of uncertainty, where I don't weigh outcomes conditional on heads any more heavily in my mind than outcomes conditional on tails.  You can call that number whatever you like, but it has to obey the probability laws on pain of stupidity.  So I don't have the slightest hesitation about calling my outcome-weighting a probability."

I side with the Bayesians.  You may have noticed that about me.

Even before a fair coin is tossed, the notion that it has an inherent 50% probability of coming up heads may be just plain wrong.  Maybe you're holding the coin in such a way that it's just about guaranteed to come up heads, or tails, given the force at which you flip it, and the air currents around you.  But, if you don't know which way the coin is biased on this one occasion, so what?

I believe there was a lawsuit where someone alleged that the draft lottery was unfair, because the slips with names on them were not being mixed thoroughly enough; and the judge replied, "To whom is it unfair?"

To make the coinflip experiment repeatable, as frequentists are wont to demand, we could build an automated coinflipper, and verify that the results were 50% heads and 50% tails.  But maybe a robot with extra-sensitive eyes and a good grasp of physics, watching the autoflipper prepare to flip, could predict the coin's fall in advance—not with certainty, but with 90% accuracy.  Then what would the real probability be?

There is no "real probability".  The robot has one state of partial information.  You have a different state of partial information.  The coin itself has no mind, and doesn't assign a probability to anything; it just flips into the air, rotates a few times, bounces off some air molecules, and lands either heads or tails.

So that is the Bayesian view of things, and I would now like to point out a couple of classic brainteasers that derive their brain-teasing ability from the tendency to think of probabilities as inherent properties of objects.

Let's take the old classic:  You meet a mathematician on the street, and she happens to mention that she has given birth to two children on two separate occasions.  You ask:  "Is at least one of your children a boy?"  The mathematician says, "Yes, he is."

What is the probability that she has two boys?  If you assume that the prior probability of a child being a boy is 1/2, then the probability that she has two boys, on the information given, is 1/3.  The prior probabilities were:  1/4 two boys, 1/2 one boy one girl, 1/4 two girls.  The mathematician's "Yes" response has probability ~1 in the first two cases, and probability ~0 in the third.  Renormalizing leaves us with a 1/3 probability of two boys, and a 2/3 probability of one boy one girl.

But suppose that instead you had asked, "Is your eldest child a boy?" and the mathematician had answered "Yes."  Then the probability of the mathematician having two boys would be 1/2.  Since the eldest child is a boy, and the younger child can be anything it pleases.

Likewise if you'd asked "Is your youngest child a boy?"  The probability of their being both boys would, again, be 1/2.

Now, if at least one child is a boy, it must be either the oldest child who is a boy, or the youngest child who is a boy.  So how can the answer in the first case be different from the answer in the latter two?

Or here's a very similar problem:  Let's say I have four cards, the ace of hearts, the ace of spades, the two of hearts, and the two of spades.  I draw two cards at random.  You ask me, "Are you holding at least one ace?" and I reply "Yes."  What is the probability that I am holding a pair of aces?  It is 1/5.  There are six possible combinations of two cards, with equal prior probability, and you have just eliminated the possibility that I am holding a pair of twos.  Of the five remaining combinations, only one combination is a pair of aces.  So 1/5.

Now suppose that instead you asked me, "Are you holding the ace of spades?"  If I reply "Yes", the probability that the other card is the ace of hearts is 1/3.  (You know I'm holding the ace of spades, and there are three possibilities for the other card, only one of which is the ace of hearts.)  Likewise, if you ask me "Are you holding the ace of hearts?" and I reply "Yes", the probability I'm holding a pair of aces is 1/3.

But then how can it be that if you ask me, "Are you holding at least one ace?" and I say "Yes", the probability I have a pair is 1/5?  Either I must be holding the ace of spades or the ace of hearts, as you know; and either way, the probability that I'm holding a pair of aces is 1/3.

How can this be?  Have I miscalculated one or more of these probabilities?

If you want to figure it out for yourself, do so now, because I'm about to reveal...

That all stated calculations are correct.

As for the paradox, there isn't one.  The appearance of paradox comes from thinking that the probabilities must be properties of the cards themselves.  The ace I'm holding has to be either hearts or spades; but that doesn't mean that your knowledge about my cards must be the same as if you knew I was holding hearts, or knew I was holding spades.

It may help to think of Bayes's Theorem:

P(H|E) = P(E|H)P(H) / P(E)

That last term, where you divide by P(E), is the part where you throw out all the possibilities that have been eliminated, and renormalize your probabilities over what remains.

Now let's say that you ask me, "Are you holding at least one ace?"  Before I answer, your probability that I say "Yes" should be 5/6.

But if you ask me "Are you holding the ace of spades?", your prior probability that I say "Yes" is just 1/2.

So right away you can see that you're learning something very different in the two cases.  You're going to be eliminating some different possibilities, and renormalizing using a different P(E).  If you learn two different items of evidence, you shouldn't be surprised at ending up in two different states of partial information.

Similarly, if I ask the mathematician, "Is at least one of your two children a boy?" I expect to hear "Yes" with probability 3/4, but if I ask "Is your eldest child a boy?" I expect to hear "Yes" with probability 1/2.  So it shouldn't be surprising that I end up in a different state of partial knowledge, depending on which of the two questions I ask.

The only reason for seeing a "paradox" is thinking as though the probability of holding a pair of aces is a property of cards that have at least one ace, or a property of cards that happen to contain the ace of spades.  In which case, it would be paradoxical for card-sets containing at least one ace to have an inherent pair-probability of 1/5, while card-sets containing the ace of spades had an inherent pair-probability of 1/3, and card-sets containing the ace of hearts had an inherent pair-probability of 1/3.

Similarly, if you think a 1/3 probability of being both boys is an inherent property of child-sets that include at least one boy, then that is not consistent with child-sets of which the eldest is male having an inherent probability of 1/2 of being both boys, and child-sets of which the youngest is male having an inherent 1/2 probability of being both boys.  It would be like saying, "All green apples weigh a pound, and all red apples weigh a pound, and all apples that are green or red weigh half a pound."

That's what happens when you start thinking as if probabilities are in things, rather than probabilities being states of partial information about things.

Probabilities express uncertainty, and it is only agents who can be uncertain.  A blank map does not correspond to a blank territory.  Ignorance is in the mind.

 

Part of the sequence Reductionism

Next post: "The Quotation Is Not the Referent"

Previous post: "Mind Projection Fallacy"

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Here is another example me, my dad and my brother came up with when we were discussing probability.

Suppose there are 4 card, an ace and 3 kings. They are shuffled and placed face side down. I didn't look at the cards, my dad looked at the first card, my brother looked at the first and second cards. What is the probability of the ace being one of the last 2 cards. For me: 1/2 For my dad: If he saw the ace it is 0, otherwise 2/3. For my brother: If he saw the ace it is 0, otherwise 1.

How can there be different probabilities of the same event? It is because probability is something in the mind calculated because of imperfect knowledge. It is not a property of reality. Reality will take only a single path. We just don't know what that path is. It is pointless to ask for "the real likelihood" of an event. The likelihood depends on how much information you have. If you had all the information, the likelihood of the event would be 100% or 0%.

So therefore a person with perfect knowledge would not need probability. Is this another interpretation of "God does not play dice?" :-)

I think this is the only interpretation of "God does not play dice."

At least in its famous context, I always interpreted that quote as a metaphorical statement of aesthetic preference for a deterministic over a stochastic world, rather than an actual statement about the behavior of a hypothetical omniscient being. A lot of bullshit's been spilled on Einstein's religious preferences, but whatever the truth I'd be very surprised if he conditioned his response to a scientific question on something that speculative.

This is more or less what I was saying, but left (perhaps too) much of it implicit.

If there were an entity with perfect knowledge of the present ("God"), they would have perfect knowledge of the future, and thus "not need probability", iff the universe is deterministic. (If there is an entity with perfect knowledge of the future of a nondeterministic reality, we have described our "reality" too narrowly - include that entity and it is necessarily deterministic or the perfect knowledge isn't).

GBM:

Q: What is the probability for a pseudo-random number generator to generate a specific number as his next output?

A: 1 or 0 because you can actually calculate the next number if you have the available information.

Q: What probability do you assign to a specific number as being it's next output if you don't have the information to calculate it?

Replace pseudo-random number generator with dice and repeat.

Even more important, I think, is the realization that, to decide how much you're willing to bet on a specific outcome, all of the following are essentially the same:

  • you do have the information to calculate it but haven't calculated it yet
  • you don't have the information to calculate it but know how to obtain such information.
  • you don't have the information to calculate it

The bottom line is that you don't know what the next value will be, and that's the only thing that matters.

Constant: The competent frequentist would presumably not be befuddled by these supposed paradoxes.

Not the last two paradoxes, no. But the first case given, the biased coin whose bias is not known, is indeed a classic example of the difference between Bayesians and frequentists. The frequentist says:

"The coin's bias is not a random variable! It's a fixed fact! If you repeat the experiment, it won't come out to a 0.5 long-run frequency of heads!" (Likewise when the fact to be determined is the speed of light, or whatever.) "If you flip the coin 10 times, I can make a statement about the probability that the observed ratio will be within some given distance of the inherent propensity, but to say that the coin has a 50% probability of turning up heads on the first occasion is nonsense - that's just not the real probability, which is unknown."

According to the frequentist, apparently there is no rational way to manage your uncertainty about a single flip of a coin of unknown bias, since whatever you do, someone else will be able to criticize your belief as "subjective" - such a devastating criticism that you may as well, um, flip a coin. Or consult a magic 8-ball.

Sudeep: If quantum mechanics is true, then ignorance/uncertainty is a part of nature and not just something that agents have.

A common misconception - Jaynes railed against that idea too, and he wasn't even equipped with the modern understanding of decoherence. In quantum mechanics, it's an objective fact that the blobs of amplitude making up reality sometimes split in two, and you can't predict what "you" will see, when that happens, because it is an objective fact that different versions of you will see different things. But all this is completely mechanical, causal, and deterministic - the splitting of observers just introduces an element of anthropic pseudo-uncertainty, if you happen to be one of those observers. The splitting is not inherently related to the act of measurement by a conscious agent, or any kind of agent; it happens just as much when a system is "measured" by a photon bouncing off and interacting with a rock.

There are other interpretations of quantum mechanics, but they don't make any sense. Making this fully clear will require more prerequisite posts first, though.

Eliezer:

"The coin's bias is not a random variable! It's a fixed fact! If you repeat the experiment, it won't come out to a 0.5 long-run frequency of heads!"

You're repeating the wrong experiment.

The correct experiment for a frequentist to repeat is one where a coin is chosen from a pool of biased coins, and tossed once. By repeating that experiment, you learn something about the average bias in the pool of coins. For a symmetrically biased pool, the frequency of heads would approach 0.5.

So your original premise is wrong. A frequentist approach requires a series of trials of the correct experiment. Neither the frequentist nor the Bayesian can rationally evaluate unknown probabilities. A better way to say that might be, "In my view, it's okay for both frequentists and Bayesians to say "I don't know.""

I think EY's example here should actually should be targeted at the probability as propensity theory of Von Mises (Richard, not Ludwig), not the frequentist theory, although even frequentists often conflate the two.

The probability for you is not some inherent propensity of the physical situation, because the coin will flip depending on how it is weighted and how hard it is flip. The randomness isn't in the physical situation, but in our limited knowledge of the physical situation.

The argument against frequentist thinking is that we're not interested in a long term frequency of an experiment. We want to know how to bet now. If you're only going to talk about long term frequencies of repeatable experiments, you're not that useful when I'm facing one con man with a biased coin.

That singular event is what it is. If you're going to argue that you have to find the right class of events in your head to sample from, you're already halfway down the road to bayesianism. Now you just have to notice that the class of events is different for the con man than it is for you, because of your differing states of knowledge, you'll make it all the way there.

Notice how you thought up a symmetrically biased pool. Where did that pool come from? Aren't you really just injecting a prior on the physical characteristics into your frequentist analysis?

If you push frequentism past the usual frequentist limitations (physical propensity, repeated experiments), you eventually recreate bayesianism. "Inside every Non-bayesian, there is a bayesian struggling to get out".

I think EY's example here should actually should be targeted at the probability as propensity theory of Von Mises (Richard, not Ludwig), not the frequentist theory, although even frequentists often conflate the two.

yep.

I always found it really strange that EY believes in Bayesianism when it comes to probability theory but many worlds when it comes to quantum physics. Mathematically, probability theory and quantum physics are close analogues (of which quantum statistical physics is the common generalisation), and this extends to their interpretations. (This doesn't apply to those interpretations of quantum physics that rely on a distinction between classical and quantum worlds, such as the Copenhagen interpretation, but I agree with EY that these don't ultimately make any sense.) There is a many-worlds interpretation of probability theory, and there is a Bayesian interpretation of quantum physics (to which I subscribe).

I need to write a post about this some time.

There is a many-worlds interpretation of probability theory, and there is a Bayesian interpretation of quantum physics (to which I subscribe).

Both of these are false. Consider the trillionth binary digit of pi. I do not know what it is, so I will accept bets where the payoff is greater than the loss, but not vice versa. However, there is obviously no other world where the trillionth binary digit of pi has a different value.

The latter is, if I understand you correctly, also wrong. I think that you are saying that there are 'real' values of position, momentum, spin, etc., but that quantum mechanics only describes our knowledge about them. This would be a hidden variable theory. There are very many constraints imposed by experiment on what hidden variable theories are possible, and all of the proposed ones are far more complex than MWI, making it very unlikely that any such theory will turn out to be true.

I think that you are saying that there are 'real' values of position, momentum, spin, etc., but that quantum mechanics only describes our knowledge about them.

I am saying that the wave function (to be specific) describes one's knowledge about position, momentum, spin, etc., but I make no claim that these have any ‘real' values.

In the absence of a real post, here are some links:

By the way, you seem to have got this, but I'll say it anyway for the benefit of any other readers, since it's short and sums up the idea: The wave function exists in the map, not in the territory.

The wave function exists in the map, not in the territory.

Please explain how you know this?

ETA: Also, whatever does exist in the territory, it has to generate subjective experiences, right? It seems possible that a wave function could do that, so saying that "the wave function exists in the territory" is potentially a step towards explaining our subjective experiences, which seems like should be the ultimate goal of any "interpretation". If, under the all-Bayesian interpretation, it's hard to say what exists in the territory besides that the wave function doesn't exist in the territory, then I'm having trouble seeing how it constitutes progress towards that ultimate goal.

Please explain how you know this?

I wouldn't want to pretend that I know this, just that this is the Bayesian interpretation of quantum mechanics. One might as well ask how we Bayesians know that probability is in the map and not the territory. (We are all Bayesians when it comes to classical probability, right?) Ultimately, I don't think that it makes sense to know such things, since we make the same physical predictions regardless of our interpretation, and only these can be tested.

Nevertheless, we take a Bayesian attitude toward probability because it is fruitful; it allows us to make sense of natural questions that other philosophies can't and to keep things mathematically precise without extra complications. And we can extend this into the quantum realm as well (which is good since the universe is really quantum). In both realms, I'm a Bayesian for the same reasons.

A half-Bayesian approach adds extra complications, like the two very different maps that lead to same predictions. (See this comment's cousin in reply to endoself.)

ETA: As for knowing what exists in the territory as an aid to explaining subjective experience, we can still say that the territory appears to consist ultimately of quark fields, lepton fields, etc, interacting according to certain laws, and that (built out of these) we appear to have rocks, people, computers, etc, acting in certain ways. We can even say that each particular rock appears to have a specific value of position and momentum, up to a certain level of precision (which fails to be infinitely precise first because the definition of any particular rock isn't infinitely precise, long before the level of quantum indeterminacy). We just can't say that each particular quark has a specific value of position and momentum beyond a certain level of precision, despite being (as far as we know) fundamental, and this is true regardless of whether we're all-Bayesian or many-worlder. (Bohmians believe that such values do exist in the territory, but these are unobservable even in principle, so this is a pointless belief).

Edit: I used ‘world’ consistently in a technical sense.

Nevertheless, we take a Bayesian attitude toward probability because it is fruitful; it allows us to make sense of natural questions that other philosophies can't and to keep things mathematically precise without extra complications. And we can extend this into the quantum realm as well

Where "extending" seems to mean "assuming". I find it more fruitful to come up with tests of (in)determinsm, such as Bell's Inequalitites.

I'm not sure what you mean by ‘assuming’. Perhaps you mean that we see what happens if we assume that the Bayesian interpretation continues to be meaningful? Then we find that it works, in the sense that we have mutually consistent degrees of belief about physically observable quantities. So the interpretation has been extended.

If the universe contains no objective probabilities, it will still contain subjective, ignorance based probabilities.

If the universe contains objective probabilities, it will also still contain subjective, ignorance based probabilities.

So the fact subjective probabilities "work" doesn't tell you anything about the universe. It isn't a test.

Aspect's experiment to test Bells theorem is a test. It tells you there isn't (local, single-universe) objective probability.

OK, I think that I understand you now.

Yes, Bell's inequalities, along with Aspect's experiment to test them, really tell us something. Even before the experiment, the inequalities told us something theoretical: that there can be no local, single-world objective interpretation of the standard predictions of quantum mechanics (for a certain sense of ‘objective’); then the experiment told us something empirical: that (to a high degree of tolerance) those predictions were correct where they mattered.

Like Bell's inequalities, the Bayesian interpretation of quantum mechanics tells us something theoretical: that there can be a local, single-world interpretation of the standard predictions of quantum mechanics (although it can't be objective in the sense ruled out by Bell's inequalities). So now we want the analogue of Aspect's experiment, to confirm these predictions where it matters and tell us something empirical.

Bell's inequalities are basically a no-go theorem: an interpretation with desired features (local, single-world, objective true value of all potentially observable quantities) does not exist. There's a specific reason why it cannot exist, and Aspect's experiment tests that this reason applies in the real world. But Fuchs et al's development of the Bayesian interpretation is a go theorem: an interpretation with some desired features (local, single-world) does exist. So there's no point of failure to probe with an experiment.

We still learn something about the universe, specifically about the possible forms of maps of it. But it's a purely theoretical result. I agree that Bell's inequalities and Aspect's experiment are a more interesting result, since we get something empirical. But it wasn't a surprising result (which might be hindsight bias on my part). There seem to be a lot of people here (although that might be my bad impression) who think that there is no local, single-world interpretation of the standard predictions of quantum mechanics (or even no single-world interpretation at all, but I'm not here to push Bohmianism), so the existence of the Bayesian interpretation may be the more surprising result; it may actually tell us more. (At any rate, it was surprising once upon a time for me.)

I have not read the latter link yet, though I intend to.

I am saying that the wave function (to be specific) describes one's knowledge about position, momentum, spin, etc., but I make no claim that these have any ‘real' values.

What do you have knowledge of then? Or is there some concept that could be described as having knowledge of something without that thing having an actual value?

From Baez:

Probability theory is the special case of quantum mechanics in which ones algebra of observables is commutative.

This is horribly misleading. Bayesian probability can be applied perfectly well in a universe that obeys MWI while being kept completely separate mathematically from the quantum mechanical uncertainty.

Probability theory is the special case of quantum mechanics in which ones algebra of observables is commutative.

This is horribly misleading. Bayesian probability can be applied perfectly well in a universe that obeys MWI while being kept completely separate mathematically from the quantum mechanical uncertainty.

As a mathematical statement, what Baez says is certainly correct (at least for some reasonable mathematical formalisations of ‘probability theory’ and ‘quantum mechanics’). Note that Baez is specifically discussing quantum statistical mechanics (which I don't think he makes clear); non-statistical quantum mechanics is a different special case which (barring trivialities) is completely disjoint from probability theory.

Of course, the statement can still be misleading; as you note, it's perfectly possible to interpret quantum statistical physics by tacking Bayesian probability on top of a many-worlds interpretation of non-statistical quantum mechanics. That is, it's possible but (I argue) unwise; because if you do this, then your beliefs do not pay rent!

The classic example is a spin-1/2 particle that you believe to be spin-up with 50% probability and spin-down with 50% probability. (I mean probability here, not a superposition.) An alternative map is that you believe that the particle is spin-right with 50% probability and spin-left with 50% probability. (Now superposition does play a part, as spin-right and spin-left are both equally weighted superpositions of spin-up and spin-down, but with opposite relative phases.) From the Bayesian-probability-tacked-onto-MWI point of view, these are two very different maps that describe incompatible territories. Yet no possible observation can ever distinguish these! Specifically, if you measure the spin of the particle along any axis, both maps predict that you will measure the spin to be in one direction with 50% probability and in the other direction with 50% probability. (The wavefunctions give Born probabilities for the observations, which are then weighted according to your Bayesian probabilities for the wavefunctions, giving the result of 50% every time.)

In statistical mechanics as it is practised, no distinction is made between these two maps. (And since the distinction pays no rent in terms of predictions, I argue that no distinction should be made.) They are both described by the same ‘density matrix’; this is a generalisation of the notion of quantum state as a wave vector. (Specifically, the unit vectors up to phase in the Hilbert space describe the pure states of the system, which are only a degenerate case of the mixed states described by the density matrices.) A lot of the language of statistical mechanics is frequentist-influenced talk about ‘ensembles’, but if you just reinterpret all of this consistently in a Bayesian way, then the practice of statistical mechanics gives you the Bayesian interpretation.

I am saying that the wave function (to be specific) describes one's knowledge about position, momentum, spin, etc., but I make no claim that these have any ‘real' values.

What do you have knowledge of then? Or is there some concept that could be described as having knowledge of something without that thing having an actual value?

This is the weak point in the Bayesian interpretation of quantum mechanics. I find it very analogous to the problem of interpreting the Born probabilities in MWI. Eliezer cannot yet clearly answer these questions that he poses:

What are the Born probabilities, probabilities of? Here's the map - where's the territory?

And neither can I (at least, not in a way that would satisfy him). In the all-Bayesian interpretation, the Born probabilities are simply Bayesian probabilities, so there's no special problems about them; but as you point out, it's still hard to say what the territory is like.

My best answer is simply what you suggest, that our maps of the universe assign probabilities to various possible values of things that do not (necessarily) have any actual values. This may seem like a counterintuitive thing to do, but it works, and we have no other way of making a map.

By the way, I've thought of a couple more references:

Baez (1993) is where I really learnt quantum statistical mechanics (despite having earlier taken a course in it), and my first (subtle) introduction to the Bayesian interpretation (not made explicit here). Note the talk about the ‘post-Everett school’, and recall that Everett is credited with founding the many-worlds interpretation (although he avoided the term ‘MWI’). The Bayesian interpretation could have been understood in the 1930s (and I have heard it argued, albeit unconvincingly, that it is what Bohr really meant all along), but it's really best understood in light of the modern understanding of decoherence that Everett started. We all-Bayesians are united with the many-worlders (and the Bohmians) in decrying the mystical separation of the universe into ‘quantum’ and ‘classical’ worlds and the reality of the ‘collapse of the wavefunction’. (That is, we do believe in the collapse of the wavefunction, but not in the territory; for us, it is simply the process of updating the map on the basis of new information, that is the application of a suitably generalised Bayes's Theorem.) We just think that the many-worlders have some unnecessary ontological baggage (like the Bohmians, but to a lesser degree).

Bartels (1998) is my first attempt to explain the Bayesian interpretation (on Usenet), albeit not a very good one. It's overly mathematical (and poorly so, since W*-algebras make a better mathematical foundation than C*-algebras). But it does include things that I haven't said here, (including mathematical details that you might happen to want). Still (even for the mathematics), if you read only one, read Baez.

Edit: I edited to use the word ‘world’ only in the technical sense of an interpretation.

I wrote:

The classic example is a spin-1/2 particle that you believe to be spin-up with 50% probability and spin-down with 50% probability.

I've begun to think that this is probably not a good example.

It's mathematically simple, so it is good for working out an example explicitly to see how the formalism works. (You may also want to consider a system with two spin-1/2 particles; but that's about as complicated as you need to get.) However, it's not good philosophically, essentially since the universe consists of more than just one particle!

Mathematically, it is a fact that, if a spin-1/2 particle is entangled with anything else in the universe, then the state of the particle is mixed, even if the state of the entire universe is pure. So a mixed state for a single particle suggests nothing philosphically, since we can still believe that the universe is in a pure state, which causes no problems for MWI. Indeed, endoself immediately looks at situations where the particle is so entangled! I should have taken this as a sign that my example was not doing its job.

I still stand by my responses to endoself, as far as they go. One of the minor attractions of the Bayesian interpretation for me is that it treats the entire universe and single particles in the same way; you don't have to constantly remind yourself that the system of interest is entangled with other systems that you'd prefer to ignore, in order to correctly interpret statements about the system. But it doesn't get at the real point.

The real point is that the entire universe is in a mixed state; I need to establish this. In the Bayesian interpretation, this is certainly true (since I don't have maximal information about the universe). According to MWI, the universe is in a pure state, but we don't know which. (I assume that you, the reader, don't know which; if you do, then please tell me!) So let's suppose that |psi> and |phi> are two states that the universe might conceivably be in (and assume that they're orthogonal to keep the math simple). Then if you believe that the real state of the universe is |psi> with 50% chance and |phi> with 50% chance, then this is a very different belief than the belief that it's (|psi> + |phi>)/sqrt(2) with 50% chance and (|psi> - |phi>)/sqrt(2) with 50% chance. Yet these two different beliefs lead to identical predictions, so you're drawing a map with extra irrelevant detail. In contrast, in the fully Bayesian interpretation, these are just two different ways of describing the same map, which is completely specified upon giving the density matrix (|psi><phi|)/2.

Edit: I changed uses of ‘world’ to ‘universe’; the former should be reserved for its technical sense in the MWI.

As a mathematical statement, what Baez says is certainly correct.

I definitely don't disagree with that.

Specifically, if you measure the spin of the particle along any axis, both maps predict that you will measure the spin to be in one direction with 50% probability and in the other direction with 50% probability.

They can give different predictions. Maybe I can ask my friend who prepared they quantum state and ey can tell me which it really is. I might even be able to use that knowledge to predict the current state of the apparatus ey used to prepare the particle. Of course, it's also possible that my friend would refuse to tell me or that I got the particle already in this state without knowing how it got there. That would just be belief in the implied invisible. "On August 1st 2008 at midnight Greenwich time, a one-foot sphere of chocolate cake spontaneously formed in the center of the Sun; and then, in the natural course of events, this Boltzmann Cake almost instantly dissolved." I would say that this hypothesis is meaningful and almost certainly false. Not that it is "meaningless". Even though I cannot think of any possible experimental test that would discriminate between its being true, and its being false.

A final possibility is that there never was a pure state; the universe started off in a mixed state. In this example, whether this should be regarded as an ontologically fundamental mixed state or just a lack of knowledge on my part depends on which hypothesis is simpler. This would be too hard to judge definitively given our current understanding.

What are the Born probabilities, probabilities of? Here's the map - where's the territory?

In MWI, the Born probabilities aren't probabilities, at least not is the Bayesian sense. There is no subjective uncertainty; I know with very high probability that the cat is both alive and dead. Of course, that doesn't tell us what they are, just what they are not.

We all-Bayesians are united with the many-worlders (and the Bohmians) in decrying the mystical separation of the world into ‘quantum’ and ‘classical’ and the reality of the ‘collapse of the wavefunction’.

I think a large majority of physicists would agree that the collapse of the wavefunction isn't an actual process.

How would you analyze the Wigner's friend thought experiment? In order for Wigner's observations to follow the laws of QM, both versions of his friend must be calculated, since they have a chance to interfere with each other. Wouldn't both streams of conscious experience occur?

They can give different predictions. [...]

I don't understand what you're saying in these paragraphs. You're not describing how the two situations lead to different predictions; you're describing the opposite: how different set-ups might lead to the two states.

Possibly you mean something like this: In situation A, my friend intended to prepare one spin-down particle, but I predict with 50% chance that they hooked up the apparatus backward and produced a spin-up particle instead. In situation B, they intended to prepare a spin-right particle, with the same chance of accidental reversal. These are different situations, but the difference lies in the apparatus, my friend's mind, the lab book, etc, not in the particle. It would be much the same if I knew that the machine always produced a spin-up particle and the up/down/right/left dial did nothing: the situations are different, but not because of the particle produced. (However, in this case, the particle is not even entangled with the dial reading.)

A final possibility is that there never was a pure state; the universe started of in a mixed state.

I especially don't know what you mean by this. The states that most people talk about when discussing quantum physics (including Eliezer in the Sequence) are pure states, and mixed states are probabilistic mixtures of these. If you're a Bayesian when it comes to classical probability (even if you believe in the wave function when it comes to purely quantum indeterminacy), then you should never believe that the real wave function is mixed; you just don't know which pure state it is. Unless you distinguish between the map where the particle is spin-up or -down with equal odds from the map where the particle is definitely in the fullymixed state in the territory? Then you have an even greater plethora of distinctions between maps that pay no rent!

How would you analyze the Wigner's friend thought experiment?

For Schrödinger's Cat or Wigner's Friend, in any realistic situation, the cat or friend would quickly decohere and become entangled in my observations, leaving it in a mixed state: the common-sense situation where it's alive/happy/etc with 50% chance and dead/sad/etc with 50% chance. (Quantum physics should reproduce common sense in situations where that applies, and killing a cat with radioactive decay or a pseudorandom coin flip doesn't matter to the cat --ordinarily.) However, if we imagine that we keep the cat or friend isolated (where common sense doesn't apply), then it is in a superposition of these instead of a mixture --from my point of view. My friend's state of knowledge is different, of course; from that point of view, the state is completely determined (with or without decoherence). And how is it determined? I don't know, but I'll find out when I open the door and ask.

I don't understand what you're saying in these paragraphs. You're not describing how the two situations lead to different predictions; you're describing the opposite: how different set-ups might lead to the two states.

I did not explain this very well. My point was that when we don't know the particle's spin, it is still a part of the simplest description that we have of reality. It should not be any more surprising that a belief about a quantum mechanical state does not have any observable consequences than that a belief about other parts of the universe that cannot be seen due to inflation does not have any observable consequences.

Unless you distinguish between the map where the particle is spin-up or -down with equal odds from the map where the particle is definitely in the fullymixed state in the territory? Then you have an even greater plethora of distinctions between maps that pay no rent!

I included this just in case a theory that implies such a thing ever turn out to be simpler than alternatives. I thought this was relevant because I mistakenly thought that you had mentioned this distinction.

And how is it determined? I don't know, but I'll find out when I open the door and ask.

What if your friend and the cat are implemented on a reversible quantum computer? The amplitudes for your friend's two possible states may both affect your observations, so both would need to be computed.

My point was that when we don't know the particle's spin, it is still a part of the simplest description that we have of reality.

Sure, the spin of the particle is a feature of the simplest description that we have. Nevertheless, no specific value of the particle's spin is a feature of the simplest description that we have; this is true in both the Bayesian interpretation and in MWI.

To be fair, if reality consists only of a single particle with spin 1/2 and no other properties (or more generally if there is a spin-1/2 particle in reality whose spin is not entangled with anything else), then according to MWI, reality consists (at least in part) of a specific direction in 3-space giving the axis and orientation of the particle's spin. (If the spin is greater than 1/2, then we need something a little more complicated than a single direction, but that's probably not important.) However, if the particle is entangled with something else, or even if its spin is entangled with some another property of the particle (such as its position or momentum), then the best that you can say is that you can divide reality mathematically into various worlds, in each of which the particle has a spin in a specific direction around a specific axis.

(In the Bohmian interpretation, it is true that the particle has a specific value of spin, or rather it has a specific value about any axis. But presumably this is not what you mean.)

As for which is the simplest description of reality, the Bayesian interpretation really is simpler. To fully describe reality as best I can with the knowledge that I have, in other words to write out my map completely, I need to specify less information in the fully Bayesian interpretation (FBI) than in MWI with Bayesian classical probability on top (MWI+BCP). This is because (as in the toy example of the spin-1/2 particle) different MWI+BCP maps correspond to the same FBI map; some additional information must be necessary to distinguish which MWI+BCP map to use.

If you're an objective Bayesian in the sense that you believe that the correct prior to use is determined entirely by what information one has, then I can't even tell how one would ever distinguish between the various MWI+BCP maps that correspond to a given FBI map. (A similar problem occurs if you define probability in terms of propensity to wager, since there is no way to settle the wagers.) Even if I ask my friend who prepared the state, my friend's choice to describe it one way rather than another way only gives me information about other things (the apparatus, my friend's mind, their lab book, etc). It may be possible to always choose a most uniform MWI+BCP map (in the toy example, a uniform probability distribution over the sphere); I'll have to think about this.

For the record, I do believe in the implied invisible, if it really is implied by the simplest description of reality. In this case, it's not.

I mistakenly thought that you had mentioned this distinction.

I certainly didn't mean to; from my point of view, that makes the MWI only more ridiculous, and I don't want to attack a straw man.

What if your friend and the cat are implemented on a reversible quantum computer? The amplitudes for your friend's two possible states may both affect your observations, so both would need to be computed.

So compute both. There are theoretical problems with implementing an observer on a reversible computer (quantum or otherwise), because Bayesian updating is not reversible; but from my perspective, I'll compute my state and believe whatever that comes out to.

Probably I don't understand what your question here really is. Is there a standard description of the problem of Wigner's friend on a quantum computer, preferably together with the WMI resolution of it, that you can link to or write down? (I can't find one online with a simple search.)

Specifically, if you measure the spin of the particle along any axis, both maps predict that you will measure the spin to be in one direction with 50% probability and in the other direction with 50% probability.

On the other hand, if the particle is spin up, the probability of observing "up" in an up-down measurement is 1, while the probability is 0 if the particle is down. So in the case of an up-down prior, observing "up" changes your probabilities, while in the case of a left-right prior, it does not.

That's a good point. It seems to me another problem with the MWI (or specifically, with Bayesian classical probability on top of quantum MWI) that making an observation could leave your map entirely unchanged.

However, in practice, followers of MWI have another piece of information: which world we are in. If your prior is 50% left and 50% right, then either way you believe that the universe is a superposition of an up world and a down world. Measuring up tells you that we are in the up world. For purposes of future predictions, you remember this fact, and so effectively you believe in 100% up now, the same as the person with the 50% up and 50% down prior. Those two half-Bayesians disagree about how many worlds there are, but not about what the up world —the world that we're in— is like.

To be precise, if your prior is 50% left and 50% right, then you generally believe that the world you are in is either a left world or a right world, and you don't know which. A left or right world itself factorises into a tensor product of (rest of the world) × (superposition of up particle and down particle). Measuring the particle along the up/down axis causes the rest of the world to be become entangled with the particle along that axis, splitting it into two worlds, of which you observe yourself to be in the 'up' one.

Of course, observing the particle along the up/down axis tells you nothing about whether its original spin was left or right, and leaves you incapable of finding out, since the two new worlds are very far apart, and it's the phase difference between those two worlds that stores that information.

The wave function exists in the map, not in the territory.

That is not an uncontroversial fact. For instance, Roger Penrose, from the Emperor's New Mind

OBJECTIVITY AND MEASURABILITY OF QUANTUM STATES Despite the fact that we are normally only provided with probabilities for the outcome of an experiment, there seems to be something objective about a quantum-mechanical state. It is often asserted that the state-vector is merely a convenient description of 'our knowledge' concerning a physical system or, perhaps, that the state-vector does not really describe a single system but merely provides probability information about an 'ensemble' of a large number of similarly prepared systems. Such sentiments strike me as unreasonably timid concerning what quantum mechanics has to tell us about the actuality of the physical world. Some of this caution, or doubt, concerning the 'physical reality' of state-vectors appears to spring from the fact that what is physically measurable is strictly limited, according to theory. Let us consider an electron's state of spin, as described above. Suppose that the spin-state happens to be |a), but we do not know this; that is, we do not know the direction a in which the electron is supposed to be spinning. Can we determine this direction by measurement? No, we cannot. The best that we can do is extract 'one bit' of information that is, the answer to a single yes no question. We may select some direction P in space and measure the electron's spin in that direction. We get either the answer YES or NO, but thereafter, we have lost the information about the original direction of spin. With a YES answer we know that the state is now proportional to |p), and with a NO answer we know that the state is now in the direction opposite to p. In neither case does this tell us the direction a of the state before measurement, but merely gives some probability information about a. On the other hand, there would seem to be something completely objective about the direction a itself, in which the electron 'happened to be spinning' before the measurement was made For we might have chosen to measure the electron's spin in the direction a -and the electron has to be prepared to give the answer YES, with certainty, if we happened to have guessed right in this way! Somehow, the 'information' that the electron must actually give this answer is stored in the electron's state of spin. It seems to me that we must make a distinction between what is 'objective' and what is 'measurable' in discussing the question of physical reality, according to quantum mechanics. The state- vector of a system is, indeed, not measurable, in the sense that one cannot ascertain, by experiments performed on the system, This objectivity is a feature of our taking the standard quantum-mechanical formalism seriously. In a non-standard viewpoint, the system might actually 'know', ahead of time, the result that it would give to any measurement. This could leave us with a different, apparently objective, picture of physical reality. precisely (up to proportionality) what that state is; but the state vector does seem to be (again up to proportionality) a completely objective property of the system, being completely characterized by the results that it must give to experiments that one might perform. In the case of a single spin-one-half panicle, such as an electron, this objectivity is not unreasonable because it merely asserts that there is some direction in which the electron's spin is precisely defined, even though we may not know what that direction is. (However, we shall be seeing later that this 'objective' picture is much stranger with more complicated systems- even for a system which consists merely of a pair of spin-one-half particles. ) But need the electron's spin have any physically defined state at all before it is measured? In many cases it will not have, because it cannot be considered as a quantum system on its own; instead, the quantum state roust generally be taken as describing an electron inextricably entangled with a large number of other particles. In particular circumstances, however, the electron (at least as regards its spin) can be considered on its own. In such circumstances, such as when its spin has actually previously been measured in some (perhaps unknown) direction and then the electron has remained undisturbed for a while, the electron does have a perfectly objectively defined direction of spin, according to standard quantum theory. resolved. The possible relevance of quantum effects to brain function will be considered in the final two chapters.

There are other interpretations of quantum mechanics, but they don't make any sense.

In you opinion. Many Worlds does not make sense in the opinions of its critics. You are entitled to back an interpretation as you are entitled to back a football team. You are not entitled to portray your favourite interpretation of quantum mechanics as a matter of fact. If interpretations were proveable, they wouldn't be called interpretations.

As I understand it, EY's commitment to MWI is a bit more principled than a choice between soccer teams. MWI is the only interpretation that makes sense given Eliezer's prior metaphysical commitments. Yes rational people can choose a different interpretation of QM, but they probably need to make other metaphysical choices to match in order to maintain consistency.

MWI distinguishes itself from Copenhagen by making testable predictions. We simply don't have the technology yet to test them to a sufficient level of precisions as to distinguish which meta-theory models reality.

See: http://www.hedweb.com/manworld.htm#unique

In the mean time, there are strong metaphysical reasons (Occam's razor) to trust MWI over Copenhagen.

In the mean time, there are strong metaphysical reasons (Occam's razor) to trust MWI over Copenhagen.

Indeed there are, but this is not the same as strong metaphysical reasons to trust MWI over all alternative explanations. In particular, EY argued quite forcefully (and rightly so) that collapse postulates are absurd as they would be the only "nonlinear, non CPT-symmetric, acausal, FTL, discontinuous..." part of all physics. He then argued that since all single-world QM interpretations are absurd (a non-sequitur on his part, as not all single-world QM interpretations involve a collapse), many-worlds wins as the only multi-world interpretation (which is also slightly inaccurate, not that many-minds is taken that seriously around here). Ultimately, I feel that LW assigns too high a prior to MW (and too low a prior to bohmian mechanics).

It's not just about collapse - every single-world QM interpretation either involves extra postulates, non-locality or other surprising alterations of physical law, or yields falsified predictions. The FAQ I linked to addresses these points in great detail.

MWI is simple in the Occam's razor sense - it is what falls out of the equations of QM if you take them to represent reality at face value. Single-world meta-theories require adding additional restrictions which are at this time completely unjustified from the data.

He still shouldn't be stating it as a fact when it based on "commitments".

Yes rational people can choose a different interpretation of QM, but they probably need to make other metaphysical choices to match in order to maintain consistency.

Aumann's agreement theorem.

Aumann's agreement theorem.

assumes common priors, i.e., a common metaphysical commitment.

However, Robin Hanson has presented an argument that Bayesians who agree about the processes that gave rise to their priors (e.g., genetic and environmental influences) should, if they adhere to a certain pre-rationality condition, have common priors.

The metaphysical commitment necessary is weaker than it looks.

This theorem (valuable though it may be) strikes me as one of the easiest abused things ever. I think Ayn Rand would have liked it: if you don't agree with me, you're not as committed to Reason as I am.