User:Johnjbarton/sandbox/ensemble

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Probability; propensity

Quantum observations are inherently statistical. For example, the electrons in a low-intensity double slit experiment arrive at random times and seemingly random places and yet eventually show an interference pattern.

Matter wave double slit diffraction pattern building up electron by electron. Each white dot represents a single electron hitting a detector; with a statistically large number of electrons interference fringes appear.^[1]

The theory of quantum mechanics offer only statistical results. Given that we have prepared a system in a state $\psi$ , the theory predicts a result $a_{j}$ as a probability distribution:

P(a_{j}|\psi )=|<a_{j}|\psi >|^{2}

.

Different approaches to probability can be applied to connect the probability distribution of theory to the observed randomness.

Popper,^[2] Ballentine^[3], Paul Humphreys,^[4], and others^[5] point to propensity as the correct interpretation of probability in science. Propensity, a form of causality that is weaker than determinism, is the tendency of a physical system to produce a result.^[6] Thus the mathematical statement

Pr(e|G)=r

means the propensity for event $e$ to occur given the physical scenario $G$ is $r$ . The physical scenario is view as weakly causal condition.

The weak causation invalidates Bayes' theorem and correlation is no longer symmetric.^[4] As noted by Paul Humphreys, many physical examples show the lack of reciprocal correlation, for example, the propensity for smokers to get lung cancer does not imply lung cancer has a propensity to cause smoking.

Propensity closely matches the application of quantum theory: single event probability can be predicted by theory but only verified by repeated samples in experiment. Popper explicitly developed propensity theory to eliminate subjectivity in quantum mechanics.^[5]

^ Bach, Roger; Pope, Damian; Liou, Sy-Hwang; Batelaan, Herman (2013-03-13). "Controlled double-slit electron diffraction". New Journal of Physics. 15 (3). IOP Publishing: 033018. arXiv:1210.6243. Bibcode:2013NJPh...15c3018B. doi:10.1088/1367-2630/15/3/033018. ISSN 1367-2630. S2CID 832961.
^ Popper, Karl R. “The Propensity Interpretation of Probability.” The British Journal for the Philosophy of Science, vol. 10, no. 37, 1959, pp. 25–42. JSTOR, http://www.jstor.org/stable/685773. Accessed 27 Aug. 2023.
^ ^a ^b Cite error: The named reference BallentinePropensity was invoked but never defined (see the help page).
^ ^a ^b Humphreys, Paul (October 1985). "Why Propensities Cannot be Probabilities". The Philosophical Review. 94 (4): 557. doi:10.2307/2185246.
^ ^a ^b Berkovitz, Joseph. “The Propensity Interpretation of Probability: A Re-Evaluation.” Erkenntnis (1975-) 80 (2015): 629–711. http://www.jstor.org/stable/24735118.
^ For inferential probability, the arguments may, in principle, be any propositions. But for ensemble probability, the first argument A must be an event, and the second argument C must describe a repeatable procedure that can generate an ensemble of outcome events.^[3]

[Bach_Pope_Liou_Batelaan_2013_p=033018-1] Bach, Roger; Pope, Damian; Liou, Sy-Hwang; Batelaan, Herman (2013-03-13). "Controlled double-slit electron diffraction". New Journal of Physics. 15 (3). IOP Publishing: 033018. arXiv:1210.6243. Bibcode:2013NJPh...15c3018B. doi:10.1088/1367-2630/15/3/033018. ISSN 1367-2630. S2CID 832961.

[2] Popper, Karl R. “The Propensity Interpretation of Probability.” The British Journal for the Philosophy of Science, vol. 10, no. 37, 1959, pp. 25–42. JSTOR, http://www.jstor.org/stable/685773. Accessed 27 Aug. 2023.

[BallentinePropensity-3] Cite error: The named reference BallentinePropensity was invoked but never defined (see the help page).

[Humphreys-4] Humphreys, Paul (October 1985). "Why Propensities Cannot be Probabilities". The Philosophical Review. 94 (4): 557. doi:10.2307/2185246.

[Berkovitz-5] Berkovitz, Joseph. “The Propensity Interpretation of Probability: A Re-Evaluation.” Erkenntnis (1975-) 80 (2015): 629–711. http://www.jstor.org/stable/24735118.

[6] For inferential probability, the arguments may, in principle, be any propositions. But for ensemble probability, the first argument A must be an event, and the second argument C must describe a repeatable procedure that can generate an ensemble of outcome events.^[3]

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