Monday Seminar: Bayesian Quantum Theory
In today's Quantum Gravity Group meeting, Thomas Marlow (who is doing his PhD research here at Nottingham but is not an "official" member of the QG Group, though he often assists to our seminars) gave us a talk based on his two recent papers. The idea is to modify Cox's axioms for Bayesian probability to bring them more in tune with quantum theory. In particular the axiom of universal comparability ("for any propositions p and q, P (p|I) is greater, equal or smaller than P (q|I)", where I is an hypothesis) is dropped. This forces us to represent probabilities by complex numbers rather than real numbers. Probabilities are assigned to histories (temporal sequences of propositions) and a new axiom is introduced saying that temporal inversion of the sequence gives a complex conjugate probability. The probability of a history is found by a generalisation of the usual trace rule in the density matrix formalism.
Thomas hopes this formalism to have applications in quantum gravity. He makes much of an an analogy between Bayesian probability and General Relativity being both "relational" in the sense that they are compatible with Leibniz's principles of sufficent reason and of identity of indiscernibles, as discussed by Smolin. My intuitive opinion (not based, I confess, in any careful reading of the enormous material existent in this subject) is that Bayesianism is a reasonable interpretation of probability as applied to daily life and to all non-quantum contexts. Normally an assignment of probability to a proposition can only be done with respect to some given information, in a relative and not absolute way. (The probability of this coin toss falling on heads is 1/2 relative to my knowledge, but may be 1 relative to a Laplacean demon who can pre-calculate the result). But in a quantum context we do seem to have an absolute notion of probability: the probability that this particular measurement of an electron (in a pure state) will give as result spin up or down seems to be defined by quantum theory independently of any background information. (Of course it depends on the state of the electron, but normally the state in which the electron is taken as an "objective property" of it in some sense). My gut feeling is thus that Bayesianism is not an adequate interpretation of probability in quantum theory, but I am open to be convinced...
Thomas hopes this formalism to have applications in quantum gravity. He makes much of an an analogy between Bayesian probability and General Relativity being both "relational" in the sense that they are compatible with Leibniz's principles of sufficent reason and of identity of indiscernibles, as discussed by Smolin. My intuitive opinion (not based, I confess, in any careful reading of the enormous material existent in this subject) is that Bayesianism is a reasonable interpretation of probability as applied to daily life and to all non-quantum contexts. Normally an assignment of probability to a proposition can only be done with respect to some given information, in a relative and not absolute way. (The probability of this coin toss falling on heads is 1/2 relative to my knowledge, but may be 1 relative to a Laplacean demon who can pre-calculate the result). But in a quantum context we do seem to have an absolute notion of probability: the probability that this particular measurement of an electron (in a pure state) will give as result spin up or down seems to be defined by quantum theory independently of any background information. (Of course it depends on the state of the electron, but normally the state in which the electron is taken as an "objective property" of it in some sense). My gut feeling is thus that Bayesianism is not an adequate interpretation of probability in quantum theory, but I am open to be convinced...
posted by Alejandro at
6:27 PM
6 Comments:
Hiyah al,
wow, im so happy that you listened to my talk so well. I cant wholly disagree with anything you've just said. You are of course right that quantum mechanics is usually used to argue that quantum probabilities are somehow more `objective' than other probabilities. Although it is a falicy to say that this must be so. Or at least I have to say I dont understand the argument that no background information goes into probability assignments in quantum theory. In quantum theory things seem necessarily defined with respect to measurement etc. This means that probabilities, if they are to be objective must, in the least, be dispositional properties that depend on prior background information about which measurements occured and when. Such depenedence on background information certainly doesnt force us to use a Bayesian interpretation as we can interpret probabilities as dispositional propensities (objective chances that are defined relative to the experiment being discussed). However, I am convinced by the Bayesian programme not because things are, it seems to me, defined relative to background information in quantum theory, but rather that we have no reason to presume that such probabilities are objective, and it is not clear that we are helped in our endevour by doing so. If quantum probabilities are objective then by presuming they are subjective we will find that we are naturally forced to assign probabilities independent of anything other than the relevant background information. If this is the case then we would have proved that an objective (or at least intersubjective) interpretation of probability is the way to go. However, this has not yet been proved, and the only way to prove it is to start out with a subjective interpretation and prove that we are naturally forced to introduce objectivity or intersubjectivity. The is one reason why I have taken to using Bayesian probabilities. Another reason is that it seems to be defined relationally, which is an approach that I identify with. The only way one can show that a theory of probability is adequate in quantum theory is simply to apply it and find out if you get consistency with physics. This is what we are trying to do. It is certainly the case that standard interpretations of probability are NOT compatible with the physics. Relative frequencies are necessarily additive, and propensities are just relative frequencies or Bayesian probabilities that are suddenly called objective without justification. Quantum theory introduces nonadditive 'probabilities' when everything in relative frequencies and Bayesian prob theory tells us that probs should be additive. Thus something is clearly going wrong. Similarly with nonlocal probabilities. Thus we believe it is necessary to start foundationally from the beginning, only introduce elements to the theory that are relationally defined, and then we might get something novel that certainly, by definition since we are doing things relationally, is compatible with the physics. We have not yet achieved this aim, but we think it is a path worth following.
Thanks again for your comments, sorry if i've ranted a little bit!
Yours,
Tom
By
Anonymous, at 9:38 AM, March 14, 2006
I mentioned another interesting Bayesian approach to quantum mechanics here, which is receiving much philosophical attention.
By
Anonymous, at 10:07 AM, March 14, 2006
Tom, Alejandro, are you aware of the work of Saul Youssef on the reinterpretation of quantum mechanics as what he calls "an exotic probability theory"?
He has also modified the Cox axioms such that probability is bayesianly defined over the complex numbers.
It would be interesting to see if there are any substantial differences between your approaches, and if you are able to derive the same results as Youssef claims, such as the path integral, the superposition principle, the expansion postulate and the schrodinger/klein-gordon equations.
Please see http://arxiv.org/abs/quant-ph/9509004 and references therefrom.
Regards,
By
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