by gill1109 » Wed Nov 24, 2021 9:01 am
FrediFizzx wrote: ↑Wed Nov 24, 2021 7:10 am
gill1109 wrote: ↑Mon Nov 22, 2021 9:26 pm
I presented a mathematical theorem. It uses the standard notation and concepts of probability theory. If you don’t know them, you might have difficulties reading what I wrote. Ask questions. Let me know what you don’t understand.
Now that is really funny because I don't think I have ever seen you do,
Theorem 1: Blah, Blah, Blah, ....
IOW, I don't think I have ever seen you make a real theorem statement like they do in most math documents. Send me a link if you have.
Take a look at my
Theorem 1 in
https://arxiv.org/abs/1207.5103
Statistics, Causality and Bell's Theorem
Abstract: Bell's [Physics 1 (1964) 195-200] theorem is popularly supposed to establish the nonlocality of quantum physics. Violation of Bell's inequality in experiments such as that of Aspect, Dalibard and Roger [Phys. Rev. Lett. 49 (1982) 1804-1807] provides empirical proof of nonlocality in the real world. This paper reviews recent work on Bell's theorem, linking it to issues in causality as understood by statisticians. The paper starts with a proof of a strong, finite sample, version of Bell's inequality and thereby also of Bell's theorem, which states that quantum theory is incompatible with the conjunction of three formerly uncontroversial physical principles, here referred to as locality, realism and freedom. Locality is the principle that the direction of causality matches the direction of time, and that causal influences need time to propagate spatially. Realism and freedom are directly connected to statistical thinking on causality: they relate to counterfactual reasoning, and to randomisation, respectively. Experimental loopholes in state-of-the-art Bell type experiments are related to statistical issues of post-selection in observational studies, and the missing at random assumption. They can be avoided by properly matching the statistical analysis to the actual experimental design, instead of by making untestable assumptions of independence between observed and unobserved variables. Methodological and statistical issues in the design of quantum Randi challenges (QRC) are discussed. The paper argues that Bell's theorem (and its experimental confirmation) should lead us to relinquish not locality, but realism.
Statistical Science 2014, Vol. 29, No. 4, 512-528
DOI: 10.1214/14-STS490
Theorem 1. Given an N × 4 spreadsheet of numbers ±1 with columns A, A′, B and B′, suppose that, completely at random, just one of A and A′ is observed and just one of B and B′ are observed in every row. Then, for any η ≥ 0, (3) [displayed equation].
The various objects defined in equation (3) are defined earlier in the paper. I will see if I can put the equation into this thread later on, as well as the needed definitions.
The proof is given in full later in the paper.
[quote=FrediFizzx post_id=298 time=1637766620 user_id=58]
[quote=gill1109 post_id=285 time=1637645213 user_id=60]
I presented a mathematical theorem. It uses the standard notation and concepts of probability theory. If you don’t know them, you might have difficulties reading what I wrote. Ask questions. Let me know what you don’t understand. [/quote]
Now that is really funny because I don't think I have ever seen you do,
Theorem 1: Blah, Blah, Blah, ....
IOW, I don't think I have ever seen you make a real theorem statement like they do in most math documents. Send me a link if you have.
[/quote]
Take a look at my [b]Theorem 1[/b] in [url]https://arxiv.org/abs/1207.5103[/url]
[i]Statistics, Causality and Bell's Theorem[/i]
Abstract: Bell's [Physics 1 (1964) 195-200] theorem is popularly supposed to establish the nonlocality of quantum physics. Violation of Bell's inequality in experiments such as that of Aspect, Dalibard and Roger [Phys. Rev. Lett. 49 (1982) 1804-1807] provides empirical proof of nonlocality in the real world. This paper reviews recent work on Bell's theorem, linking it to issues in causality as understood by statisticians. The paper starts with a proof of a strong, finite sample, version of Bell's inequality and thereby also of Bell's theorem, which states that quantum theory is incompatible with the conjunction of three formerly uncontroversial physical principles, here referred to as locality, realism and freedom. Locality is the principle that the direction of causality matches the direction of time, and that causal influences need time to propagate spatially. Realism and freedom are directly connected to statistical thinking on causality: they relate to counterfactual reasoning, and to randomisation, respectively. Experimental loopholes in state-of-the-art Bell type experiments are related to statistical issues of post-selection in observational studies, and the missing at random assumption. They can be avoided by properly matching the statistical analysis to the actual experimental design, instead of by making untestable assumptions of independence between observed and unobserved variables. Methodological and statistical issues in the design of quantum Randi challenges (QRC) are discussed. The paper argues that Bell's theorem (and its experimental confirmation) should lead us to relinquish not locality, but realism.
Statistical Science 2014, Vol. 29, No. 4, 512-528
DOI: 10.1214/14-STS490
Theorem 1. Given an N × 4 spreadsheet of numbers ±1 with columns A, A′, B and B′, suppose that, completely at random, just one of A and A′ is observed and just one of B and B′ are observed in every row. Then, for any η ≥ 0, (3) [displayed equation].
The various objects defined in equation (3) are defined earlier in the paper. I will see if I can put the equation into this thread later on, as well as the needed definitions.
The proof is given in full later in the paper.