The Amelioration of Uncertainty

Bayes Theorem and Quantum Mechanics

I was reminded of this old post by Andrew Gelman about whether Quantum Mechanics requires a change in the axioms of probability theory. Without weighing in I’ll just point out that Bayesian Statistics is more general than Bayes Theorem and this affects the controversy.

With some poetic license we can say Bayesians take the axioms of probability theory (sum and product rule) as fundamental and we’d rather solve every problem with them than invent new ad-hoc principles like p-values. So when it comes to judging hypothesis x based on data D we use the product rule to derive Bayes Theorem:

    equation

But this only applies in certain contexts, or at least, it’s only usable in certain contexts. Under different conditions the axioms imply quite different formulas. If for example there is a nuisance parameter equation they lead to this updating formula:

    equation

If D only affects x through an intermediary equation then they imply the correct updating formula is:

    equation

If x is determined by averaging over models equation then they are related by the following system of equations:

    equation

The particular examples aren’t important. The point is that there are an infinite number of contexts in statistics and so the axioms imply an infinite number of “updating formulas”. Bayes Theorem is merely the simplest.

So every researcher has two choices. They can believe we understand the context in QM and therefore need a weird new form of probability theory. Or they can keep the old axioms and search for the context which generates the peculiar formulas seen in QM.

Which research path is chosen seems to depend largely on philosophy and temperament. Everyone has to make their choice and live with the consequences though, since it’s doubtful they’re equally fruitful. Sometimes philosophy really does make a difference.

November 2, 2013
4 comments »
  • November 3, 2013Rasmus Bååth

    The comments on the post by Gelman is really an interesting read and so is the list of diffrent interpretations of QM on Wikipedia :)

    http://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics

    It’s a really long list and I’m just wondering, in your opinion, which are the worst offenders of probability as extended logic? Which interpretation, if any, do you favour?

  • November 3, 2013Joseph

    Rasmus,

    I don’t know what to think about QM. I only know which things I’m willing to spend time working on and which I’m not. For example, I’m unwilling to spend even 1 second of my life thinking about the many worlds interpretation of QM.

    Those kinds of decisions are highly personal though.

  • November 4, 2013Daniel Lakeland

    I’m with you on the Many Worlds interpretation. As far as I’m concerned we have enough crazy stuff going on in this world for a lifetime ;-)

  • November 4, 2013Joseph

    Indeed. There’s this blog:

    http://www.math.columbia.edu/~woit/wordpress/

    Which I love for no other reason than because it confirms all my prejudices about physics. I wonder if Gelman knows Peter Woit since they’re both at Columbia.

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