Joy Christian wrote:We are all born ignorant, but one must work hard to remain stupid
FrediFizzx wrote:@Joy LOL!! Ignorance is curable but stupid is forever.
gill1109 wrote:FrediFizzx wrote:@Joy LOL!! Ignorance is curable but stupid is forever.
What an incredibly high level of intellectual debate. Very helpful, Fred. Thanks again.
gill1109 wrote:My favourite proof of the CHSH inequality is given in my short paper https://arxiv.org/pdf/2103.00225.pdf
Comment on “Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space” (Now at version 3). No spreadsheets. Just some basic probability theory. Some simple mathematical assumptions involving eight binary random variables A, B, X, Y, X_1, X_2, Y_1, Y_2
A and B take values in {1, 2}. The other variables take values in {-1,+1}
Independence: A, B is statistically independent of X_1,X_2,Y_1,Y_2
LHV: X = X_A, Y = Y_B
In an experiment, one observes (many times) realisations of the quadruple (X, Y, A, B) (outcomes and settings).
X_1, X_2, Y_1, Y_2 are hidden variables (or if you prefer, functions of hidden variables).
One estimates four conditional correlations rho(i,j) := E(XY | A = i, B = j)
I’m interested in criticisms of the mathematics.
minkwe wrote:gill1109 wrote:My favourite proof of the CHSH inequality is given in my short paper https://arxiv.org/pdf/2103.00225.pdf
Comment on “Bell’s Theorem Versus Local Realism in a Quaternionic Model of Physical Space” (Now at version 3). No spreadsheets. Just some basic probability theory. Some simple mathematical assumptions involving eight binary random variables A, B, X, Y, X_1, X_2, Y_1, Y_2
A and B take values in {1, 2}. The other variables take values in {-1,+1}
Independence: A, B is statistically independent of X_1,X_2,Y_1,Y_2
LHV: X = X_A, Y = Y_B
In an experiment, one observes (many times) realisations of the quadruple (X, Y, A, B) (outcomes and settings).
X_1, X_2, Y_1, Y_2 are hidden variables (or if you prefer, functions of hidden variables).
One estimates four conditional correlations rho(i,j) := E(XY | A = i, B = j)
I’m interested in criticisms of the mathematics.
Richard, come-on, I'm surprised after so many years on this forum you still present a proof like that in a paper.
You are talking about "In an experiment" so please explain the experiment that this expression represents:
Each term is equivalent to a pair of coin tosses essentially. How many pairs of coin tosses are involved in the above expression?
Do you really not understand what it means to factor out random variables like
How many pairs of coin tosses are involved in the above expression?
gill1109 wrote:
A big majority of the reviewers think the paper should be published. ... All referees but one agree that Christian's maths is wrong. If anyone is interested I am happy to confidentially show you the reports. ... If my paper is published then in fact all referee reports will be published too.
gill1109 wrote:The expression does not "represent" an experiment. "Z" is a random variable. It is a function from a probability space (Omega, F, P) to the real line. No "coin tosses" are represented or used in that expression.
I am considering a certain possible real world experiment. A thought experiment, easy to implement in a school classroom.
I have given talks to high school children explaining the proof, they loved it.
I have given talks to Buddhist scholars and neuroscientists explaining the proof, they loved it. It should not be difficult. The question is finding the right language to use when explaining it to different kinds of people.
gill1109 wrote:The expression does not "represent" an experiment. "Z" is a random variable.
gill1109 wrote:That probability distribution is described in terms of some further random variables (X1, X2, Y1, Y2). The eight altogether have a joint probability distribution which satisfies the conditions which I have written down.
minkwe wrote:gill1109 wrote:The expression does not "represent" an experiment. "Z" is a random variable.
How many independent random variables make up Z?gill1109 wrote:That probability distribution is described in terms of some further random variables (X1, X2, Y1, Y2). The eight altogether have a joint probability distribution which satisfies the conditions which I have written down.
Okay so at least you admit that there are 4 independent random variables () in Z. Now please answer how many independent random variables are there in the outcomes of a Bell test experiment? Of course you won't answer. You'll instead start talking about high-school students and Buddhists.
Heinera wrote:Perhaps to make this clearer to @minkwe: This is basically the model of the CHSH urn experiment.
gill1109 wrote:1) In my mathematical model, Z is one random variable, built up by adding four products of pairs of other random variables.
2) In one trial of a Bell experiment one gets, per trial, to generate two settings and to observe two outcomes. That’s four binary variables. Since settings can be chosen by auxiliary random experiments, I model the four together as four random variables. Often, A and B are engineered to appear like independent fair coin tosses. So typically, A and B are statistically independent of one another; the pair (X,Y) is obviously not statistically independent of (A,B).
minkwe wrote:gill1109 wrote:1) In my mathematical model, Z is one random variable, built up by adding four products of pairs of other random variables.
Simple answer: There are 4 random variables (), no need for weaselly words.2) In one trial of a Bell experiment one gets, per trial, to generate two settings and to observe two outcomes. That’s four binary variables. Since settings can be chosen by auxiliary random experiments, I model the four together as four random variables. Often, A and B are engineered to appear like independent fair coin tosses. So typically, A and B are statistically independent of one another; the pair (X,Y) is obviously not statistically independent of (A,B).
Simple answer: There are 8 random variables in a Bell test Experiment no need to play word games.
There is no such thing as a CHSH urn experiment.
gill1109 wrote:minkwe wrote:gill1109 wrote:1) In my mathematical model, Z is one random variable, built up by adding four products of pairs of other random variables.
Simple answer: There are 4 random variables (), no need for weaselly words.2) In one trial of a Bell experiment one gets, per trial, to generate two settings and to observe two outcomes. That’s four binary variables. Since settings can be chosen by auxiliary random experiments, I model the four together as four random variables. Often, A and B are engineered to appear like independent fair coin tosses. So typically, A and B are statistically independent of one another; the pair (X,Y) is obviously not statistically independent of (A,B).
Simple answer: There are 8 random variables in a Bell test Experiment no need to play word games.
There is no such thing as a CHSH urn experiment.
Yes there is. I defined it mathematically. This mathematical model is a mathematical model of Heinera’s urn experiment, it’s a mathematical model of a Bell simulation model using LHV and satisfying freedom (no conspiracy), it’s a mathematical model of one trial of a real loophole-free Bell experiment assuming local realism.
In fact it is just Bell’s model of local hidden variables, augmented with explicit random selection of settings.
Start with a probability space which is a product space of two independent components. (Omega, F, P) with Omega = Omega1 x Omega2, F = F1 x F2, P = P1 x P2 (I won’t bore you with the definitions of product of two sigma-algebras and product of two probability measures - these are not Cartesian products of sets).
Omega1 is Lambda, the space of the hidden variable lambda, P1 is Bell’s probability measure rho.
On it are defined X1, X2, Y1, Y2; these are Bell’s functions A with settings fixed at a1, a2, and B with settings b1, b2. Four functions of lambda.
Omega2 is the space on which are defined A, B
I don’t think “random variable” is a weasely word. It’s a mathematical term. https://en.m.wikipedia.org/wiki/Random_variable
minkwe wrote:gill1109 wrote:minkwe wrote:gill1109 wrote:1) In my mathematical model, Z is one random variable, built up by adding four products of pairs of other random variables.
Simple answer: There are 4 random variables (), no need for weaselly words.2) In one trial of a Bell experiment one gets, per trial, to generate two settings and to observe two outcomes. That’s four binary variables. Since settings can be chosen by auxiliary random experiments, I model the four together as four random variables. Often, A and B are engineered to appear like independent fair coin tosses. So typically, A and B are statistically independent of one another; the pair (X,Y) is obviously not statistically independent of (A,B).
Simple answer: There are 8 random variables in a Bell test Experiment no need to play word games.
There is no such thing as a CHSH urn experiment.
Yes there is. I defined it mathematically. This mathematical model is a mathematical model of Heinera’s urn experiment, it’s a mathematical model of a Bell simulation model using LHV and satisfying freedom (no conspiracy), it’s a mathematical model of one trial of a real loophole-free Bell experiment assuming local realism.
In fact it is just Bell’s model of local hidden variables, augmented with explicit random selection of settings.
Start with a probability space which is a product space of two independent components. (Omega, F, P) with Omega = Omega1 x Omega2, F = F1 x F2, P = P1 x P2 (I won’t bore you with the definitions of product of two sigma-algebras and product of two probability measures - these are not Cartesian products of sets).
Omega1 is Lambda, the space of the hidden variable lambda, P1 is Bell’s probability measure rho.
On it are defined X1, X2, Y1, Y2; these are Bell’s functions A with settings fixed at a1, a2, and B with settings b1, b2. Four functions of lambda.
Omega2 is the space on which are defined A, B
I don’t think “random variable” is a weasely word. It’s a mathematical term. https://en.m.wikipedia.org/wiki/Random_variable
Admit it, there are 4 independent random variables in your proof and 8 in an actual Bell test experiment. Will you admit this simple obvious fact? Or will you try to obscure it with irrelevant nonsense.
I ask a question about random variables and you talk about "hidden variables" and every type of variable except the one I asked about. Then when I call you out on it, you imply I'm saying "random variable" is a weasely word. Yet you can't accept the simple fact that there are 4 independent random variables in your proof and 8 in a Bell test experiment.
minkwe wrote:There is no such thing as a CHSH urn experiment.
Heinera wrote:minkwe wrote:There is no such thing as a CHSH urn experiment.
The delusions of the Bell deniers are now reaching comical proportions. It's like flat earthers claiming "satelites do not exist."
Heinera wrote:minkwe wrote:There is no such thing as a CHSH urn experiment.
The delusions of the Bell deniers are now reaching comical proportions. It's like flat earthers claiming "satelites do not exist."
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