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Does QM ever violate classical probability theory?

  1. Sep 8, 2011 #1

    BWV

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    reading this http://uk.arxiv.org/abs/1004.2529 about supposed parallels between the mathematical structure of probability in QM and some problems in economics

    question is that are there really any violations of classical probability theory, such as Pr(A) > Pr(A [itex]\cup[/itex] B) in QM? The supposed examples all seem to point to interference effects which to my thinking are not violations of Probability theory as you could construct a similar situation with classical waves
     
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  3. Sep 8, 2011 #2

    xts

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    Reading that - haven't you got the imperssion, that it would be one more great example for Sokal and Bricmont?
    (http://en.wikipedia.org/wiki/Fashionable_Nonsense)

    I haven't spotted any.
     
  4. Sep 8, 2011 #3

    BWV

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    Thanks

    although to be fair the linked article is only attempting to use the mathematics of QM for other applications, rather than postulating some link between the actual physics and economics.

    But the whole formal structure of "disjunctions" in the paper does not make much sense to me - the whole phenomenon could just as easily be explained by cognitive errors of individuals responding to polls.
     
  5. Sep 8, 2011 #4

    xts

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    I would not call it 'attempting to use mathematics... for..."
    I would rather call it: 'attempting to impress social-science reader with mathematic symbols she do not understand'
     
  6. Sep 8, 2011 #5

    BWV

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    true enough, but not as egregious as using curvature tensors to explain arbitrage relationships

    http://arxiv.org/abs/hep-th/9710148

    Have had a lot of fun at the office giving this paper to newly minted MBAs as "required study material"
     
  7. Sep 8, 2011 #6

    Fredrik

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    I haven't really thought about it, but the probability measures in QM are (generalized) probability measures on a non-distributive lattice, not probability measures on σ-algebras. You can probably derive some "violations" from the non-distributivity, but I'm not going to think about that tonight.

    I think your ">" should be a "≤". [itex]P(A\cup B)=P(A\cup(B-A))=P(A)+P(B-A)\geq P(A)[/itex]

    I haven't read the article, but the journal reference says "International Journal of Theoretical Physics", and one of the authors is Diederik Aerts, who I believe is a leading expert on quantum logic.
     
  8. Sep 8, 2011 #7

    xts

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    So try to read it!
    I am looking forward to see your defense of this babble: please try to use non-adjectival logic.
     
  9. Sep 8, 2011 #8
    [itex]P(.): \mathcal{T} \rightarrow \mathbb{R}[/itex] is a mapping from the field of all events [itex]\mathcal{T}[/itex] to the set of real numbers with the following properties (due to Kolmogorov):
    Non-negativity:

    [tex]
    \left(\forall A \in \mathcal{T}\right) \, P(A) \ge 0
    [/tex]

    Additivity:
    [tex]
    A \cap B = \emptyset \Rightarrow P(A \cup B) = P(A) + P(B)
    [/tex]

    Normalization:
    [tex]
    P(\Omega) = 1
    [/tex]
    where [itex]\Omega[/itex] is the certain event.

    If you use the set relations:
    [tex]
    A \cup B = A \cup (B - A), \; A \cap (B - A) = \emptyset
    [/tex]
    and
    [tex]
    B = (A \cap B) \cup (B - A), \; (A \cap B) \cap (B - A) = \emptyset
    [/tex]
    and use the additivity of probability:
    [tex]
    P(A \cup B) = P(A) + P(B - A)
    [/tex]
    [tex]
    P(B) = P(A \cap B) + P(B - A)
    [/tex]
    to eliminate [itex]P(B - A)[/itex], we get:
    [tex]
    P(A \cup B) = P(A) + P(B) - P(A \cap B)
    [/tex]

    Then, if what you claim is true:

    it would imply:
    [tex]
    P(A \cap B) > P(B), \; \forall A, B \in \mathcal{T}
    [/tex]
    Then, taking any [itex]A \subseteq B[/itex], we would have [itex]A \cap B = B[/itex] and we would arrive at a contradiction:
    [tex]
    P(B) > P(B) \ \bot
    [/tex]
     
  10. Sep 8, 2011 #9

    BWV

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    I was posting a violation of classical prob listed in the link. The example given was polling data of categorizations - for example respondents were asked something like does x belong in category a or in categories (a or b). There are common cognitive biases that violate probability laws - the author seems to be saying that you need a "non-classical" statistics to deal with them, which seems like a dubious proposition to me
     
  11. Sep 8, 2011 #10
    How does the author measure [itex]P(A)[/itex] or [itex]P(A \cup B)[/itex]?
     
  12. Sep 9, 2011 #11

    Fra

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    I didn't read the paper just skimmed the abstract and first page fast last night. Just some thouhgts on the topic, less specific to the paper.

    1) Their conceptual analogy between QM and decision theory is to ve very intuitive and interesting. I myself use decision theory analogies alot since for me they are highly natural. We all make decisions everday, one deos not need to study it formally to understand it.

    2) It's not right to say that they violate classical probability - of course they don't. That' just mathematics. What they do claim is this: The cognitive decisions made by test subjects FAILS to be MODELLED by classical probability (classical logic), and that it's better modelled by quantum logic. This is not surprising to me.

    This is exactly on par with that you can't explain quantum interference in terms of classical bayesian probability and classic logic. But this we know already.

    3) I think the whole point is that whatever you call it "cognitve errors" or something else, the task is to model it. To predict how a subject responds to input, means to understand its' decision process. The point is that whatever happens in the decision process, that is due to nature, so it does not make much sense to call it "errors" if "errors" are integral part of the decision process.

    The analogy with physics is:

    Determine how a system (say a piece of matter) RESPONDS/ACTS on perturbation/input, and quantum mechanically we model this as the system "considering all paths etc" which are then "added" under interference effects, this is very analogous to a decision process!

    Determing how a subject responds/acts to input, which of course can be tought of as the subject choosing and action after some cognitive reflections to make a decision on how to rationally respond to input.

    Here a big difference that leads to a different insight is that decisions does not have the purpse of beeing what might be thought of as "logically correct", instead decisions are tactical decisions, that are expect to yield maximum return. This is why sometimes decisions that turn out "wrong" is retrospect, are still maximally rational. This is why one should be careful to make statements of what in a deciusion process that are "errors".

    /Fredrik
     
  13. Sep 9, 2011 #12

    Fra

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    As I understand it, from trials of test subjecst (humans) that get to answer verbal questions. Replacing logical operations with and and or. So the measurement = observing what human subjects "decide" to answer on given questions.

    So what the try to model is the human decision process. All they found is that contrary to what we think is "logical" the subjects fail to make decsions as per classical logic. That's their only point. If we call this an "error" or logical fallacy, that's partly right, but that's beyond the point. The quest is still to modell it, with "fallacies" and all. I think the onjceture from cognitive psychology is that there exists a rational explanation for the fallacies, which involves how the decisions process in the brain actually works - it apparently does not work with simple classical logic.

    /Fredrik
     
  14. Sep 9, 2011 #13

    xts

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    So maybe you understand why Bell's theorem and EPR paradox were mentioned in introduction to the paper as important to it? Look a bit later - how impressive mathematical formalism!

    If you find it interesting - you should follow references to check the methodology they used to obtain data. Actually, they didn't do any experiments - they reinterprete 20-years old data collected by J.Hampton (http://www.staff.city.ac.uk/hampton/PDF files/Hampton Disjunction1988b.pdf) on an impressive sample of 40 students, answering junctive questions and disjuntive ones a week later. Hampton somehow forgot to test, what would be a correlation between answers if exactly the same questions would be asked twice. Whole this "theory" is built upon inconsistency of answers given by two persons.
    Other data comes from the experiment, where 2 groups of 10 persons each were asked disjunctive question, why other group of 20 were asked junctive one. Quick excercise for you on "classical" probability theory: what is a probability that 25% or less of 20 people give one answer and 30% or more of 10 people gives the same answer if those are independent samples? That is the "evidence" (25% > 30%) upon which the whole "theory" is built.

    Again. 40 person sample with no estimation of expected errors, or comparison made on statistics: 5/20 vs. 3/10 I would be rather cautious using capitalics here.

    Great! But - again:
    - data they use do not justify rejection of simplest theory: "human cognition follows Boolean logic"
    - even if cognition do not follow classical logic, there is no relation with EPR, Bell, von Neumann, and Quantum Mechanics.

    But "Quantum" makes papers more sexy! Not only papers - Calgonit Quantum - the best dishwasher tabs!
     
    Last edited: Sep 9, 2011
  15. Sep 9, 2011 #14

    Fra

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    I'm not particularly interested nor impressed by that paper as such, but I interpreted the OP as questioning what possibly reasonable relation there exists between decision theory and physics, in their logic.

    I think the connection is there, but this was known to be before, that paper doesn't seem present anything new as I see.

    You are possible right that "it's sexy", that's possibly a factor :) But that doesn't make the connection wrong.

    I think it was mentioned since in physics bell's theorem and the inequalities more or less serves as the no-go theorem for locally realistic theories; which essentially means they are explained by classical logic.

    Not that I bother care much, but they seem to have some idea about some no-go teorem for decisions theory, which you could tell if it can be explained by classicla logic or not.
    Not the paper itself, but the general connection. But I surely don't need that paper to know that. In particular the connection between decision process and physical interactions is profound IMO. This you can even tell from most of my posts on here. But that papers isn't a physics paper so I don't care, I just wanted to stand up for the connection.

    /Fredrik
     
  16. Sep 9, 2011 #15

    The reason I was asking this is because if they made a statistical estimate of the probabilities, then they can only give confidence interval for the difference [itex]P(A) - P(A \cup B)[/itex]. If this interval happens to contain the zero, then, any statistical test of the hypothesis for a particular sign of this difference will be rejected at the given level of confidence. As the sample size (40 students) is pretty small, I would assume the 95% confidence interval (standard in natural sciences) to be pretty wide. I do not intend to perform any calculation for this, however.
     
  17. Sep 9, 2011 #16

    xts

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    You should make! Data used to build the "theory" presented in this paper are not significant even on one standard deviation level.
     
  18. Sep 9, 2011 #17

    xts

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    Of course, I can't prove their idea is wrong. I may only say that the justification they provided is wrong.
    I may also say that the physical theories they quote are not related by any means to the subject they touch.
    Such 'parallels' are not more justified than claim about quantum nature of day and night (which is cyclic, but the light emitted by excited atom is also cyclic, and is described by quantum mechanics).
    There are millions crap ideas you can't prove to be wrong.
    What I (and every sceptical person, stick to Occam's rule) expect from scientific paper is not to present "theories", which (however not falsifable) are not justified by any evidence, especially not by the evidence cited by the particular paper.
     
    Last edited: Sep 9, 2011
  19. Sep 9, 2011 #18

    Fra

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    Very possible! I honestly didn't read it carefuly, and I don't even care to.

    I have other justifications that's more than good enough for me and I don't need more. However my htinking is different, rather than trying to apply QM formalism to decisions theory, I think that DEEPER insights into how decisions work, will help us find an instrinsic measurement theory that's helpful for QG and unification.

    This is why that paper itself doesn't interest med much. (Not enough to even read it properly)

    /Fredrik
     
  20. Sep 9, 2011 #19

    xts

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    Honestly: me too. I just browsed it to point out several absurds and followed the bib-link to see what data they are based on.

    Wow! Would you say few words more how the human decision theory is related to Quantum Gravity?
     
  21. Sep 9, 2011 #20

    DrDu

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    In the theory of weak measurements there appear probabilities >1 and <0, so they violate classical probability explicitly.
     
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