As a student of theoretical physics I'm well acquainted with the multitude of crackpot ideas attempting to circumvent Bell's theorem regarding local hidden variable theories in quantum physics.

Recently, however, I've been working on my master's thesis regarding Bayesian probability, and I came across a very interesting paper by Jaynes on precisely the subject of Bell's theorem (E.T. Jaynes, Clearing Up Mysteries - The Original Goal, In: Proceedings, Maximum Entropy and Bayesian Method, 1989).

Jaynes writes about what he calls the Mind Projection Fallacy and its prevalence in quantum mechanics. He claims the fallacy is a result of failing to appreciate probabilities as representations of states of knowledge (epistemological), as opposed to as fundamental properties of nature (ontological); clearly, Jaynes advocates the Bayesian perspective on probability.

Using his 'Bayesian inference as extended logic' approach, Jaynes derives a number of - to me - impressive results in this paper and others. More to the point, on pages 7-16 he explains two objections to Bell's results:

  1. Bell didn't appreciate the difference between the epistemological nature of probability in making predictions and the ontological nature of causality. This lead him to propose the wrong probability distribution for his class of hidden variable theories; one which is indeed (trivially) violated by quantum mechanics.

  2. Bell did not include all local hidden variable theories. For instance, his choice excludes those where the hidden variables are time-dependent.

These objections don't read crackpot in my opinion, and as demonstrated in the linked papers there is a slight historic tendency for the Bayesian perspective to make one see old results in a new light, particularly in other fields of physics.

I've heard that Jaynes is adept at making himself seem obviously right and others obviously wrong - so I may have fallen for that trap - but this argument struck me as something that should've gotten a lot more attention than I'm aware it has. That is, I was still taught the Copenhagen interpretation complete with Bell's theorem ruling out local determinism, which seems to imply that this argument has either not gotten mainstream attention or has been thoroughly debunked.

Are there any obvious counters to Jaynes' viewpoint that I'm not aware of?

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    $\begingroup$ This blog post comes from a similar position as you and provides references to work that (nominally) answers your question. $\endgroup$ Feb 2, 2016 at 0:07
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    $\begingroup$ But then Jaynes did not give an example of a locally deterministic theory where Bell's theorem is actually violated. $\endgroup$ Feb 2, 2016 at 0:08
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    $\begingroup$ Great question! +1 $\endgroup$
    – Thriveth
    Feb 2, 2016 at 0:42
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    $\begingroup$ @DerekElkins: I'm still rather confused on one issue. Jaynes claims that Bell's assumptions are wrong, and the paper by Colbeck and Renner mentions similarly that the assumptions are too strong, in a way that is reminiscent of Jaynes' logical vs physical causality argument. On the other hand, they explain their assumptions are weaker because they want to exclude more theories, not because Bell is wrong. Jaynes is wrong about a more informative theory existing (either classical or quantum), but his argument regarding Bell's assumptions still seems compelling. I don't know what to make of this. $\endgroup$
    – Timsey
    Feb 2, 2016 at 18:40
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    $\begingroup$ Yes, e.g. 't Hooft has been working on locally deterministic models, he claims that the superdeterminism loophole is going to make his effort not futile. But what ultimately matters is whether or not his proposed models will end up working or not. $\endgroup$ Feb 2, 2016 at 19:29

4 Answers 4


Bell's argument actually fails in a deterministic universe. The argument's fundamental assumption is that the outcome of a measurement on one particle can't depend on the measurement basis chosen by the other experimenter for the other particle at a spacelike separated location. This is usually described as a locality assumption, but there's more to it than that. If the experimenter's choice is actually an inevitable consequence of the state of the universe at earlier times, then it's possible that it's an inevitable consequence of the state at the place and time that the entangled pair of particles was created. If so, the particles could decide then and there how to respond to the measurements that inevitably will happen later, reproducing the prediction of quantum mechanics without any spooky long-distance communication.

It's hard to imagine how elementary particles could know in advance the result of a physical process as complicated as the choice of a measurement basis by an experimenter, nor how that choice could be predictable solely from information localized in a small part of its past light cone, and as far as I know no one has actually proposed a local hidden variable theory in which this happens. But Bell didn't prove it impossible.

So if Jaynes is taking the hardline position that the universe is deterministic and all probabilities are inherently about our lack of knowledge of its state, I think he's at least morally correct, and he may be literally correct in his entire argument. He's right that Bell's argument assumes a Y-causes-X relationship in the probabilities P(X|Y), and he's right that there is no such relationship in subjective probabilities. You need some kind of objective unpredictability in the world for Bell's argument to go through – not quantum unpredictability, but some sort of free choice or true classical randomness at the spacelike separated locations of the measurements.

To the extent that Jaynes acts like the mystery of Bell's theorem is totally resolved just because he can model the quantum prediction in his classical probabilistic logic, I think he's wrong. A deterministic model where particles know the future is not logically impossible, but it would be really frickin' weird, and I don't expect any such model to appear. (Although, quantum mechanics is also weird, and I never would have predicted it, so I suppose I should expect to be surprised.)

  • $\begingroup$ "and as far as I know no one has actually proposed a local hidden variable theory in which this happens." Gerardus 't Hooft, physics nobel prize winner, as been working on this arxiv.org/abs/2103.04335 $\endgroup$ Oct 10, 2022 at 18:58
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    $\begingroup$ @MiltonSilva I think all he does in that and similar papers is argue that the outcomes of all future measurements can be encoded into the initial conditions, which has the same problem as arguing that God created fake fossils 6000 years ago: there's no reason for fake fossils to seem to support biological evolution unless God deliberately chose to deceive us. 't Hooft doesn't know how to make a superdeterministic theory that naturally behaves like QM. $\endgroup$
    – benrg
    Oct 13, 2022 at 18:21

I'm a fan of Jaynes and doubt that there is a legitimate refutation to his objection of the lack of time variance, though I think that should obviously be extended to space-time.

The natural conditioned variables for deterministic events in space-time for something with a spacetime wavefunction would have to include the space-time of each event, detector orientation, and any quantum interactions of the measurement devices themselves. Maybe the last doesn't play much of a part, but for a deterministic theory, the rest are obvious.

M1(d1| st0, st1, or1, lambda(st0)) M2(d2| st0, st2, or2, lambda(st0))

You would detect/not detect based on OR1 alignment relative to particle state at st1 after it's split at st0 and state lambda(st0).

Jaynes' objection is straightforward. Bell didn't analyze conditioned on these other parameters, hence did not rule out a hidden variable theory based on them.

The problem with any "there is no such theory that" is that it has to assume some class of theories. If Bell didn't condition on the space-time of each event, then he didn't. If others didn't, then they didn't.

Also, mathematics advances. You can't cover all possible mathematics. We expand the use of mathematical structures as we find an effective use of them. Saw some paper mentioning imaginary valued probabilities. What does that mean? Who the hell knows? But I bet Bell didn't include imaginary probabilities in his class of possible solutions to analyze for a deterministic theory.

From the description in the blog post, the Gill paper seems a convoluted special case, introducing "free choice" analysis confusing matters and besides the point, and less likely to produce a valid probabilistic analysis than Bell's original scenario.

I wish there was a source of raw data of detector counts from which we could attempt to construct a deterministic theory.


This sounds more like a mild crackpot, the further I read. He accuses Bohr of assuming that all instruments are subject to the uncertainty principle (fourth paragraph, page 8). This, however, was an experimental observation, not an assumption. Page 9, end of paragraph 2, he accuses Bohr of confusing limitations on QM theory with limitations on the validity of lab measurements. This is a rather dubious point, as theory is needed to interpret experimental results. Further on page 9, 5th paragraph, he essentially refutes the orthodox QM view by defining it as incorrect, stating that it violates his "necesarry division of labor" in theoretical physics.

At the top of page 13, the author makes a distinction between a physical influence faster than light and a logical inference, which is charactaristic of the intangibility of the "spooky action at a distance". It does not, however, change the result of the reasoning of the EPR experiment.

The last part, where it looks like he's getting into a possible loophole, he mentions time-varying hidden variables, but really doesn't say clearly how this would work. His closing statement sums his views up well.

It is very common for crackpots to object to relativity or quantum mechanics on "philosophical" grounds, (which I put in quotes to avoid insulting philosophers) and treat their own philosophy as axiomatic. This seems to be what he's doing.

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    $\begingroup$ Jaynes possessed hallmarks of a crackpot - a prolific writer who expressed himself in acerbic, polemical language, proselytized about his ideas, and had a somewhat conspiratorial view of science and mistrust of authorities. But unlike a crackpot, Jaynes made significant, strikingly original contributions in several fields in physics and remains greatly respected in fields he influenced. $\endgroup$
    – innisfree
    Sep 24, 2016 at 6:00

Bell was simply wrong, did not understand basic probability math, and the experiments actually show no difference between all the ideas and Einstein–Podolsky–Rosen's (EPR) simple hidden variable. Remember Occam's, and please choose the simpler explanation. The math is explained properly by Jaynes, but I'll make it very simple with an example. Two particles are created such that they have opposing spins. This is a given, but we will gloss over the fact that the spins are not necessarily exactly opposite at every point in time in their futures. We only require that some physical measurement (e.g. angular momentum) is conserved according to what current theory or measurement says so. But let's for the moment assume they are always opposite. If we assume EPR, then there is some 3D vector showing the spin poles of each particle. We don't know what it is yet, so we measure it. (By the way, this is identical to what would happen with quantum mechanics (QM), that the actual value only exists at measurement time.) But how? We have this apparatus that shows if some set angle of a piece of equipment is aligned with the spin of the particle. Well, not exactly, as we could never get it aligned exactly, so our apparatus just tells us if the angles are off by less than 90 (spin UP) or more than 90 (spin DOWN) from the the angle we chose. So our unknown value, as measured in the 2D plane of our rotating test, can only tell us that the actual spin value of particle A was somewhere $\pm$ 90 degrees to our test angle. This is the same regardless of EPR or QM. The actual value, whether it collapsed just now, or existed from $t_0$, we can only know the value was in this half of the potential solution space. Now we know with certainty that the other particle B must be in the other 180 degree part of the circle. If the other measurement was 90 degrees off from the first, we would expect 50/50 chance of UP or DOWN, as the known 180 degree possible area is divided exactly in half. If the chosen angles were the same, there is 100% chance they will be opposite. However, if measurement B was 120 degrees off from A, there is a smaller possibility they will have the same result! We are simply revealing the underlying real physical value. The expositions of the form, "look at these 9 possible outcomes of 11, 12, 13, 21, etc" do not correctly describe that each of these 9 buckets have different probabilities as a function of how we are measuring, and what angles we choose. The 5/9ths ratio is false! And now that I have shown you how easy it is to see with an example, you should use the correct math equations as Jayne does. The expected outcome of measurement B changes when we have information from measurement A. No need to separate with a long distance. No transfer of state. Just a little more info gained. And it makes no difference! These experiments can never reveal what is really going on. We can only know that there is a fuzzy value of some measurement that tells us which half of a circle had the spin of the thing when we measured it. I like Occam; no nonsense required.

  • $\begingroup$ The model you describe doesn't violate Bell's inequality. If the spin direction is chosen uniformly at random, then the model gives a triangle-shaped correlation function that you can see in many discussions of Bell's theorem (e.g. the red curve here), which saturates the inequality. Jaynes knew that, and wasn't making the argument that you think he was making. $\endgroup$
    – benrg
    Nov 13, 2022 at 2:43
  • $\begingroup$ @galen your edit has significantly changed the tone of the post. It is certainly improved but I suspect the OP was deliberately provocative so your edit goes against the intent of the OP. $\endgroup$ Nov 13, 2022 at 4:15
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    $\begingroup$ I do appreciate the edit, as my tone was uncalled for and I offer my apologies. I will study this more. but it does seem almost impossible to discuss without actual data, if any is available in a simple xls file. Also, without specifying what the measurment devices actually do, it is likewise difficult to discuss. My assumed method is likely far off., as I have written a simple C++ program that passes the 3 machine Randi challenge with this assumption quite easily. $\endgroup$ Nov 14, 2022 at 0:11

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