Understanding Bell's logic

Gordon Watson

I am hoping it may be helpful to separate Bell's logic from Bell's mathematics
http://www.physicsforums.com/showthread.php?t=406372.

Understanding one may better help us understand the other.

Thank you Bill.

In the language that is evolving at "Understanding Bell's mathematics", http://www.physicsforums.com/showthread.php?t=406372, we have Alice with outcomes G or R (detector oriented a), Bob with outcomes G' or R' (detector oriented b).

H specifies an EPR-Bell experiment.

λ represents Bell's supposed [page 13] variables "which, if only we knew them, would allow decoupling ... " [of the outcomes].

Question: Why would Bell want to decouple outcomes which are correlated? Is he too focussed on separating variables?

Bell's λ would allow Bell to write -- consistent with with his (11) --

(11a) (P(GG'|H,a,b,λ) = P1(G|H,a,λ) P2(G'|H,b,λ).

So Bell's logic, as cited above in bold, leads him to suggest that

(11b) (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a)

would avoid some well-known inequalities.

I do not follow Bell's logic. I do not see that his move avoids any inequalities.

Note 1: a and b are not signals.

Note 2: Probability theory, widely seen as the logic of science, would have --

(11c) (P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a,G).

So, by comparison [Bell's (11b) with (11c)], Bell's (11b) and his logic is equivalent to dropping G from the conditionals on G'.

Which is equivalent to saying that G and G' are not correlated?

ThomasT

Ok, so Bell's logic was flawed. This was demonstated in the thread "Understanding Bell's Mathematics".

This has been known, and demonstrated, years ago.

Bottom line, few people care. If Bell's logic was flawed and if violations of Bell inequalities don't tell us anything about nature then ... so what.

Gordon Watson

Do doubters understand the full implication of Bell's lambda, and therefore the full implication of his logic?

Do supporters?

Probabilistic refutations do not impress his myriad supporters.

The difference might be in how one views the logic attached to Bell's lambda.


Ether-logic was flawed. Stomach-ulcer logic was flawed. ...

Much was learnt from the related experiments.

Including that the logic was flawed.

C'est la vie.

That's what.

ThomasT

Not to devalue your efforts, but my apprehension of the view of the physics community at large (garnered from conversations with dozens of working physicists over the years) is that Bell's theorem just isn't important.

If Bell was right then we have nonlocal or ftl influences that can't be detected or used for any conceivable purpose. If Bell was wrong, well, then he was just wrong. Nothing is affected either way (except wrt the agendas of a very small minority of physics professionals).

Nevertheless, it is satisfying to periodically revisit and dispell myths. And, I think that you and billschnieder have done a nice job in that regard.

I sensed that there was something not quite right about Bell's LR ansatz from the first time I saw it. But, lacking the requisite skills to communicate this clearly, I was only able to talk about my apprehension of it in rather vague terms.

So, I thank you. And don't let my previous post in this thread tarnish your efforts, or diminish the admiration I have wrt your ability to elucidate something which I intuitively saw but was unable communicate.

Gordon Watson

Dear Thomas,

Ok. Thank you. No worries at all. And please ...

Do not devalue your own efforts.

You and your P(AB|H) are the catalysts that prompted me to present my similar intuition, backed by some knowledge of probability theory, etc.

So thank you again,

and prepare for the storm,

Jenni

DrChinese

No, there is no known flaw in his logic. Please provide a peer reviewed reference that states this if you believe I am wrong. As I say over and over again, you must read it in the context he wrote it. If you don't like his derivation, there are plenty of other peer reviewed versions of it available. For example, Mermin. Or Aspect. Or Zeilinger.

Or even better, derive it for yourself. You will see that you can do it a variety of ways. You always use some variation of the following:

a) The setting at a does not affect the outcome at B, and vice versa.
b) P(A)+P(~A)=100%, and all variations of this with A, B and C simultaneously.
c) The QM prediction is cos^(theta).

Folks, please get a grip on this subject. Genovese does a review of Bell tests periodically, and his last review had over 500 peer-reviewed references in 100+ pages.

Research on Hidden Variable Theories: a review of recent progresses,
Marco Genovese (2005)
http://arxiv.org/abs/quant-ph/0701071

Do you seriously think that they just happened to overlook these "flaws" in Bell? If you do, publish a paper on it. Otherwise, I am going to point you back to Forum guidelines on personal theories. If you have a question, ask it. But quit making statements that are your pet opinions.

JesseM

No, it's not equivalent to saying G and G' are not correlated in a general sense, it just means that if you already know λ then learning G will give you no further information about the probability of G'. For example, if G and G' are correlated, but this correlation is explained entirely by a common cause that lies in the region where the past light cones of the two measurements overlap (the common cause might be that the two particles sent to either experimenter are always created with an identical set of hidden variables by the source), then if you already have the information about the common cause contained in λ (which could detail the set of hidden variables associated with each particle, for example) then in this case it will be true that learning G won't change your estimate of the probability of G'. This is precisely the sort of physical logic that Bell was using, and my argument in which λ was made to stand for all facts in the past light cones of the measurement events was an attempt to make this a bit more rigorous.

billschnieder

You are wrong, and JenniT is correct dropping G from
(P(GG'|H,a,b,λ) = P1(G|H,a,λ,b) P2(G'|H,b,λ,a,G)
means clearly that in the probability space defined by (H, a,b,λ) G and G' are not correlated. In other words under a given set of specific conditions ("H", "a","b","λ"), there will be no correlation between G and G'. It is a simple exercise to see if this is consistent with the EPR situation Bell was attempting to model. Your misunderstanding is fueled by a confusion between functional notation and probability notation. P(G'|H,b,λ,a,G) does not mean P2 is a function of (H,b,λ,a,G). It simply means the specific conditions (H,b,λ,a,G) define the probability space in which P(G') is calculated.

JesseM

Yes, but there can still be a correlation in the total probability space even if there is no correlation in any subset of trials where ("H", "a","b","λ") all have some fixed value. You already showed that you understood this distinction in this post on your old thread when you said:
In the same way, P(GG') may be different than P(G)*P(G') in our probability space (so the 'marginal probabilities' of G and G' are correlated), while at the same time P(GG'|H,a,b,λ) = P(G|H,a,b,λ)*P(G'|H,a,b,λ).
In the EPR situation only the marginal probabilities, along with conditional probabilities which condition on observable conditions like the detector settings, are actually measurable. The λ is defined to represent hidden-variable states so conditional probabilities involving that term cannot be directly observed, although we can reason theoretically about some general properties of these conditional probabilities that must be true under the theoretical assumption of local realism.
What misunderstanding would that be? You disagree that Bell's equation allows there to be a (marginal) correlation between G and G'? If not, that's all I was saying, and it should have been quite obvious from the context that I was talking about a marginal correlation and not a correlation conditioned on λ.
I never used any words like "is a function of", so I have no idea what this criticism is referring to. And I don't know that "function of" has some precise definition in probability theory that forbids you from saying that the expression P(A|B) "is a function of" A and B (even if there was this would be more of a semantic quibble than a substantive critique). Also, I think it is legitimate to say that the probability space used to calculate P(G'), which includes events where H,b,λ,a,G take different values, is the same as the probability space used to calculated P(G'|H,b,λ,a,G), where we assume H,b,λ,a,G all have some known values. It's just that the expression P(G'|H,b,λ,a,G) indicates we must look at a subset of events in the larger sample space where H,b,λ,a,G take these known values, and look at the the frequency of G' within that subset.

Gordon Watson

Thank you JesseM. I appreciate this detail. I have some basic questions.

1. Could you define for me (briefly) and distinguish Bell's use of the words observable and beable? Is Bell's lambda an observable or a beable or something else -- like what? What size set might it be?


2. If Bell's lambda were an infinite set of spinors (because we want a realistic general "Bell" vector that applies to both bosons and fermions), then wouldn't we need aG to define the infinite subset of spinors that were relevant to the applicable conditional? You seem to require that we would know a priori which of that infinite set satisfied this subset aG conditional? This a priori subset being the lambda you would require here?


3. Beside which, if aG were implicit in your lambda, its restatement/extraction by me would be superfluous and not change the outcome that attaches to the disputed conditional? Note that you seem to require lambda to be an undefined infinite set, perhaps not recognizing that it is an infinite subset (selected by the condition aG, out of your undefined infinite set) which is relevant here?

4. As with the ether experiments and their outcome, don't Bell-tests show that Bell's supposition re Bell's lambda is false?

Thank you.

billschnieder

I'm not sure you understand the point at all.

if A and B are correlated marginally, then P(AB) > P(A)P(B)

If you collect data such that your data samples the entire probability space (that is what marginal probability is) , then the above expression is true. It is no different that defining "Z = All possible facts in the universe", and writing P(AB|Z). You are still dealing with a marginal probability.

Now, if there exists a certain factor C within Z such that the set (C, notC) is the same as Z, then if we say C is the cause of the marginal correlaction between A and B, it means within C, under certain circumstances it maybe correct to write P(AB|C) = P(A|C)P(B|C). It means that C screens-off the marginal correlation between A and B.

However, and please pay attention to this part, this means if data is collected in the full universe fairly sampling both situations where C is true and situations where C is not true (or notC is true), a correlation will be observed in the data, and if data is collected only under situations where C is True, there will be no correlation in the data.

1) You see therefore why it makes no sense to define C as vaguely as you are defining it
2) If hidden elements of reality C exist, then it is impossible to collect data under situations where C is not True, because C will define the actual context of the data. So no matter how hard you try, all your observed frequencies will always be conditioned on the actual contexts that created the data, whether you like it or not.
3) Following from (2), if hidden elements of reality exist, then the correlations observed in experiments exist even when conditioned on C. Because it is impossible to not condition the results on C. Therefore equations such as P(AB|C) = P(A|C)P(B|C) are not accurate. For example if hidden element C is always present and C=42, then P(AB) is not different from P(AB|C=42)
4) For the type of situation Bell is modelling, where he is assuming that hidden elements of reality exist. Marginal probabilities do not come into the picture because the existence of hidden elements of reality MUST always be a conditioning factors.
5) Therefore I hope it is clear to you now why it makes no sense to say the observed EPR correlations are caused by the hidden variables and yet write an equation such as P(AB|C) = P(A|C)P(B|C) in which means if the hidden elements of reality C are realized, no correlation between will be observed between A and B.

Again, just in case it wasn't clear the first time, by writing P(AB|C) = P(A|C)P(B|C), you are saying if the hidden elements of reality C exist, then no correlation will be observed between A and B. Yet Bell starts out by assuming that hidden elements of reality exists. Just because you drop C from the LHS of P(AB|C) does not enable you to escape this trap. The only escape is for you to show how it is possible in a real experiment to collect data fairly for situations where C is true and also for situations where C is not True.

In case you still insist on your approach, could you answer one simple question.

Are the correlations calculated in Aspect type experiments marginal or conditional on the real physical situation which produced them?

JesseM

No, I don't. If local realism is true, then the random variable representing "the state of all local physical variables in the past light cones of the measurements" will have a perfectly well-defined value on each measurement. Are you planning on answering the question I asked you in my most recent post to you (post #80) on the "Understanding Bell's Mathematics" thread? Again:
Of course, if you want a simpler example of a "C" you could also consider post #18 from the thread where we first got into the Bell discussion, either the scratch lotto card analogy or the flashlight analogy. Do you disagree that in both those examples, there would be a correlation in the marginal probabilities of different measurement outcomes, but if C represented the value of the "hidden" facts on each trial (the hidden fruits behind the cards in the lotto analogy, the fact about whether Alice got flashlight X or flashlight Y in the flashlight example), then conditioned on C there would be no correlation in measurement outcomes?
C is a random variable which can take multiple values on different trials. The simplest type of hidden-variables theory would just say that on each trial, the particles have hidden variables that predetermine their spins on each of the measurement settings. For example, if there are three measurement settings a=0 degrees, b=120 degrees, and c=240 degrees, then on each trial the random variable C might take any one of the 8 values c1, c2, c3, c4, c5, c6, c7, c8, defined as:

c1: spin-up on a, spin-up on b, spin-up on c
c2: spin-up on a, spin-up on b, spin-down on c
c3: spin-up on a, spin-down on b, spin-up on c
c4: spin-up on a, spin-down on b, spin-down on c
c5: spin-down on a, spin-up on b, spin-up on c
c6: spin-down on a, spin-up on b, spin-down on c
c7: spin-down on a, spin-down on b, spin-up on c
c8: spin-down on a, spin-down on b, spin-down on c

(note that these are directly analogous to the eight possible hidden-fruit states on the cards in the scratch lotto card analogy)

According to this type of hidden-variables theory, do you deny that on each trial C would have one of these values, and the complete sample space would include trials with all possible values of C?
Do the "actual contexts" include hidden variables? For example, consider again the flashlight analogy:
So, would "C" include the fact about whether H1 or H2 obtain on each trial? If we do define it this way, do you agree that P(AB|C) = P(A|C)*P(B|C), even though P(AB) is not equal to P(A)*P(B)?
I think you've confused yourself with purely verbal, nonmathematical arguments. If you actually examine one of my examples that involve elements hidden from the experimenters (and which help determine the measurement outcomes), you'll see that your general verbal arguments are giving you incorrect conclusions when applied to these examples.
Again, the whole idea is that the variable can take different values on different trials, like in the flashlight example where the random variable H could take value H1 or H2 on different trials? Do you disagree that this was meant to be true of Bell's λ, since he actually integrated over all possible values of λ in equation (2) in his paper?
C can take different values, and for any specific value, if you look only at the subset of trials where C took that trial, there will be no correlation between A and B, but if you look at the total collection of trials, there will be a correlation. Of course this is a theoretical conclusion based on the assumption that the universe obeys local hidden variables, since C represents hidden variables, even if such a theory was correct there would be no way for us to actually know the value of C on each trial (which is why it is helpful to think of all equations involving hidden variables as having precise values that would be known by an imaginary omniscient observer).
Nope, a marginal correlation will be observed between A and B. By writing that equation I'm only saying that if hidden variables exist, then there would be no correlation between A and B in any subset of trials where the hidden variables all took the same value.
In reality, or under the assumption that we live in a universe with local realist laws? Bell's whole approach is to derive certain inequalities from the assumption of local realism, show these inequalities conflict with actual quantum-mechanical results, and therefore conclude (proof by contradiction) that real-world quantum physics is inconsistent with local realism.

If you're talking about reality, I think Bell's reasoning is correct and quantum mechanics rules out local realism, so I don't think there are any real local hidden variables you can condition measurement outcomes on to make the correlations disappear. If you're talking about what would be true theoretically in a universe obeying local realist laws, then in that case all correlations between spacelike-separated measurements could only be marginal ones, and conditioned on a sufficiently large set of local physical facts in the past light cones of the measurements there could be no correlations between measurements.

JesseM

beables represent local hidden variables that are supposed to explain correlations seen in observables, and observables are just facts we can actually measure, like whether a particle gives result "spin-up" or "spin-down" when passed through a Stern-Gerlach device oriented at some angle. Take a look at my scratch lotto card analogy in this post (beginning with the paragraph that starts 'Suppose we have a machine that generates pairs...')--the observables would be the cherries or lemons that Alice and Bob actually see when they pick a single square to scratch, the beables would be the complete set of hidden fruits behind all three squares, which are used to explain why it is that they always find the same fruit whenever they scratch the same box (the assumption being that on each trial, the two cards have the same set of hidden fruits).
I don't know much about relativistic quantum theory which is where I think "spinors" appear--my question here would be, are spinors actually local variables associated with a single point in spacetime, or are they defined in some more abstract "space" like Hilbert space?
Not clear what you mean by "this subset aG conditional", can you elaborate?
What do you mean by "implicit in"? Do you mean that the measurement a and the result G can be determined from the value of lambda? If so, I'm not sure why you think that, the measurement can be random and I told you in post #82 on this thread that the probabilities of different outcomes may be other than 0 or 1 in a probabilistic local realist theory.
Nothing "undefined" about it, as I said to billschnieder:
Like I said to billschnieder in the last post, the basic logic of Bell's argument is a proof-by-contradiction. He starts only by assuming that the universe obeys local realist laws, and then shows that they produce predictions about the statistics of Aspect-type experiments that contradict the predictions (and experimental results) of QM, and so concludes that QM is incompatible with local realism (so if QM's predictions hold up to experimental tests, our own universe must not obey local realist laws).

billschnieder

JesseM,
With all due respect, I do not take you seriously because you completely ignore everything I say. You keep repeating points I have debunked and expect me to keep repeating myself. You keep dragging tangential discussions from thread to thread and I don't bother going down that rabbit trail because it hijacks the thread. You redefine everything I say so that it means something different and then you use the strawman to purport to be arguing against what I said. The recent one is your claim that C is a random variable. It is NOT.


My simple response to everything in your last post is that C is NOT a random variable so you are arguing against yourself. At best, A and B may be considered random variables but C is definitely positively NOT a random variable. It is a specific conditioning factor. You keep repeating the fautly idea that C has multiple values. C as it appears in the equation I wrote, is a specific set of elements of reality. C is NOT all possible sets of elements of reality. It can not be because some of those sets will be mutually exclusive and you can not condition a probability on mutually exclusive factors. As I have explained, in calculating a conditional probability everything after the "|" is assumed to be true simultaneously. It is therefore fallacious to suggest that a probability can be conditioned on mutually exclusive factors at the same time. Until you understand this simple point, you will be totally confused by everything I'm saying.

You did not answer. Clearly correlations are calculated in Aspect type experiments otherwise there will be nothing to compare to Bell's inequalities. Are those correlations marginal or conditional on the hidden elements of reality causing them. The answer should be simple, although it is a trick question.

Clearly you seem to be confused about what it means to make an assumption. Once you make the assumption, that the universe is local realistic, there is (and should) no longer any be any distinction between reality and local reality in all your equations. If you continue to make a distinction and have one equation for reality and a different one for local reality, then your claim of having assumed local reality is false.

So let us try again.

JesseM

You never gave a clear definition of what C is. You did say "If hidden elements of reality C exist" which suggests it should be a random variable, since the value of the hidden variables would differ from one trial to another. But then you said "C will define the actual context of the data" which perhaps suggests you intended C to be a mere specification of the sample space of possible combinations of values that could obtain on any trial, similar to the way you were defining "z" in posts 55, 70, and 70 on the other thread. If you want C not to be a random variable, but simply a specification of the sample space which should be the same on every trial, then it's exceedingly weird notation to actually include that as a symbol in your equations, in any standard probability equation the sample space will be defined beforehand and it'll then be implicit in all the equations rather than represented using a symbol. And if your C is not a random variable, then it has nothing to do with Bell's λ or with the symbols that have played a similar role in my equations like H, since those symbols were random variables--do you disagree?

If you agree those were defined as random variables that could take different values on different trials, then perhaps you can see why your whole discussion becomes a totally irrelevant tangent: you triumphantly declared that P(AB|C)=P(A|C)*P(B|C) also implies P(AB)=P(A)*P(B) as if this somehow discredited my earlier arguments about there being a marginal correlation but no conditioned correlation, but while this might be true under your definition of C, it in no way shows there is anything wrong with my argument that P(GG'|H,a,b,λ)=P(G|H,a,λ,b)*P(G'|H,b,λ,a,G) and yet P(GG') is not equal to P(G)*P(G') (i.e. G and G' are marginally correlated but uncorrelated conditioned on H,a,b,λ), since here λ is obviously meant to be a random variable. Nor does it show there is anything wrong with Bell's equation (2) in his original paper, where λ was also a random variable. So sure, I agree with your statement that if C is a non-variable that simply represents the sample space, then your arguments in post #11 are correct, but it would be completely incoherent to use those arguments to try to discredit my arguments or Bell's, since your C is defined in a completely different way than the λ and H that appeared in the equations.
"Faulty" only under your definition of C, which you did not actually make clear in your previous post. So, now I agree that your C cannot have multiple values, but hopefully you agree that Bell's λ is a random variable that does take multiple values (as made clear by the fact that he is integrating over all values of λ in equation 2), and likewise that when I and other defenders of Bell write equations like P(AB|H)=P(A|H)*P(B|H), the symbols playing a similar role there like H are also intended to be random variables that can take different values on different trials (different points in the sample space). Do you disagree with this?
Hold on, are you saying that regardless of your own personal definition of C, there is something incorrect in general about writing a conditional probability equation where the conditioning factor is a random variable that can take different values on different trials? For example, if H is a random variable that can take values H1 and H2 (as in my flashlight example), and A is another random variable that can take values A1 and A2, are you claiming it would then be incorrect to write the equation P(A and H) = P(A|H)*P(H)? If so you are badly confused, when a probability equation is written with random variables, all that means is that the equation should hold for each possible combination of specific values of the random variables--for example, P(A and H) = P(A|H)*P(H) is true as long as it's true that P(A1 and H1)=P(A1|H1)*P(H1) and P(A1 and H2)=P(A1|H2)*P(H2) and P(A2 and H1)=P(A2|H1)*P(H1) and P(A2 and H2)=P(A2|H2)*P(H2). If all four of those equations involving all possible combinations of specific values of A and H are true, that means the general equation P(A and H) = P(A|H)*P(H) is also true.
I made clear that the question wasn't sufficiently well-defined when I asked for the clarification "In reality, or under the assumption that we live in a universe with local realist laws?"
Yes, this would be "reality".
The correlations measured in real experiments may be conditional on detector settings, but they are not conditional on any hidden elements of reality, regardless of whether such hidden elements exist or not (since even if they do exist we don't know their value on each trial so we can't measure a frequency which is conditioned on them).

Bell's approach is to start from the theoretical assumption of local realism, then use that to derive predictions about what correlations should be seen when you don't condition on the hidden variables. Since these predictions differ from the correlations predicted by QM and also from those observed in real experiments, this is taken as evidence that local realism is false in our universe.
None of the equations in Bell's derivation of the Bell inequality involve statements about what is true in our reality, they are only about what should theoretically be true in a universe obeying local realist laws. Some of these equations actually involve the hidden local variables, while others involve only things that would be observed by theoretical experimenters in such a theoretical universe (which can then be compared with the observations of real experimenters in the real universe), but either way we are assuming a universe whose laws are locally realistic. If you ever thought I was saying anything different, it's you who was confused about the basic logic of Bell's proof.

DrChinese

Well said, JesseM! I wish more people would listen to these words.

The local realist is making an "extra" assumption (or two). If the term local realist is to mean anything, then such assumption(s) should be spelled out. It is then subject to verification or rejection... or in this case to be shown to be incompatible with something else (QM).

I think any reasonable local realist can come up with a mathematical constraint or requirement that models locality and realism. Once that is agreed upon, I think the Bell program can be applied and the conclusion will simply match Bell. On the other hand, failure to provide such constraints for locality and realism would be tantamount to accepting the result prima facie.

billschnieder

Did you completely ignore my statement that the set ("C", "notC") is equivalent to Z. Since you insist that C must have multiple values, can you explain what "notC" will represent according to your understanding of what C is supposed to entail. Give a short example of the different values you think could represent C and at the same time specify clearly what "notC" represents in your example -- please no 15-page scratch lotto cards examples that will require me to respond to every sentence because I will just ignore it.


Yes I disagree.

Go back and read it. I said no such thing.

λ in Bell's paper is supposed to represent the EPR "elements of reality" which cause in the observed correlations. EPR elements of reality are not random variables no matter how loudly you shout that they are.
That is circular reasoning and I can use the same point against you -- each term in the integral can only have a single value of λ so in fact by integrating you are adding P(AB|a,b,λ1) + P(AB|a,b,λ2) + ... + P(AB|a,b,λn) where n represents the number of possible realizations of your λ. You still can not escape the fact that the conditioning elements can never be as broad as λ. In each case in which a joint conditional probability is calculated, λn is specific and definitely not a random variable. That is why I told you repeatedly that in calculating a conditional probability you can not condition on a vague concept such as λ with multiple values. This is the same reason why in such a case, the LHS of Bell's equation (2) can not be a probability conditioned on the vaquely defined λ. It clearly looks like a marginal probability.

So the claim that his inequalities derived from such an expression are based on the assumption that hidden variables exist is specious.

Absolutely disagree, see my response above, H can not have multiple values in that expression. If you want to write it as P(AB)=P(A|H1)*P(B|H1) + P(A|H2)*P(B|H2) + ... + P(A|Hn)*P(B|Hn) go ahead, but don't deceive yourself and others that you are calculating P(AB|H).

BTW The sample space for H1 is different from the sample space for H2 etc. They are not part of the same sample space. If H1 and H2 are mutually exclusive, your so-called H-sample space is undefined.

I am definitely saying saying if H can take on multiple values H1 and H2 it is OK to write P(AB|H1) and P(AB|H2), but when you write P(AB|H) the only meaning here is that H is a placeholder for a specific value of H not all possible values of H simultaneously. ie, each concrete value of P(AB|H) you could ever calculate can only be valid for a specific H not the vague concept of being conditioned on "the H variable" or all values of H. For example if H represents the face of a coin and has two possible realizations in a toss "heads" or "tails", writing P(...|H) in which H includes all possible values is no different than writing P(...|heads, tails) But since heads and tails are mutually exclusive, your probability is undefined and meaningless if you insist on that definition. If H1 and H2 are mutually exclusive P(AB|H1H2) is undefined and meaningless. Therefore P(AB|H) can not imply that H is a variable with multiple values in a single expression.

Can you explain to me how Aspect and others made sure in their experiments that IF hidden elements of reality exist, then the measured data will not depend on the their presence.

In other words, is it possible for hidden elements of reality to exist and not exist at the same time? Isn't it obvious that IF hidden elements of reality exist, then they govern the results observed in Aspect type experiments?

IF hidden elements of reality exist, then it is impossible for Aspect et al to collect data under circumstances in which hidden variables do not exist. Therefore your statement that the correlations they observed is "regardless of whether such hidden elements exist or not" is far off base.

In other words you are saying the correlations Bell is calculating are those that should be seen in our universe if experiments are performed such that those variables (which we have assumed exist), should not affect the results.
Now can you point me to an experiment in which the experimenters made sure that IF hidden elements of reality exist, they should not affect the data measured? By your own admission, those are the only data that are comparable to Bell's inequalities.

Could you explain how Aspect et al made sure their data was collected in such a way that IF hidden elements of reality exist, they should not influence the results.

Once the assumption is made that our universe is local realistic, the distinction you are trying to make is artificial.
Until and unless you can demonstrate that the "theoretical experiments" are comparable to actual experiments performed in our universe, you can not use Bell's equations to say anything about our universe.