Re: Jaynes and Bell's Theorem (QM Interpretations)

[nobody <nobody@mscan2.ucar.edu>]

On Tue, 5 Feb 2002 Al Tino wrote:

> I have read several of E.T. Jaynes' interesting papers on Bayesian
> inference.  In particular, I have read Jaynes' "Clearing up
> Mysteries...", which Gordon Pusch references,

[ URL: http://bayes.wustl.edu/etj/articles/cmystery.pdf
   or  http://bayes.wustl.edu/etj/articles/cmystery.ps.gz ]

> but I'm having trouble understanding Jaynes' (Bayesian) critique of
> Bell's Theorem.

That's because it is not a terribly cogent critique.

> Is Jaynes saying that Bell made an error in coming up with his
> theorem, and therefore it is invalid and ignorable?  And that
> empirical violations of the Bell inequalities are not particularly
> meaningful?

I think that Jaynes is arguing that Bell's physical assumptions that
serve to define his notion of a local hidden variable theory are not
meaningful.  But Jaynes completely evades the question of what a
proper characterization of a local hidden variable theory might be.

With the indulgence of the moderators, I'd like to quote your excellent
summary of Bell's physical assumptions:

> Let p(A|a) represent the probability of measuring "up" on particle 1
> when its detector is set to angle "a"; similarly, p(B|b) for particle
> 2. The joint probability for measurements on the pair is then
> p(AB|abh), where "h" represents the "hidden variable" of the local
> realistic model that Bell assumed.
> 
> Jaynes objects to Bell's factorization of the joint probability for
> measurements on the correlated EPR particles:
>  
>    p(AB|abh) = p(A|ah)p(B|bh)
>  
> Jaynes states that the fundamentally correct factorization is:
> 
>    p(AB|abh) = p(A|Babh)p(B|abh) [equivalently, p(B|Aabh)p(A|abh)]
> 
> This is certainly true, but uninteresting.  To deduce something
> interesting, some physical assumptions must be introduced.

Entirely correct.

> While Bell lumped his assumptions into something he called "locality",
> later investigators (e.g., Jarrett, Shimony) have broken Bell's
> "locality" into two separate, independent pieces:
> 
>    1) parameter independence: p(B|abh) = p(B|bh); this says that the
>       probability of the result at particle 2 does not depend on the
>       detector angle of particle 1.  Were this not the case, special
>       relativity could be violated.  [Of course, QM satisfies this.]
> 
>    2) outcome independence: p(A|Babh) = p(A|abh); this says that if my
>       "realistic" model is "complete", my measurement "outcome" for
>       particle 1 does not depend on the outcome for particle 2. [QM
>       does not satisfy this.]
> 
> These two together allow Bell's factorization and, thus, Bell's theorem.

Indeed.

> Could someone familiar with the Bayesian view tell me why Jaynes would
> object to these assumptions?  How would Jaynes represent these
> physical assumptions?

Those are very good questions.  As to the first, Jaynes does explicitly
accuse Bell of making a logical error:

% How do we get out of this?  Just as Bell revealed hidden assumptions in
% von Neumann's argument, so we need to reveal the hidden assumptions in
% Bell's argument.  There are at least two of them  both of which require
% the Jeffreys viewpoint about probability to recognize:
% 
% (1) As his words above show Bell took it for granted that a conditional
%     probability P(X|Y) expresses a physical causal influence exerted by
%     Y on X.  But we show presently that one cannot even reason correctly
%     in so simple a problem as drawing two balls from Bernoulli's Urn
%     if he interprets probabilities in this way.  Fundamentally,
%     consistency requires that conditional probabilities express logical
%     inferences, just as Harold Jeffreys saw.  Indeed  this is also the
%     crucial point that Bohr made in his reply to EPR in words that Bell
%     quoted and which we repeat below.

In my view, this accusation is simply false.  What Bell assumes is that
_if_ there is a causal influence exerted by Y on X _then_ the probability
P(X|Y) must reflect it in a particular way.  There is nothing that argues
against an epistemologic view of probabilities in this:  indeed, if Y
exerts a causal influence on X, and if something is known about Y, then
Bayesian philosophy _demands_ that such information be taken into account
in assigning probabilities to X.

As to the second question, I think that Jaynes simply does not understand
the issue.  He says:

% (2) The class of Bell theories does not include all local hidden
%     variable theories;  it appears to us that it excludes just the class
%     of theories that Einstein would have liked most.  Again  we need to
%     learn from Jeffreys the distinction between the epistemological
%     probabilities of the QM formalism and the ontological frequencies
%     that we measure in our experiments.  A hidden variable theory need
%     not reproduce the mathematical form of the QM probabilities in the
%     manner of [Bell's physical assumptions] in order to predict the same
%     observable facts that QM does.
% 
% The spooky superluminal stuff would follow from Hidden Assumption (1)
% but that assumption disappears as soon as we recognize  with Jeffreys
% and Bohr that what is traveling faster than light is not a physical
% causal influence but only a logical inference.

It is certainly true that one can construct hidden variable theories
that do not satisfy Bell's physical assumptions and do reproduce the
predictions of QM.  But are they local hidden variable theories?  Jaynes
asserts that they can be.  But while he declares Bell's mathematical
characterization of locality to be invalid, he offers offers no
alternative characterization and seems to dismiss the matter of such
characterization as irrelevant.  This suggests to me that he did not
really understand the issue.

nobody

P.S.  Incidentally, there are other serious flaws in this paper.
For instance, the derivation of the diffusion equation is completely
bogus.  It can be done (and done correctly) from the assumptions Jaynes
makes by the methods outlined in (e.g.) R. F. Pawula, IEEE Trans.
Inform. Theory 13, 33-41 (1967).