Are different interpretations of probability equivalent?

[kered rettop]

There appear to be at least three concepts of probability. In my words
frequencies in a history (theretical and measured)
reasonable expectation
propensity to outcomes
There may be more.
I am wondering whether these are actually different meanings, which could affect how probability is used in an argument, or whether they can be shown to be equivalent?

 

[Dale]

As long as any given interpretation of probability is consistent with the Kolmogorov axioms then they will be equivalent in the sense that you can use the same proofs and theorems.

 

[.Scott]

Each of these concepts of probability is loosely tied to different methods for computing the probability. So they can be different.

The historical frequency is a measured or estimated value - and the resulting probability value is described as the result of the documented methodology behind that measurement or estimation. One can then judge for oneself how appropriate it would be use for predicting similar events in the future.

"Reasonable expectation" and "Propensity to outcomes" are both future-looking values. "Reasonable expectation" (Bayesian) are generally evidence based on historical frequency. The notion is that uncertainty of the result is due to a lack of complete information - and so an estimate is based on what information is available. Notice that "reasonable" and "expectations" are human-centered concepts.

"Propensity of outcomes" implies that a model can be made of what is being measured and that that model explains the distribution of outcomes. So, for example, a fair coin will turn up heads 50% of the time and tails 50% of the time. If the historic frequency is several standard deviations from this result, then propensity of outcomes challenges whether those experiments were properly conducted and scored and whether a fair coin was used.

 

[kered rettop]

As long as any given interpretation of probability is consistent with the Kolmogorov axioms then they will be equivalent in the sense that you can use the same proofs and theorems.

I should have known better than to ask on a maths forum. The question arose because a certain interpretation of quantum mechanics has been criticised for the way it uses probability. I haven't the faintest idea whether the model is consistent with the Kolmorogov axioms or even how to go about finding out. Thanks for trying though.

 

[FactChecker]

I should have known better than to ask on a maths forum.

If you ask a probability question, expect a probability theory answer. Komogorov formalized the subject. As far as I know, there is no other answer that would withstand serious scrutiny. Did you want a serious answer or just some vague hand-waving?

 

[Dale]

I should have known better than to ask on a maths forum. The question arose because a certain interpretation of quantum mechanics has been criticised for the way it uses probability. I haven't the faintest idea whether the model is consistent with the Kolmorogov axioms or even how to go about finding out. Thanks for trying though.

Well, the first axiom is that $P(E)\ge 0$, so the QM model is different. Many of the theorems used in normal probability will not transfer.

 

[kered rettop]

If you ask a probability question, expect a probability theory answer. Komogorov formalized the subject. As far as I know, there is no other answer that would withstand serious scrutiny. Did you want a serious answer or just some vague hand-waving?

What I wanted was a serious answer that I would undertand. What I found out was that the answer was beyond my ability to understand right now. So what I wanted turned out to be impossible. To me, that's a good outcome, so I thanked Dale for the answer.

Your point?

 

[kered rettop]

Well, the first axiom is that $P(E)\ge 0$, so the QM model is different. Many of the theorems used in normal probability will not transfer.

Hmm. Does QM actually require negative probabilities? I don't know any complex functions whose modulus-squared is negative! I know of some arguments (which I read ages ago) which attempt to break down a single probability into terms representing "familiar" possibilities which then need a correction term which is negative. But that term isn't for a possible outcome. I fancy the example I saw was for entanglement and EPR correlations. Is this what you meant?

 

[Dale]

Does QM actually require negative probabilities? I don't know any complex functions whose modulus-squared is negative!

Hmm, you could be right. My statistics is much better than my QM. So if probability in QM does satisfy the axioms then it is equivalent. If it doesn’t then it isn’t equivalent.

 

[FactChecker]

What I wanted was a serious answer that I would undertand. What I found out was that the answer was beyond my ability to understand right now. So what I wanted turned out to be impossible. To me, that's a good outcome, so I thanked Dale for the answer.

Your point?

Sorry. I misinterpreted your post.

 

[kered rettop]

Hmm, you could be right. My statistics is much better than my QM. So if probability in QM does satisfy the axioms then it is equivalent. If it doesn’t then it isn’t equivalent.

Fair enough. I guess I'll have to work through Kolmogorov for myself. I am not optimistic about the outcome.

 

[Dale]

Fair enough. I guess I'll have to work through Kolmogorov for myself. I am not optimistic about the outcome.

Oh, I should have posted this in the beginning:
https://en.m.wikipedia.org/wiki/Probability_axioms

 

[kered rettop]

Oh, I should have posted this in the beginning:
https://en.m.wikipedia.org/wiki/Probability_axioms

Yeah, thanks. I still have difficulty relating axiomatic probability theory to the arguments about probability which vex physicists. I'm going to have to drop the subjecrt or I'll spend far too long on it than the family will tolerate.

 

[gentzen]

Hmm. Does QM actually require negative probabilities? I don't know any complex functions whose modulus-squared is negative! I know of some arguments (which I read ages ago) which attempt to break down a single probability into terms representing "familiar" possibilities which then need a correction term which is negative. But that term isn't for a possible outcome. I fancy the example I saw was for entanglement and EPR correlations. Is this what you meant?

There is a reason why Kolmogorov introduced his probability axioms in 1933: Eugene Wigner had introduced his Wigner quasiprobability distribution in 1932. This is a "partial" Fourier transform of the density operator $\rho(x,x')$: $$W(x,p):=\int \rho(x+\frac{y}{2},x-\frac{y}{2})\exp(ipy)dy$$ It is a real function (not complex, unlike $\rho$), which satisfies all properties of a probability distribution, except non-negativity. Feynman famously tried to interpret those negative probabilities. In fact, one motivation for his proposal of quantum computation was to clarify whether this could work at all. The MWI aspect of quantum computation really had to wait for David Deutsch:

Comment #231 December 8th, 2023 at 5:57 am
After watching the interview, I decided to read Feynman’s Simulating Physics with Computers from June 1982. Wow, I didn’t expect that he would spend so much time talkling about negative probabilities (and even less that this predates his longer exposition from 1987). Looks like when it comes to interpretations of quantum mechanics, Feynman is on the “let’s try to better understand quantum mechanics” (and especially how its “probabilities” differ from probabilities) side

we always have had a great deal of difficulty in understanding the world view that quantum mechanics represents. At least I do, because I’m an old enough man that I haven’t got to the point that this stuff is obvious to me. Okay, I still get nervous with it. And therefore, some of the younger students … you know how it always is, every new idea, it takes a generation or two until it becomes obvious that there’s no real problem. It has not yet become obvious to me that there’s no real problem. I cannot define the real problem, therefore I suspect there’s no real problem, but I’m note sure there’s no real problem. So that’s why I like to investigate things. Can I learn anything from asking this question about computers–about this may or may not be mystery as to what the world view of quantum mechanics is?

and not so much on the many-worlds side

There are all kinds of questions like this, and what I’m trying to do is to get you people who think about computer-simulation possibilities to pay a great deal of attention to this, to digest as well as possible the real answers of quantum mechanics, and see if you can’t invent a different point of view than the physicists have had to invent to describe this. In fact the physicists have no good point of view. Somebody mumbled something about a many-world picture, and that many-world picture says that the wave function ψ is what’s real, and damn the torpedos if there are so many variables, NR. All these different worlds and every arrangement of configurations are all there just like our arrangement of configurations, we just happen to be sitting in this one. It’s possible, but I’m not very happy with it.

And that is basically what I wanted to learn from reading it, namely whether this close connection between MWI and QC was there already before David Deutsch.

So Feynman's "only" publications on negative probabilities are from 1982 and 1987. Both publications are easy to find online.

 

[kered rettop]

Sorry. I misinterpreted your post.

No problem.

 

[kered rettop]

There is a reason ...

That's interesting but it's not so much another interpretation of probability, it's actually quasi probability. It may work the same way in the maths, but cannot be used in an ontic model. Which is what I wanted to know about, as you have probably guessed! Thanks for the info.

 

[gentzen]

That's interesting but it's not so much another interpretation of probability, it's actually quasi probability.

My post was not intended as an answer to your initial question, but as a clarification regarding the relation between negative probabilities and QM, which came up in your exchange with Dale on whether quantum mechanics is consistent with the Kolmorogov axioms:

Well, the first axiom is that $P(E)\ge 0$, so the QM model is different. Many of the theorems used in normal probability will not transfer.

Does QM actually require negative probabilities? I don't know any complex functions whose modulus-squared is negative!

Hmm, you could be right. My statistics is much better than my QM. So if probability in QM does satisfy the axioms then it is equivalent. If it doesn’t then it isn’t equivalent.

It may work the same way in the maths, but cannot be used in an ontic model. Which is what I wanted to know about, as you have probably guessed! Thanks for the info.

You mean quasi probability may work the same way in the maths as probability? Well, some theorems will work for both, some won't. (Even so it is true that the Wigner function allows to give a formulation of QM equivalent to more familiar formulations, I guess that is not what you ment.)

But why do you mention "an ontic model"? This seems to stray even farther from your initial question. Your reaction to my post indicates that something about it made you "Sad", but it is unclear to me, whether

• you are sad about Feynman giving more importance to negative probabilities than to MWI
• you are sad about my questionable suggestions regarding motivations of Kolmogorov and Feynman
• you are sad because my post didn't answer your initial question
• you are sad because my post didn't answer what your really wanted to know about QM, deep inside

 

[kered rettop]

My post was not intended as an answer to your initial question, but as a clarification regarding the relation between negative probabilities and QM, which came up in your exchange with Dale on whether quantum mechanics is consistent with the Kolmorogov axioms:

I realized that. And yes it did.

But why do you mention "an ontic model"? This seems to stray even farther from your initial question.

Thank you for raising the question. Having considered it, I agree with your implication. The fact that a negative probability has no direct physical meaning, does not preclude its being used. However, I still don't know whether the axiomatic theory is another interpretation of probability. The wiki article just plunges straight into talking about probability as if everyone knows what it means.

Your reaction to my post indicates that something about it made you "Sad", but it is unclear to me, whether

• you are sad about Feynman giving more importance to negative probabilities than to MWI
• you are sad about my questionable suggestions regarding motivations of Kolmogorov and Feynman
• you are sad because my post didn't answer your initial question
• you are sad because my post didn't answer what your really wanted to know about QM, deep inside

I genuinely cannot remember. I suspect it was the fifth option. I've removed it.

 

[Dale]

The wiki article just plunges straight into talking about probability as if everyone knows what it means.

Probability means anything that satisfies those axioms. That is the great thing about axioms, in general.

For example, there are some axioms for vector spaces. Anything that follows those axioms is a vector. So acceleration is a vector, but so are polynomials. When talking about vectors you don’t have to worry about if you are talking about acceleration or polynomials.

Same with all interpretations of probability that are consistent with the axioms. It doesn’t matter if you are talking about frequencies or propensities.

 

[romsofia]

I think the first chapter of this book will help you understand, as the first paragraph does address your question as a negative, and he goes on to explain the different ideals: https://www.arielcaticha.com/my-book-entropic-physics

But note that Caticha has his own interpretation of QM, but at least he does attempt to express the confusion of probability in this text that is still a work in progress. The first chapter is only 7 pages long, and you could probably skip the last section on entropic physics.

 

[kered rettop]

Hmm, you could be right. My statistics is much better than my QM. So if probability in QM does satisfy the axioms then it is equivalent. If it doesn’t then it isn’t equivalent.

Probability means anything that satisfies those axioms. That is the great thing about axioms, in general.

For example, there are some axioms for vector spaces. Anything that follows those axioms is a vector. So acceleration is a vector, but so are polynomials. When talking about vectors you don’t have to worry about if you are talking about acceleration or polynomials.

Same with all interpretations of probability that are consistent with the axioms. It doesn’t matter if you are talking about frequencies or propensities.

So, would it be correct to say that axiomatic probability theory cannot be invoked to distinguish between different interpretations (except in the sense of questioning whether a particular interpretation actually does talk about probabilities at all)?

 

[Dale]

So, would it be correct to say that axiomatic probability theory cannot be invoked to distinguish between different interpretations (except in the sense of questioning whether a particular interpretation actually does talk about probabilities at all)?

I would agree with that.

 

[kered rettop]

I would agree with that.

Well, that's very satisfactory. Thanks. Back to the thread that prompted my question!

 

[kered rettop]

I think the first chapter of this book will help you understand, as the first paragraph does address your question as a negative, and he goes on to explain the different ideals: https://www.arielcaticha.com/my-book-entropic-physics

But note that Caticha has his own interpretation of QM, but at least he does attempt to express the confusion of probability in this text that is still a work in progress. The first chapter is only 7 pages long, and you could probably skip the last section on entropic physics.

Well the first chapter really only sets out the motivation for constructing a framework. I can hardly disagree. But whether having a bullet-proof framework is necessary before one can offer coherent and uncontroversial answers to questions like mine is not so clear. I can imagine a future discussion in PF, in the year 2068, going like this:

Q.
I've just measured some spins, and got 4,782,223 "ups" and 5,217,777 "downs". The experiment was to test a theory that postulates an eight physical force. It should result in a 50/50 ratio if the force exists and an all-"up" one (determined by the equipment) if not. Unfortunately, the argument relies on probability being frequencies. But I cannot formulate the theory in terms of anything other than propensities. Can I safely assume that propensities and frequencies are equivalent in meaning so this is not a potential problem? I just want to be sure that I can use classical statistics to justify concluding that the force is probably real.

A.
Sorry, we're still waiting for Caticha to finish his book.

 

[Dale]

Can I safely assume that propensities and frequencies are equivalent in meaning so this is not a potential problem?

I wouldn’t say that there is no potential problem. While accelerations and polynomials are both vector spaces and they are equivalent in terms of all of the theorems of vectors that doesn’t mean that you can use an acceleration anywhere a polynomial is needed. Nor does it mean that you can substitute one for the other in every case without any problem. You have to decide what “equivalent” means for your purpose. That is why I was clear about my meaning above.

 

[kered rettop]

I wouldn’t say that there is no potential problem.

Nor would I.

While accelerations and polynomials are both vector spaces and they are equivalent in terms of all of the theorems of vectors that doesn’t mean that you can use an acceleration anywhere a polynomial is needed. Nor does it mean that you can substitute one for the other in every case without any problem. You have to decide what “equivalent” means for your purpose. That is why I was clear about my meaning above.

Yes, of course. Don't confuse my hypothetical questioner's worries with anything I may think! You can safely assume that while the fictional user is needlessly fretting about applying probability theory to his experimental results in 2048, I'm just asking my own questions here.