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The Bayesian interpretation of quantum mechanics is correct. So there!
Toby, I see you are editing this very minute, so maybe the following may be resolved once you see this here.
But presently when I look at the entry it seems to me that the very explanation of what is Bayesian about it all is currently missing. Currently the section “Formalism” describes in which sense a suitable algebra may be thought of as a non-associative analog of an algebra dual to a space of probability measures.
There ought to be a line following this about what specifially Bayes is meant to have to do with this.
I added some more to Bayesian reasoning.
Where the text says
People have implied […] that this is what Niels Bohr meant all along when he put forth the Copenhagen interpretation
I have appended
(for more on this suggestion see also at Bohr topos).
I still feel the entry presently does not say what its title announces. I suppose you want to add a line roughly like the following, at the end of the section “Formalism”:
This allows to systematically think of quantum observables as random variables. There have been many debates about what that means. In hidden variable theory one supposes that it means that there is an underlying ensemble of which these probability variables are coarse grained expectation values. This would give an actual “frequentist” interpretation to the probabilities appearing here. But it has also been argued that these probabilities are nothing but estimates of the available subjective knowledge of the system, and this may be related to the Bayesian interpretation to probability.
Or something like this.
My editing is done for today. What you were missing is now under Interpretation, although it is rather too brief if one isn't already familiar with Bayesianism in classical probability.
Is it really defensible to say that “the state vector is the map and not the territory” when there is no territory? For instance, saying “this collapse takes place in the map, not the territory; it is not a physical process” implies that “physical process” means something that takes place in the territory rather than the map; but then doesn’t this argument show that there are no “physical processes” at all?
Mike,
it is typical for these debates to remain inconclusive, but let’s just isolate what the point of saying “Bayesian” here is supposed to be:
So the point is that a) when a Bayesian makes a measurement that decides the actual value of a probability variable, then he “updates his prior” to be a probability distribution delta-peaked at that value and then prodeeds by time-evolving this probability distribution from there on by Bayes’ law and b) this may seem to be exactly what happens to physicists when they see their “wavefunction collapse” and then proceed time-evolving it with Schrödinger’s equation from there. So the idea is that b) ist just an instance of a).
(I am not sure if this “interpretation of wave function collapse” derserves to be called an “interpretation of quantum mechanics”, since the reason why there are just probabilties in the first place is untouched by invoking Bayes and this is what is mostly what makes people invoke “interpretations of quantum mechanics”.)
On the other hand, it is known that coupling a big quantum mechanical system (e.g. an observer) to a tiny one (i.e. an particle in a detector) DOES lead to a dynamical collapse-like reduction of the tiny system’s state to a “classical mixture”, and this happens purely dynamically by Schrödinger’s equation applied to both the system and its observer at once, as it should be. This insight goes by the name “decoherence” and when it was figured out long ago some people thought this settles the issue. But debates about this do still continue, remain inconclusive and maybe always will. The Wikipedia entry Wave function collapse is actually pretty good. It explains this decoherence-explanation of the “collapse”.
I should add that decoherence only explains the reduction (collapse) from quantum probability to classical probability. To the classical probability distribution that remains after decoherence one will of course tend to want to apply somethingy like Bayesian reasoning again.
I’m sorry, but I don’t really see how what you said answers my question. (-: How about this for a more specific question: according to the point of view presented on the page, what is an example of a “physical process”?
I didn’t try to answer your question, I tried to clarify what this thread is about (or ought to be about, judging from its title), namely the idea/suggestion that ’wave function collapse’ is an instance of updating Bayesian priors.
For a decent discussion of what a physical process in (quantum) theory is I would invoke a decent formal foundations of (quantum) physics in higher topos theory and then see what that tells us.
Ok. Maybe Toby can answer my question, since he wrote the bit that confused me.
Reading only Mike #6 so far: I like to say that there is a territory, but it cannot be described as we would like, as a function from the space of observables to the real line.
One might say, if the territory is not a function from the space of observables to the real line, then what is it? But of course, even in the days of classical physics (or today using a different interpretation of quantum physics), when you could describe it with such a function, you wouldn't say that the territory is this function, or at least you might not. Just because you have a perfect mathematical description of reality (even if you really do have one), that doesn't mean that reality is that mathematics. (To be sure, some people, such as Max Tegmark, take precisely the opposite position here.) Even a perfect map is not the territory.
So it's very interesting that an obvious idea as to how to describe reality completely doesn't exist, but that doesn't mean that we don't have reality. Reality is what it is, meanwhile we are just doing our best to understand it.
Then again, perhaps the territory can still be described using a different mathematical construct, say something that lives in a Bohr topos (which I say only because I haven't really figured out how a Bohr topos works yet). Even so, that would still be a map, not the territory.
Edit to answer Mike #9: all of the usual things that we think of as physical processes are still physical processes. Although I usually prefer the Heisenberg picture when thinking philosophically, ‘process’ evokes time, so let's use the Schrödinger picture, as Urs was doing. Then we can say that physical processes comprise the time evolution of the state. However, changing which state you use when you gain information is not that.
Edit: I just realized that this comment used the word ‘map’ in two subtly similar but fundamentally different ways. If this had been intentional, then it would have been a pun, but I'm afraid that it must have just been confusing. So for one sense I have replaced it with ‘function’.
I've decided that the formalism on this page is too precise. It's very natural, since probability theory is linked to measure spaces which are linked to their real $L^\infty$ spaces, which are $W^*$-algebras, or better $JBW$-algebras since only that structure is relevant even in the quantum case. But it's also fairly obscure, in two ways: using the Jordan multiplication of the real-valued observables instead of the full $W^*$-algebra with its commutators is rare (since the commutators are used to describe time evolution, if nothing else), and even using a $W^*$-algebra rather than an arbitrary $C^*$-algebra is rare (even if you don't restrict yourself to operators on a Hilbert space). So I should move that to JBW-algebras in quantum mechanics or something like that, then make the Formalism here more general.
Edit: Done.
Thanks for this. Toby, clicking through to localisable measurable space and measurable locale I am surprised not to find to connection to valuations on locales nor to the sheaf theoretic approach to measure theory. We have some information at Boolean topos and at Measurable spaces. Maybe we should discuss a bit before revising.
That rearrangement makes very good sense to me know. I have added cross-links vigorously.
It can be nice to describe the kinematics of a quantum system using a JBW-algebra.
Not the greatest of opening lines here in JBW-algebraic quantum mechanics.
What’s trying to be said? That there’s some kind of advantage to this formalism? Can we be specific?
That's why I wrote
This is hastily copied from elsewhere and minimally edited. More work should be done to spell this out. Also motivation.
What you're asking for is the motivation. I've put some in now.
Sorry, it’s just that after all these years I still have the words of my primary school teacher in my head, “Always look for an alternative to the word ’nice’.” She wasn’t too fond of ’got’ either.
Well, now that you've read the motivation, maybe you can replace it with a better word? That would be nice. (^_^)
Re: #12, the current wording on the page implies to me that collapse is “not a physical process” because it “takes place in the map, not the territory”. However, if the quantum state is a map and not the territory, then it seems to me that the time evolution of that state is also something taking place in the map and not the territory, and thus also ought not to be considered a “physical process”. That’s what I was trying to say.
The idea from the Bayesian point of view is that the evolution of that state/of the probability distribution is that which most closely accounts for the actual process. One uses Bayes’ law to update a prior given certain information, such as to stay as close to reality as possbible, and the idea is that this is what Schrödinger’s equation does for you in quantum mechanics, to update the best possible of your knowledge of the actual system, to keep the map as close to the territory as possible. That’s the idea.
I prefer the Heisenberg picture, in which the state doesn't evolve but the observables do. Then the time evolution is in the territory.^{1} Even in the Schrödinger picture, the time evolution there is merely the automatic reflection of this physical process in the map, rather than a change of the map itself. Collapse, in contrast, reflects a change in the observer's information, a real updating of their map.
Technically, philosophically, the algebra of observables is also part of the map, but it's a part that we generally pretend is perfect for the sake of doing physics. But we have to remember that it's also part of the map if what we're testing is the physical theory itself. Still, the time evolution that appears there is also supposed to be a reflection of a physical process in the territory, while collapse is not. ↩
Okay, that makes a tiny bit of sense. Maybe the page could be edited to clarify this?
Wait, should we maybe distinguish between the Bayesian and the Bartelsian interpretation of quantum mechanics?
The Bayesian interpretation is simply this: the idea that a) quantum states are thought of as analogous to “Bayesian probabilities” (priors), that b) time evolution of quantum states is the analog of Bayes’ rule for updating probabilities (for instance on slides 19-21 here) and that c) collapse of the wave function is the analog of a Bayesian updating his prior to be delta-peaked.
Time evolution by the Schrödinger equation is not an analogue of Bayes's Rule; collapse is. Slide 19 says ‘evolution due to measurement’, which is unfortunately ambiguous between collapse and evolution by the time-dependent Schrödinger equation incorporating a physical interaction with a measurement device. The simplest way to tell the difference, in my opinion, is to consider the (von Neumann) entropy. Applying Bayes's Rule classically reduces the entropy of a probability distribution; collapse also reduces the entropy. However, time evolution by the Schrödinger equation is unitary and so conserves total entropy.^{1}
Also, collapse is not generally collapse to a delta distribution; in fact, in quantum mechanics, there typically is no delta distribution (that's what makes it quantum rather than classical). Of course, if you measure an observable $O$ to have a particular value, then the probability distribution for that observable collapses to a delta distribution, but the distributions for the other observables generally won't. (And indeed, if $P$ does not commute with $O$, then the distributions for $O$ and $P$ cannot simultaneously be deltas.) Maybe this is all that you meant; but although people usually talk about measuring a particular value of an observable, you might also only measure that the value is within a certain range, and that still triggers the application of Bayes's Rule and a corresponding collapse of the wavefunction. Of course, measuring the value of $O$ to lie within the measurable set $E$ is equivalent to measuring the observable $\chi_E(O)$ to have the value $1$, so something is still collapsing to a delta; but I think that describing this as updating to a delta-peaked posterior^{2} is misleading.
The physical entropy that increases is not the von Neumann entropy but a coarse-grained entropy, as described at entropy#physical. ↩
This is not the prior, since it comes after the measurement, although it will be the prior for the next measurement. ↩
So the Bayesian interpretation as I understand it (following Urs's points a,b,c) is this:
Maybe I should put this in the article.
The phrase “analogous to” makes it make a whole lot more sense. I thought at first the claim was that quantum mechanics actually is a Bayesian probability system.
@ Mike #27:
But keep in mind that ‘analogous to’ is an understatement. I took care to write ‘generalization’ wherever I could (which was at least once per item). So it is not merely an analogy. The Bayesian interpretation of classical probability theory is literally a special case of the Bayesian interpretation of quantum mechanics, not just something analogous.
Or another way to avoid having merely an analogy: Given a state and an observable, you get an honest-to-goodness probability distribution (not merely an analogue) on the spectrum of the observable. So in particular, you get a real number between $0$ and $1$ from a state $\psi$, an observable $O$, and a measurable set $E$, which is (at least naively) the probability that the value of $O$ belongs to $E$, given that the world is in the state $\psi$. The Bayesian interpretation says that this number is to be interpreted directly as a Bayesian probability (not just an analogue).
It actually says more than this, because so far we have left open the nature of $\psi$ itself. The Bayesian interpretation goes on to say that $\psi$ is nothing more than a record of all of the associated probabilities as we vary $O$ and $E$ (rather than, say, an objective feature of external reality that nevertheless gives these probabilities by some law of nature and epistemology).
@Justin: Thanks! We should consider renaming the page, or least inserting some discussion of the literature on this point.
@Toby: Sure, but I think it’s important to avoid giving the impression of the converse, i.e. that the “Bayesian interpretation of quantum mechanics” is a special case of the previously existent notion of Bayesian probability. The point is that seeing the way in which quantum mechanics generalizes Bayesian probability can help us to develop an intuition for QM based on our previous intuition for Bayesian probability — not that QM is somehow “explained” in terms of notions from ordinary probability, which the word “interpretation” may suggest.
I disagree (in part) with Justin #28, but my computer just lost it all. (I forgot the lesson that I give others, to always use Itsalltext!) So I will have to rewrite it another time.
References for lost comment: Letter to Schack, objective Bayesians in 1994.
A really good exposition of psi-ontic vs. psi-epistemic and the import of the PBR Theorem is at Matt Leifer’s blog.
Here is where I disagree with Justin #28:
We have strict containments
$Quantum\; Bayesianism \subset Bayesian\; interpretation\; of\; quantum\; mechanics \subset epistemic\; interpretation\; of\; quantum\; mechanics .$(I'm leaving off the $\psi$ as hopefully understood.)
The article as written is about the middle of these. Justin seems to be claiming that it's about the one on the right and should be retitled, but it's not. The difference is that an epistemic interpretation allows for any epistemic interpretation of probability, while the article specifically requires a Bayesian interpretation of probability. (In fact, you could even take an ontic interpretation of probability, recovering an ontic interpretation of quantum mechanics, but presumably this is ruled out by saying “epistemic”.)
But perhaps Justin's point is simply that the article could be easily generalized to cover epistemic interpretations generally. This may be true, and then it probably should be, but I don't feel qualified to judge. Somebody else could edit it thus.
Justin also seems to be claiming that the containment on the left is not strict. That depends on what one means by “Quantum Bayesianism”, but with the capital letters (at least, the capital “Q”), I recognize that as a name used specifically by Christopher Fuchs and his collaborators. (Or they will even say “QBism”, making the term even more theirs, or perhaps even specifically Fuchs's.) As Justin says, this is based on “very specific philosophical commitments”, including (but not limited to) subjective Bayesianism. But as Bayesian interpretation of probability is much broader than this, so is Bayesian interpretation of quantum mechanics.
Again, perhaps Justin's claim is simply that the term “Bayesian”, in the context of interpretation of quantum mechanics, should be reserved for QBism. But then there is no way to identify an interpretation of quantum mechanics more specific than epistemic and less specific than QBism. As an objective Bayesian myself, and one whose interpretation of quantum mechanics is a generalization of my objective Bayesian interpretation of probability, I need such a term! And while I'm willing to let Fuchs et al have “QBism” (and even “Quantum Bayesianism” if capitalized), I won't surrender the term “Bayesian” to them entirely, not even in this context.
Anyway, here's the relevance of the references that I saved above:
In the letter to Schack that appears in the middle of page 12 of Quantum States: What the Hell Are They?, Fuchs says that a “lack of free choice” about what probability to assign in a state of knowledge renders something “non-Bayesian”. Since objective Bayesianism exists, this is not true. (The thing in question appears to be a relative-state interpretation of quantum mechanics, and while I certainly agree that the many-worlds interpretation is not Bayesian, I can't accept this argument for that conclusion; I'm not even sure that there isn't a way to make a relative-state interpretation Bayesian, since $many\; worlds \subset relative\; state$ is also a proper inclusion.)
In the 1994 Usenet conversation on Bayesianism in quantum mechanics, Bayesian interpreation of quantum mechanics is discussed, predating by several years the 2001 papers of Caves, Fuchs, and Schack. There are people in that conversation who explicitly identify as objective Bayesians. So subjective Bayesianism has no priority to the term “Bayesian” in the context of interpretation of quantum mechanics.
Actually, even CF&S refer to “states of knowledge” in their 2001 papers. Quantum States is largely Fuchs's reflections on why he now rejects that interpretation. But he published it once, and so he can hardly claim that the only Bayesianism is subjective Bayesianism, even in quantum mechanics. (And for what it's worth, I don't think that he really argues that, only that subjective Bayesian is somehow more authentically Bayesian (with which I disagree) and that he has come to believe in subjective Bayesianism himself.)
Above I objected to the last paragraph of Justin #28. But I like the other three paragraphs (as well as Justin #32), and we should perhaps put this in interpretation of quantum mechanics.
I've also some stuff to Bayesian interpretation of quantum mechanics about how its topic relates to the other things in my proper containment diagram above.
@Mike #30:
not that QM is somehow “explained” in terms of notions from ordinary probability, which the word “interpretation” may suggest
Of course QM as a whole cannot be thereby explained, but there is a sense in which I want to say that the interpretation is reduced entirely to the interpretation of ordinary probability. Specifically, the probability distributions $O_*\psi$ (for $O$ an observable and $\psi$ a state) are to be interpreted as ordinary probability distributions (which, for me, are interpreted as indicating one's knowledge of something, in this case of the value of $O$). The difference, of course, is that these cannot be combined into a single joint probability distribution; this makes QM non-classical. Thus, QM is not given by a notion in ordinary probability theory; nevertheless, QM is described by net of such notions.
I like the perspective that
QM is described by a net of … notions [in ordinary probability theory]
but I don’t think it implies that
the interpretation [of QM] is reduced entirely to the interpretation of ordinary probability.
The point is exactly that a net of notions is harder to interpret than a single one.
Since the interpretation in any case comes down to interpreting what $O_*\psi$ means (what I know about the value of the observable $O$ when my state of knowledge is given by $\psi$), the same interpretation seems to me to work for the whole net. But I understand that this might not satisfy others.
Sorry for the belated reaction.
Re Toby’s #25, #26 that only the collapse is analogous to Bayes law: true, all right.
One other quibble: does it really make sense to speak of “Bayesian probability”? It’s no different from “Kolmogorov probability”, except that one tends to accompany it by more of a story.
Yes, the term should really be ‘Bayesian interpretation of probability’ or ‘probability, with a Bayesian interpreration,’ or something like that.
Or even a link to the abstract page
Link quantum probability
added pointer to the recent
also to
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