Deriving the Born Rule

[Bruce Kellett]

The meaning of probability and the origin of the Born rule has been seem
as one of the outstanding problems for Everettian quantum theory.
Attempts by Carroll and Sebens, and Zurek to derive the Born rule have
considered probability in terms of the outcome of a single experiment
where the result is uncertain. This approach is can be seen to be
misguided, because probabilities cannot be determined from a single
trial. In a single trial, any result is possible, in both the
single-world and many-worlds cases -- probability is not manifest in
single trials.

It is not surprising, therefore, the Carroll and Zurek have concentrated
on the basic idea that equal amplitudes have equal probabilities, and
have been led to break the original state (with unequal amplitudes) down
to a superposition of many parts having equal amplitude, basically by
looking to "borrow" ancillary degrees of freedom from the environment.
The number of equal amplitude components then gives the relative
probabilities by a simple process of branch counting. As Carroll and
Sebens write:

"This route to the Born rule has a simple physical interpretation. Take
the wave function and write it as a sum over orthonormal basis vectors
with equal amplitudes for each term in the sum (so that many terms may
contribute to a single branch). Then the Born rule is simply a matter of
counting -- every term in that sum contributes an equal probability."
(arxiv:1405.7907 [gr-qc])

Many questions remain as to the validity of this process, particularly
as it involves an implementation of the idea of self-selection: of
selecting which branch one finds oneself on. This is an even more
dubious process than branch counting, since it harks back to the
"many-minds" ideas of Albert and Loewer, which even David Albert now
finds to be "bad, silly, tasteless, hopeless, and explicitly dualist."

Simon Saunders (in his article "Chance in the Everett Interpretation"
(in  "Many Worlds: Everett, Quantum Theory, & Reality" Edited by
Saunders, Barrett, Kent and Wallace (OUP 2010)) points out that
probabilities can only be measured (or estimated) in a series of
repeated trials, so it is only in sequences of repeated trials on an
ensemble of similarly prepared states that we can see how probability
emerges. This idea seemed promising, so I came up with the following
argument.

If, in classical probability theory, one has a process in which the
probability of success in a single Bernoulli trial is p, the probability
of getting M successes in a sequence of N independent trials is p^M
(1-p)^(N-M). Since there are many ways in which one could get M
successes in N trials, to get the overall probability of M successes we
have to sum over the N!/M!(N-M)! ways in which the M successes can be
ordered. So the final probability of getting M successes in N
independent trials is

      Prob of M successes = p^M (1-p)^(N-M) N!/M!(N-M)!.

We can find the value of p for which this probability is maximized by
differentiating with respect to p and finding the turning point. A
simple calculation gives that p = M/N maximizes this probability (or,
alternatively, maximizes the amplitude for each sequence of results in
the above sum). This is all elementary probability theory of the
binomial distribution, and is completely uncontroversial.

If we now turn our attention to the quantum case, we have a measurement
(or sequence of measurements) on a binary quantum state

     |psi> = a|0> + b|1>,

where |0> is to be counted as a "success", |1> represents anything else
or a "fail", and a^2 + b^2 = 1. In a single measurement, we can get
either |0> or 1>, (or we get both on separate branches in the Everettian
case). Over a sequence of N similar trials, we get a set of 2^N
sequences of all possible bit strings of length N. (These all exist in
separate "worlds" for the Everettian, or simply represent different
"possible worlds" (or possible sequences of results) in the single-world
case.) This set of bit strings is independent of the coefficients 'a'
and 'b' from the original state |psi>, but if we carry the amplitudes of
the original superposition through the sequence of results, we find that
for every zero in a bit string we get a factor of 'a', and for every
one, we get a factor of 'b'.

Consequently, the amplitude multiplying any sequence of M zeros and
(N-M) ones, is a^M b^(N-M). Again, differentiating with respect to 'a'
to find the turning point (and the value of 'a' that maximizes this
amplitude), we find

    |a|^2 = M/N,

where we have taken the modulus of 'a' since a is, in general, a complex
number. Again, there will be more than one bit-string with exactly M
zeros and (N-M) ones, and summing over these gives the additional factor
of N!/M!(N-M)!, as above.

If we now compare the quantum result over N measurements with the
classical probability result for N independent Bernoulli trials, we find
that the amplitude for M successes in N trials is maximized when the
modulus squared of the original amplitude equals the relative number of
successes, M/N, which is the classical probability. Thus, the
probability for measuring 'zero' in a single trial is just given by the
modulus squared of the amplitude for that component of the original
state. This is the Born rule.

Furthermore, if in the quantum case we square the amplitude for the sum
over bit-strings with exactly M zeros, we get

    |a|^(2M) (1 - |a|^2)^(N-M) N!/M!(N-M)!,

which, with the identification |a|^2 = M/N, is just the probability for
M successes in N Bernoulli trials with probability for success p = M/N,
as above in the standard binomial case.

Now this result could be viewed as a derivation of the Born rule for the
quantum case. Or it might be no more than a demonstration of the
consistency of the Born rule, if it is already assumed. I am not sure
either way. If nothing else, it demonstrates that the mod-squared
amplitude plays the role of the probability in repeated measurements
over an ensemble of similarly prepared quantum systems, given that one
observes only one of the possible sequences of results, either as a
single world or as one's particular 'relative state'.

Note also, that this argument is independent of any Everettian
considerations -- one can always take a modal interpretation of the
bit-strings other than the one actually observed, and see them as
corresponding to 'other possible worlds', or sequences of results that
could (counterfactually) have been obtained, but weren't. In other
words, there is no necessity for the Everettian assumption that all
sequences of results obtain in some actual world or the other. The modal
single-world interpretation has the distinct advantage that it avoids
the ugly locution that "low probability sequences certainly exist".

Bruce

[Brent Meeker]

This is what you previously argued was not part of the Schroedinger
equation and was a cheat to slip the Born rule in.  It's what I said was
Carroll's "weight" or splitting of many pre-existing worlds.

>
> Consequently, the amplitude multiplying any sequence of M zeros and
> (N-M) ones, is a^M b^(N-M). Again, differentiating with respect to 'a'
> to find the turning point (and the value of 'a' that maximizes this
> amplitude), we find
>
>     |a|^2 = M/N,

Maximizing this amplitude, instead of simply counting the number of
sequences with M zeroes as a fraction of all sequences (which is
independent of a) is effectively assuming |a|^2 is a probability
weight.  The "most likely" number of zeroes, the number that occurs most
often in the 2^N sequences, is is N/2.

Brent

[Bruce Kellett]

On Wed, May 13, 2020 at 2:06 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/12/2020 7:12 PM, Bruce Kellett wrote:

> If we now turn our attention to the quantum case, we have a
> measurement (or sequence of measurements) on a binary quantum state
>
>      |psi> = a|0> + b|1>,
>
> where |0> is to be counted as a "success", |1> represents anything
> else or a "fail", and a^2 + b^2 = 1. In a single measurement, we can
> get either |0> or 1>, (or we get both on separate branches in the
> Everettian case). Over a sequence of N similar trials, we get a set of
> 2^N sequences of all possible bit strings of length N. (These all
> exist in separate "worlds" for the Everettian, or simply represent
> different "possible worlds" (or possible sequences of results) in the
> single-world case.) This set of bit strings is independent of the
> coefficients 'a' and 'b' from the original state |psi>, but if we
> carry the amplitudes of the original superposition through the
> sequence of results, we find that for every zero in a bit string we
> get a factor of 'a', and for every one, we get a factor of 'b'.

This is what you previously argued was not part of the Schroedinger
equation and was a cheat to slip the Born rule in.  It's what I said was
Carroll's "weight" or splitting of many pre-existing worlds.

This is not what Carroll does. He looks at a single measurement, and boosts the number of components of the wave function so that all have the same amplitude. That, I argue, is a mistake.

>
> Consequently, the amplitude multiplying any sequence of M zeros and
> (N-M) ones, is a^M b^(N-M). Again, differentiating with respect to 'a'
> to find the turning point (and the value of 'a' that maximizes this
> amplitude), we find
>
>     |a|^2 = M/N,

Maximizing this amplitude, instead of simply counting the number of
sequences with M zeroes as a fraction of all sequences (which is
independent of a) is effectively assuming |a|^2 is a probability
weight.  The "most likely" number of zeroes, the number that occurs most
often in the 2^N sequences, is is N/2.

I agree that if you simply look for the most likely number of zeros, ignoring the amplitudes, then that is N/2. But I do not see that maximising the amplitude for any particular value of M is to effectively assume that it is a probability. When you do this, you can see by analogy that it is a probability, but one did not assume this at the start.

Bruce

[Brent Meeker]

I think it is.  How would you justify ".. the amplitude multiplying any sequence of M zeros and (N-M) ones, is a^M b^(N-M)..." except by saying a is a probability, so a^M is the probability of M zeroes.  If it's not a probability why should it be multiplied into and expression to be maximized?

In any case though, I don't see the form of the Born rule as something problematic.  It's getting from counting branches to probabilities.  Once you assume there is a probability measure, you're pretty much forced to the Born rule as the only consistent probability measure.

Brent

When you do this, you can see by analogy that it is a probability, but one did not assume this at the start.

Bruce

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[Bruce Kellett]

On Wed, May 13, 2020 at 3:30 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/12/2020 10:08 PM, Bruce Kellett wrote:

On Wed, May 13, 2020 at 2:06 PM 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

> Consequently, the amplitude multiplying any sequence of M zeros and
> (N-M) ones, is a^M b^(N-M). Again, differentiating with respect to 'a'
> to find the turning point (and the value of 'a' that maximizes this
> amplitude), we find
>
>     |a|^2 = M/N,

Maximizing this amplitude, instead of simply counting the number of
sequences with M zeroes as a fraction of all sequences (which is
independent of a) is effectively assuming |a|^2 is a probability
weight.  The "most likely" number of zeroes, the number that occurs most
often in the 2^N sequences, is is N/2.

I agree that if you simply look for the most likely number of zeros, ignoring the amplitudes, then that is N/2. But I do not see that maximising the amplitude for any particular value of M is to effectively assume that it is a probability.

I think it is.  How would you justify ".. the amplitude multiplying any sequence of M zeros and (N-M) ones, is a^M b^(N-M)..." except by saying a is a probability, so a^M is the probability of M zeroes.  If it's not a probability why should it be multiplied into and expression to be maximized?

Trivially, without any assumptions at all. The original state has amplitudes a|0> + b|1>. If you carry the coefficients through at each branch, the branch containing a new |0> carries a weight a, and similarly, the branch containing a new |1> carries a weight b. One does not have to assume that these are probabilities to do this -- each repeated trial is a branch point, so each is another measurement of an instance of the initial state, so automatically has the coefficients present. I don't see anything sneaky here.

As to the question as to why it should be maximised? Well, why not? I am simply maximising the carried through coefficients to find if this has any bearing on the proportion M of zeros. The argument for probabilities then proceeds by means of the analogy with the traditional binomial case. I agree, this may not count as a derivation of the Born rule for probabilities, but it is certainly a good explication of the same.

 

In any case though, I don't see the form of the Born rule as something problematic.  It's getting from counting branches to probabilities.

I think my issue here is that counting branches is not the thing to do, because the branches are not in proportion to the coefficients (which turn out to be probabilities). And counting branches for probabilities requires the self-location assumption, and that is intrinsically dualist (as David Albert points out).

 Once you assume there is a probability measure, you're pretty much forced to the Born rule as the only consistent probability measure.

I agree. And that is the argument Everett made in his 1957 paper -- once you require additivity, the fact that states are normalised, screams for the coefficients to be treated as probabilities. The squared amplitudes obey all the Kolmogorov axioms, after all.

Bruce

[Philip Thrift]

On Wednesday, May 13, 2020 at 12:30:44 AM UTC-5, Brent wrote:

In any case though, I don't see the form of the Born rule as something problematic.  It's getting from counting branches to probabilities.  Once you assume there is a probability measure, you're pretty much forced to the Born rule as the only consistent probability measure.

Brent

 

Once one approaches the domain of 'quantum phenomena' as a probability/measure theorist would do, then all roads (formulations of the underlying measure space) should lead to Born.

A measure theory on the appropriately-defined measure space underlies both probability theory and what has been called quantum-probability theory.

Schwinger’s picture of Quantum Mechanics

https://arxiv.org/pdf/2002.09326.pdf

A gentle introduction to Schwinger’s formulation of quantum mechanics: The groupoid picture

https://www.researchgate.net/publication/325907723_A_gentle_introduction_to_Schwinger's_formulation_of_quantum_mechanics_The_groupoid_picture

Quantum measures and the coevent interpretation

https://arxiv.org/abs/1005.2242

cf.

Probabilities on Algebraic Structures

Ulf Grenander

review: https://projecteuclid.org/download/pdf_1/euclid.aoms/1177700302

Derivation of the Schrödinger equation from the Hamilton-Jacobi equation in Feynman's path integral formulation of quantum mechanics

J.H.Field

https://arxiv.org/abs/1204.0653

Feynman’s path integral formulation of quantum mechanics is based on the following two postulates [11]:

1. If an ideal measurement is performed to determine whether a particle has a path lying in a region of spacetime, the probability that the result will be affirmative is the absolute square of a sum of complex contributions, one from each path in the region.

2. II The paths contribute equally in magnitude but the phase of their contribution is the classical action (in units of ¯h) i.e. the time integral  along the path.

[11] Feynman R.P. 1948 Rev. Mod. Phys. 20 367.

@philipthrift

[Bruno Marchal]

> On 13 May 2020, at 04:12, Bruce Kellett <bhke...@optusnet.com.au> wrote:
>
> The meaning of probability and the origin of the Born rule has been seem as one of the outstanding problems for Everettian quantum theory. Attempts by Carroll and Sebens, and Zurek to derive the Born rule have considered probability in terms of the outcome of a single experiment where the result is uncertain. This approach is can be seen to be misguided, because probabilities cannot be determined from a single trial. In a single trial, any result is possible, in both the single-world and many-worlds cases -- probability is not manifest in single trials.
>
> It is not surprising, therefore, the Carroll and Zurek have concentrated on the basic idea that equal amplitudes have equal probabilities, and have been led to break the original state (with unequal amplitudes) down to a superposition of many parts having equal amplitude, basically by looking to "borrow" ancillary degrees of freedom from the environment. The number of equal amplitude components then gives the relative probabilities by a simple process of branch counting. As Carroll and Sebens write:
>
> "This route to the Born rule has a simple physical interpretation. Take the wave function and write it as a sum over orthonormal basis vectors with equal amplitudes for each term in the sum (so that many terms may contribute to a single branch). Then the Born rule is simply a matter of counting -- every term in that sum contributes an equal probability." (arxiv:1405.7907 [gr-qc])
>
> Many questions remain as to the validity of this process, particularly as it involves an implementation of the idea of self-selection: of selecting which branch one finds oneself on.

Only in the relative way. I mean it is not the Bayesian kind of self-selection from “nowhere” that we encounter in Leslie and Carter’s doomsday argument.

> This is an even more dubious process than branch counting, since it harks back to the "many-minds" ideas of Albert and Loewer, which even David Albert now finds to be "bad, silly, tasteless, hopeless, and explicitly dualist."
>
> Simon Saunders (in his article "Chance in the Everett Interpretation" (in "Many Worlds: Everett, Quantum Theory, & Reality" Edited by Saunders, Barrett, Kent and Wallace (OUP 2010)) points out that probabilities can only be measured (or estimated) in a series of repeated trials, so it is only in sequences of repeated trials on an ensemble of similarly prepared states that we can see how probability emerges.

I agree with this for computationalism and for quantum mechanics, but there are domains in which probabilities makes bayesian sense, and do not require any trials.

> This idea seemed promising, so I came up with the following argument.
>
> If, in classical probability theory, one has a process in which the probability of success in a single Bernoulli trial is p, the probability of getting M successes in a sequence of N independent trials is p^M (1-p)^(N-M). Since there are many ways in which one could get M successes in N trials, to get the overall probability of M successes we have to sum over the N!/M!(N-M)! ways in which the M successes can be ordered. So the final probability of getting M successes in N independent trials is
>
> Prob of M successes = p^M (1-p)^(N-M) N!/M!(N-M)!.

OK.

>
> We can find the value of p for which this probability is maximized by differentiating with respect to p and finding the turning point. A simple calculation gives that p = M/N maximizes this probability (or, alternatively, maximizes the amplitude for each sequence of results in the above sum). This is all elementary probability theory of the binomial distribution, and is completely uncontroversial.

OK.

>
> If we now turn our attention to the quantum case, we have a measurement (or sequence of measurements) on a binary quantum state
>
> |psi> = a|0> + b|1>,
>
> where |0> is to be counted as a "success", |1> represents anything else or a "fail", and a^2 + b^2 = 1. In a single measurement, we can get either |0> or 1>, (or we get both on separate branches in the Everettian case). Over a sequence of N similar trials, we get a set of 2^N sequences of all possible bit strings of length N. (These all exist in separate "worlds" for the Everettian, or simply represent different "possible worlds" (or possible sequences of results) in the single-world case.) This set of bit strings is independent of the coefficients 'a' and 'b' from the original state |psi>, but if we carry the amplitudes of the original superposition through the sequence of results, we find that for every zero in a bit string we get a factor of 'a', and for every one, we get a factor of 'b'.
>
> Consequently, the amplitude multiplying any sequence of M zeros and (N-M) ones, is a^M b^(N-M). Again, differentiating with respect to 'a' to find the turning point (and the value of 'a' that maximizes this amplitude), we find
>
> |a|^2 = M/N,
>
> where we have taken the modulus of 'a' since a is, in general, a complex number. Again, there will be more than one bit-string with exactly M zeros and (N-M) ones, and summing over these gives the additional factor of N!/M!(N-M)!, as above.
>
> If we now compare the quantum result over N measurements with the classical probability result for N independent Bernoulli trials, we find that the amplitude for M successes in N trials is maximized when the modulus squared of the original amplitude equals the relative number of successes, M/N, which is the classical probability. Thus, the probability for measuring 'zero' in a single trial is just given by the modulus squared of the amplitude for that component of the original state. This is the Born rule.

That is more or less how I interpret Paulette Février, and Preskill. I agree, but it leads to some difficulties for the singlet state, if we take the idea of “world” too much seriously. But with the consistent histories of Griffith and Omnes, there is some clarification on this (and on Feynman Integral).

>
> Furthermore, if in the quantum case we square the amplitude for the sum over bit-strings with exactly M zeros, we get
>
> |a|^(2M) (1 - |a|^2)^(N-M) N!/M!(N-M)!,
>
> which, with the identification |a|^2 = M/N, is just the probability for M successes in N Bernoulli trials with probability for success p = M/N, as above in the standard binomial case.
>
> Now this result could be viewed as a derivation of the Born rule for the quantum case. Or it might be no more than a demonstration of the consistency of the Born rule, if it is already assumed. I am not sure either way. If nothing else, it demonstrates that the mod-squared amplitude plays the role of the probability in repeated measurements over an ensemble of similarly prepared quantum systems, given that one observes only one of the possible sequences of results, either as a single world or as one's particular 'relative state'.
>
> Note also, that this argument is independent of any Everettian considerations -- one can always take a modal interpretation of the bit-strings other than the one actually observed, and see them as corresponding to 'other possible worlds', or sequences of results that could (counterfactually) have been obtained, but weren't. In other words, there is no necessity for the Everettian assumption that all sequences of results obtain in some actual world or the other. The modal single-world interpretation has the distinct advantage that it avoids the ugly locution that "low probability sequences certainly exist”.

OK. Nice.

Bruno

>
> Bruce
>
>
>
>
>
>
>
>
>
>
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[Lawrence Crowell]

There is nothing wrong formally with what you argue. I would though say this is not entirely the Born rule. The Born rule connects eigenvalues with the probabilities of a wave function. For quantum state amplitudes a_i in a superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an observable O obeys

⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.

Your argument has a tight fit with this for O_i = ρ_{ii}.

The difficulty in part stems from the fact we keep using standard ideas of probability to understand quantum physics, which is more fundamentally about amplitudes which give probabilities, but are not probabilities. Your argument is very frequentist. The argument by Carroll and Sebens, using a concept of the wave function as an update mechanism, is somewhat Bayesian. This is curious since Fuchs developed QuBism as a sort of ultra-ψ-epistemic interpretation, and Carroll and Sebens are appealing to the wave function as a similar device for a ψ-ontological interpretation.

I do though agree if there is a proof for the Born rule that is may not depend on some particular quantum interpretation. If the Born rule is some unprovable postulate then it would seem plausible that any sufficiently strong quantum interpretation may prove the Born rule or provide the ancillary axiomatic structure necessary for such a proof. In other words maybe quantum interpretations are essentially unprovable physical axioms that if sufficiently string provide a proof of the Born rule.

LC

[Bruce Kellett]

On Sat, May 16, 2020 at 11:04 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

There is nothing wrong formally with what you argue. I would though say this is not entirely the Born rule. The Born rule connects eigenvalues with the probabilities of a wave function. For quantum state amplitudes a_i in a superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an observable O obeys

⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.

Your argument has a tight fit with this for O_i = ρ_{ii}.

The difficulty in part stems from the fact we keep using standard ideas of probability to understand quantum physics, which is more fundamentally about amplitudes which give probabilities, but are not probabilities. Your argument is very frequentist.

I can see why you might think this, but it is actually not the case. My main point is to reject subjectivist notions of probability:  probabilities in QM are clearly objective -- there is an objective decay rate (or half-life) for any radioactive nucleus; there is a clearly objective probability for that spin to be measured up rather than down in a Stern-Gerlach magnet; and so on.

The argument by Carroll and Sebens, using a concept of the wave function as an update mechanism, is somewhat Bayesian.

It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I consider probabilities to be intrinsic properties -- not further analysable. In other words, I favour a propensity interpretation. Relative frequencies are the way we generally measure probabilities, but they do not define them.

This is curious since Fuchs developed QuBism as a sort of ultra-ψ-epistemic interpretation, and Carroll and Sebens are appealing to the wave function as a similar device for a ψ-ontological interpretation.

I do though agree if there is a proof for the Born rule that is may not depend on some particular quantum interpretation. If the Born rule is some unprovable postulate then it would seem plausible that any sufficiently strong quantum interpretation may prove the Born rule or provide the ancillary axiomatic structure necessary for such a proof. In other words maybe quantum interpretations are essentially unprovable physical axioms that if sufficiently string provide a proof of the Born rule.

I would agree that the Born rule is unlikely to be provable within some model of quantum mechanics -- particularly if that model is deterministic, as is many-worlds. The mistake that advocates of many-worlds are making is to try and graft probabilities, and the Born rule, on to a non-probabilistic model. That endeavour is bound to fail. (In fact, many have given up on trying to incorporate any idea of 'uncertainty' into their model -- this is what is known as the "fission program".) One of the major problems people like Deutsch, Carroll, and Wallace encounter is trying to reconcile Everett with David Lewis's "Principal Principle", which is the rule that one should align one's personal subjective degrees of belief with the objective probabilities. When these people essentially deny the existence of objective probabilities, they have trouble reconciling subjective beliefs with anything at all.

Bruce

[Philip Thrift]

This is 100% what I said two decades ago on Atoms and the Void, and continuing on Free Thinkers Physics Discussion Group.

I will make this subjective guess: Two decades from now when some of us will be in our 90s this same debate will be still going on, the same words will be repeated (as they are now from 20 years ago), and no one will really change their idea of what QM "means".

@philipthrift 

[scerir]

I vaguely remember that von Weizsaecker wrote (in 'Zeit und Wissen') that probability is 'the expectation value of the relative frequency'.

[Bruno Marchal]

On 17 May 2020, at 11:39, 'scerir' via Everything List <everyth...@googlegroups.com> wrote:

I vaguely remember that von Weizsaecker wrote (in 'Zeit und Wissen') that probability is 'the expectation value of the relative frequency'.

That is the frequency approach to probability. Strictly speaking it is false, as it gives the wrong results for the “non normal history” (normal in the sense of Gauss). But it works retire well in the normal world (sorry for being tautological).

At its antipode, there is the bayesian “subjective probabilities”, which makes sense when complete information is available . So it does not make sense in many practical situation.

Remark: the expression “subjective probabilities” is used technically for this Bayesian approach, and is quite different from the first person indeterminacy that Everett call “subjective probabilities”. The “subjective probabilities” of Everett are “objective probabilities”, and can be defined trough a frequency operator in the limit. The same occur in arithmetic, where the subjective (first person) probabilities are objective (they obey objective, sharable, laws).

Naïve many-worlds view are not sustainable, but there is no problem with consistent histories, and 0 worlds.

Bruno

Bruce wrote:

It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I consider probabilities to be intrinsic properties -- not further analysable. In other words, I favour a propensity interpretation. Relative frequencies are the way we generally measure probabilities, but they do not define them.

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[Brent Meeker]

On 5/17/2020 3:31 AM, Bruno Marchal wrote:

On 17 May 2020, at 11:39, 'scerir' via Everything List <everyth...@googlegroups.com> wrote:

I vaguely remember that von Weizsaecker wrote (in 'Zeit und Wissen') that probability is 'the expectation value of the relative frequency'.

That is the frequency approach to probability. Strictly speaking it is false, as it gives the wrong results for the “non normal history” (normal in the sense of Gauss). But it works retire well in the normal world (sorry for being tautological).

At its antipode, there is the bayesian “subjective probabilities”, which makes sense when complete information is available . So it does not make sense in many practical situation.

Remark: the expression “subjective probabilities” is used technically for this Bayesian approach, and is quite different from the first person indeterminacy that Everett call “subjective probabilities”. The “subjective probabilities” of Everett are “objective probabilities”, and can be defined trough a frequency operator in the limit.

That's questionable.  For the frequencies to be correct the splitting must the uneven.  But there's nothing in the Schoedinger evolution to produce this.  If there are two eigenvalues and the Born probabilities are 0.5 and 0.5 then it works fine.  But it the Born probabilities are 0.501 and 0.499 then there must be a thousand new worlds,  yet the Schroedinger equation still only predicts two outcomes.

Brent

The same occur in arithmetic, where the subjective (first person) probabilities are objective (they obey objective, sharable, laws).

Naïve many-worlds view are not sustainable, but there is no problem with consistent histories, and 0 worlds.

Bruno

Bruce wrote:

It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I consider probabilities to be intrinsic properties -- not further analysable. In other words, I favour a propensity interpretation. Relative frequencies are the way we generally measure probabilities, but they do not define them.

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[smitra]

Deriving the Born rule within the context of QM seems to me a rather
futile effort as you still have the formalism of QM itself that is then
unexplained. So, I think one has to tackle QM itself. It seems t me
quite plausible that QM gives an approximate description of a multiverse
of algorithms. So, we are then members of such a multiverse, this then
includes alternative versions of us who found different results in
experiments, but the global structure of this multiverse is something
that QM does not describe adequately.

QM then gives a local approximation of this multiverse that's valid in
the neighborhood of a given algorithm, That algorithm can be an observer
who has found some experimental result, and the local approximation
gives a description of the "nearby algorithms" that are processing
alternative measurement results. The formalism of QM can then arise due
to having to sum over all algorithms that fall within the criterion of
being close to the particular algorithm that is processing some
particular data. This is then a constrained summation over all possible
algorithms. One can then replace such a constrained summation by an
unrestricted summation and implement the constraint by including phase
factors of the form exp(i u constraint function) where constraint
function = 0 for the terms of the original constrained summation. One
can then write the original summation as an integral over u.

Saibal

[Bruce Kellett]

On Mon, May 18, 2020 at 5:18 AM smitra <smi...@zonnet.nl> wrote:

Deriving the Born rule within the context of QM seems to me a rather
futile effort as you still have the formalism of QM itself that is then
unexplained. So, I think one has to tackle QM itself. It seems t me
quite plausible that QM gives an approximate description of a multiverse
of algorithms. So, we are then members of such a multiverse, this then
includes alternative versions of us who found different results in
experiments, but the global structure of this multiverse is something
that QM does not describe adequately.

QM then gives a local approximation of this multiverse that's valid in
the neighborhood of a given algorithm, That algorithm can be an observer
who has found some experimental result, and the local approximation
gives a description of the "nearby algorithms" that are processing
alternative measurement results. The formalism of QM can then arise due
to having to sum over all algorithms that fall within the criterion of
being close to the particular algorithm that is processing some
particular data. This is then a constrained summation over all possible
algorithms. One can then replace such a constrained summation by an
unrestricted summation and implement the constraint by including phase
factors of the form exp(i u constraint function) where constraint
function = 0 for the terms of the original constrained summation. One
can then write the original summation as an integral over u.

And that is all hopelessly ad hoc, without a shred of evidence.

Bruce

[Lawrence Crowell]

On Sunday, May 17, 2020 at 1:57:19 AM UTC-5, Bruce wrote:

On Sat, May 16, 2020 at 11:04 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

There is nothing wrong formally with what you argue. I would though say this is not entirely the Born rule. The Born rule connects eigenvalues with the probabilities of a wave function. For quantum state amplitudes a_i in a superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an observable O obeys

⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.

Your argument has a tight fit with this for O_i = ρ_{ii}.

The difficulty in part stems from the fact we keep using standard ideas of probability to understand quantum physics, which is more fundamentally about amplitudes which give probabilities, but are not probabilities. Your argument is very frequentist.

I can see why you might think this, but it is actually not the case. My main point is to reject subjectivist notions of probability:  probabilities in QM are clearly objective -- there is an objective decay rate (or half-life) for any radioactive nucleus; there is a clearly objective probability for that spin to be measured up rather than down in a Stern-Gerlach magnet; and so on.

Objective probabilities are frequentism. The idea from a probability perspective is that one has a sample space with a known distribution. Your argument, which I agree was my first impression when I encountered the Bayesian approach to QM by Fuchs and Schack, who I have had occasions to talk to. My impression right way was entirely the same; we have operators with outcomes and they have a distribution etc according to Born rule. However, we have a bit of a sticking point; is Born's rule really provable? We most often think in a sample space frequentist manner with regards to the Born rule. However, it is at least plausible to think of the problem from a Bayesian perspective, and where the probabilities have become known is when the Bayesian updates have become very precise. 

However, all this talk of probability theory may itself be wrong. Quantum mechanics derives probabilities or distributions or spectra, but it really is a theory of amplitudes or the density matrix. The probabilities come with modulus square or the trace over the density matrix. Framing QM around an interpretation of probability may be wrong headed to begin with.

 

The argument by Carroll and Sebens, using a concept of the wave function as an update mechanism, is somewhat Bayesian.

It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I consider probabilities to be intrinsic properties -- not further analysable. In other words, I favour a propensity interpretation. Relative frequencies are the way we generally measure probabilities, but they do not define them.

I could I suppose talk to Fuchs about this. He regards QM as having this uncertainty principle, which we can only infer probabilities with a large number of experiments where upon we update Bayesian priors. Of course a frequentists, or a system based on relative frequencies, would say we just make lots of measurements and use that as a sample space. In the end either way works because QM appears to be a pure system that derives probabilities. In other words, since outcome occur acausally or spontaneously there are not meddlesome issues of incomplete knowledge. Because of this, and I have pointed it out, the two perspective end up being largely equivalent.

 

This is curious since Fuchs developed QuBism as a sort of ultra-ψ-epistemic interpretation, and Carroll and Sebens are appealing to the wave function as a similar device for a ψ-ontological interpretation.

I do though agree if there is a proof for the Born rule that is may not depend on some particular quantum interpretation. If the Born rule is some unprovable postulate then it would seem plausible that any sufficiently strong quantum interpretation may prove the Born rule or provide the ancillary axiomatic structure necessary for such a proof. In other words maybe quantum interpretations are essentially unprovable physical axioms that if sufficiently string provide a proof of the Born rule.

I would agree that the Born rule is unlikely to be provable within some model of quantum mechanics -- particularly if that model is deterministic, as is many-worlds. The mistake that advocates of many-worlds are making is to try and graft probabilities, and the Born rule, on to a non-probabilistic model. That endeavour is bound to fail. (In fact, many have given up on trying to incorporate any idea of 'uncertainty' into their model -- this is what is known as the "fission program".) One of the major problems people like Deutsch, Carroll, and Wallace encounter is trying to reconcile Everett with David Lewis's "Principal Principle", which is the rule that one should align one's personal subjective degrees of belief with the objective probabilities. When these people essentially deny the existence of objective probabilities, they have trouble reconciling subjective beliefs with anything at all.

Bruce

Maybe a part of the issue here is the role of counter-factual statements. I think that may be more of an issue here than with the objective vs subjective perspective on QM. Most interpretations of QM are not counter-factual definite. In fact the only one I think is CF definite is deBroglie-Bohm. The Lewis idea was that CF statements are modal logical and thus there exists these alternative worlds. I am not sure to what degree MWI upholds that idea. However, that is also not necessarily an argument against the idea of these branching worlds.

LC

[Bruce Kellett]

On Mon, May 18, 2020 at 11:20 AM Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Sunday, May 17, 2020 at 1:57:19 AM UTC-5, Bruce wrote:

On Sat, May 16, 2020 at 11:04 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

There is nothing wrong formally with what you argue. I would though say this is not entirely the Born rule. The Born rule connects eigenvalues with the probabilities of a wave function. For quantum state amplitudes a_i in a superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an observable O obeys

⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.

Your argument has a tight fit with this for O_i = ρ_{ii}.

The difficulty in part stems from the fact we keep using standard ideas of probability to understand quantum physics, which is more fundamentally about amplitudes which give probabilities, but are not probabilities. Your argument is very frequentist.

I can see why you might think this, but it is actually not the case. My main point is to reject subjectivist notions of probability:  probabilities in QM are clearly objective -- there is an objective decay rate (or half-life) for any radioactive nucleus; there is a clearly objective probability for that spin to be measured up rather than down in a Stern-Gerlach magnet; and so on.

Objective probabilities are frequentism.

Rubbish. Popper's original propensity ideas may have had frequentist overtones, but we can certainly move beyond Popper's outdated thinking. An objective probability is one that is an intrinsic property of an object, such as a radio-active nucleus. One can use relative frequencies or Bayesian updating to estimate these intrinsic probabilities experimentally. But neither relative frequencies nor Bayesian updating of subjective beliefs actually define what the probabilities are in quantum mechanics.

The idea from a probability perspective is that one has a sample space with a known distribution.

These are consequences of the existence of probabilities -- not a definition of them.

Your argument, which I agree was my first impression when I encountered the Bayesian approach to QM by Fuchs and Schack, who I have had occasions to talk to. My impression right way was entirely the same; we have operators with outcomes and they have a distribution etc according to Born rule. However, we have a bit of a sticking point; is Born's rule really provable? We most often think in a sample space frequentist manner with regards to the Born rule. However, it is at least plausible to think of the problem from a Bayesian perspective, and where the probabilities have become known is when the Bayesian updates have become very precise. 

Again, you are confusing measuring or estimating the probabilities with their definition. Propensities are intrinsic properties, not further analysable. Whether or not the Born rule can be derived from simpler principles is far from clear. I don't think the attempts based on decision theory (Deutsch, Wallace, etc) succeed, and attempts based on self-location (Carroll, Zurek) are far from convincing, since they are probably intrinsically dualist.

However, all this talk of probability theory may itself be wrong. Quantum mechanics derives probabilities or distributions or spectra, but it really is a theory of amplitudes or the density matrix. The probabilities come with modulus square or the trace over the density matrix. Framing QM around an interpretation of probability may be wrong headed to begin with.

The basic feature of quantum mechanics is that it predicts probabilities. You confuse the way this is expressed in the theory with the actuality of how probability is defined.

The argument by Carroll and Sebens, using a concept of the wave function as an update mechanism, is somewhat Bayesian.

It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I consider probabilities to be intrinsic properties -- not further analysable. In other words, I favour a propensity interpretation. Relative frequencies are the way we generally measure probabilities, but they do not define them.

I could I suppose talk to Fuchs about this. He regards QM as having this uncertainty principle, which we can only infer probabilities with a large number of experiments where upon we update Bayesian priors. Of course a frequentists, or a system based on relative frequencies, would say we just make lots of measurements and use that as a sample space. In the end either way works because QM appears to be a pure system that derives probabilities. In other words, since outcome occur acausally or spontaneously there are not meddlesome issues of incomplete knowledge. Because of this, and I have pointed it out, the two perspective end up being largely equivalent.

But these are ways of estimating probabilities -- probability is not necessarily defined in this way.

This is curious since Fuchs developed QuBism as a sort of ultra-ψ-epistemic interpretation, and Carroll and Sebens are appealing to the wave function as a similar device for a ψ-ontological interpretation.

I do though agree if there is a proof for the Born rule that is may not depend on some particular quantum interpretation. If the Born rule is some unprovable postulate then it would seem plausible that any sufficiently strong quantum interpretation may prove the Born rule or provide the ancillary axiomatic structure necessary for such a proof. In other words maybe quantum interpretations are essentially unprovable physical axioms that if sufficiently string provide a proof of the Born rule.

I would agree that the Born rule is unlikely to be provable within some model of quantum mechanics -- particularly if that model is deterministic, as is many-worlds. The mistake that advocates of many-worlds are making is to try and graft probabilities, and the Born rule, on to a non-probabilistic model. That endeavour is bound to fail. (In fact, many have given up on trying to incorporate any idea of 'uncertainty' into their model -- this is what is known as the "fission program".) One of the major problems people like Deutsch, Carroll, and Wallace encounter is trying to reconcile Everett with David Lewis's "Principal Principle", which is the rule that one should align one's personal subjective degrees of belief with the objective probabilities. When these people essentially deny the existence of objective probabilities, they have trouble reconciling subjective beliefs with anything at all.

Bruce

Maybe a part of the issue here is the role of counter-factual statements.

You are getting side-tracked. Quantum mechancs is not counterfactually definite, so that is a side issue as far as the origin of probability is concerned.

I think that may be more of an issue here than with the objective vs subjective perspective on QM. Most interpretations of QM are not counter-factual definite. In fact the only one I think is CF definite is deBroglie-Bohm. The Lewis idea was that CF statements are modal logical and thus there exists these alternative worlds.

Lewis explicated counterfactual statements in terms of possible worlds, not in terms of existent alternative worlds.

Bruce

[Brent Meeker]

On 5/17/2020 6:20 PM, Lawrence Crowell wrote:

On Sunday, May 17, 2020 at 1:57:19 AM UTC-5, Bruce wrote:

On Sat, May 16, 2020 at 11:04 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

There is nothing wrong formally with what you argue. I would though say this is not entirely the Born rule. The Born rule connects eigenvalues with the probabilities of a wave function. For quantum state amplitudes a_i in a superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an observable O obeys

⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.

Your argument has a tight fit with this for O_i = ρ_{ii}.

The difficulty in part stems from the fact we keep using standard ideas of probability to understand quantum physics, which is more fundamentally about amplitudes which give probabilities, but are not probabilities. Your argument is very frequentist.

I can see why you might think this, but it is actually not the case. My main point is to reject subjectivist notions of probability:  probabilities in QM are clearly objective -- there is an objective decay rate (or half-life) for any radioactive nucleus; there is a clearly objective probability for that spin to be measured up rather than down in a Stern-Gerlach magnet; and so on.

Objective probabilities are frequentism.

No necessarily.  Objective probabilities may be based on symmetries and the principle of insufficient reason.  I agree with Bruce; just because you measure a probability with frequency, that doesn't imply it must be based on frequentism.

The idea from a probability perspective is that one has a sample space with a known distribution. Your argument, which I agree was my first impression when I encountered the Bayesian approach to QM by Fuchs and Schack, who I have had occasions to talk to. My impression right way was entirely the same; we have operators with outcomes and they have a distribution etc according to Born rule. However, we have a bit of a sticking point; is Born's rule really provable? We most often think in a sample space frequentist manner with regards to the Born rule. However, it is at least plausible to think of the problem from a Bayesian perspective, and where the probabilities have become known is when the Bayesian updates have become very precise. 

However, all this talk of probability theory may itself be wrong. Quantum mechanics derives probabilities or distributions or spectra, but it really is a theory of amplitudes or the density matrix. The probabilities come with modulus square or the trace over the density matrix. Framing QM around an interpretation of probability may be wrong headed to begin with.

But if it's not just mathematics, there has to be some way to make contact with experiment...which for probabilistic predictions usually means frequencies.

Brent

 

The argument by Carroll and Sebens, using a concept of the wave function as an update mechanism, is somewhat Bayesian.

It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I consider probabilities to be intrinsic properties -- not further analysable. In other words, I favour a propensity interpretation. Relative frequencies are the way we generally measure probabilities, but they do not define them.

I could I suppose talk to Fuchs about this. He regards QM as having this uncertainty principle, which we can only infer probabilities with a large number of experiments where upon we update Bayesian priors. Of course a frequentists, or a system based on relative frequencies, would say we just make lots of measurements and use that as a sample space. In the end either way works because QM appears to be a pure system that derives probabilities. In other words, since outcome occur acausally or spontaneously there are not meddlesome issues of incomplete knowledge. Because of this, and I have pointed it out, the two perspective end up being largely equivalent.

 

This is curious since Fuchs developed QuBism as a sort of ultra-ψ-epistemic interpretation, and Carroll and Sebens are appealing to the wave function as a similar device for a ψ-ontological interpretation.

I do though agree if there is a proof for the Born rule that is may not depend on some particular quantum interpretation. If the Born rule is some unprovable postulate then it would seem plausible that any sufficiently strong quantum interpretation may prove the Born rule or provide the ancillary axiomatic structure necessary for such a proof. In other words maybe quantum interpretations are essentially unprovable physical axioms that if sufficiently string provide a proof of the Born rule.

I would agree that the Born rule is unlikely to be provable within some model of quantum mechanics -- particularly if that model is deterministic, as is many-worlds. The mistake that advocates of many-worlds are making is to try and graft probabilities, and the Born rule, on to a non-probabilistic model. That endeavour is bound to fail. (In fact, many have given up on trying to incorporate any idea of 'uncertainty' into their model -- this is what is known as the "fission program".) One of the major problems people like Deutsch, Carroll, and Wallace encounter is trying to reconcile Everett with David Lewis's "Principal Principle", which is the rule that one should align one's personal subjective degrees of belief with the objective probabilities. When these people essentially deny the existence of objective probabilities, they have trouble reconciling subjective beliefs with anything at all.

Bruce

Maybe a part of the issue here is the role of counter-factual statements. I think that may be more of an issue here than with the objective vs subjective perspective on QM. Most interpretations of QM are not counter-factual definite. In fact the only one I think is CF definite is deBroglie-Bohm. The Lewis idea was that CF statements are modal logical and thus there exists these alternative worlds. I am not sure to what degree MWI upholds that idea. However, that is also not necessarily an argument against the idea of these branching worlds.

LC

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[smitra]

That's true for everything that has ever been said about this subject.
If there were evidence then the matter would have been settled already.

Saibal

[Alan Grayson]

We don't need no stinkin' evidence. AG 

[Philip Thrift]

On Sunday, May 17, 2020 at 8:20:01 PM UTC-5, Lawrence Crowell wrote:

However, all this talk of probability theory may itself be wrong. Quantum mechanics derives probabilities or distributions or spectra, but it really is a theory of amplitudes or the density matrix. The probabilities come with modulus square or the trace over the density matrix. Framing QM around an interpretation of probability may be wrong headed to begin with.

 

LC

Physicists frequently  talk as if QM was defined by God writing its formulation into stone in front of Moses, like religious fundamentalists talk.

There are measure/probability theoretic formulations of QM  that do not come from any "density matrix" or Hilbert space but from their measure/probability spaces.

@philipthift

[Bruno Marchal]

On 17 May 2020, at 20:59, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/17/2020 3:31 AM, Bruno Marchal wrote:

On 17 May 2020, at 11:39, 'scerir' via Everything List <everyth...@googlegroups.com> wrote:

I vaguely remember that von Weizsaecker wrote (in 'Zeit und Wissen') that probability is 'the expectation value of the relative frequency'.

That is the frequency approach to probability. Strictly speaking it is false, as it gives the wrong results for the “non normal history” (normal in the sense of Gauss). But it works retire well in the normal world (sorry for being tautological).

At its antipode, there is the bayesian “subjective probabilities”, which makes sense when complete information is available . So it does not make sense in many practical situation.

Remark: the expression “subjective probabilities” is used technically for this Bayesian approach, and is quite different from the first person indeterminacy that Everett call “subjective probabilities”. The “subjective probabilities” of Everett are “objective probabilities”, and can be defined trough a frequency operator in the limit.

That's questionable.  For the frequencies to be correct the splitting must the uneven.  But there's nothing in the Schoedinger evolution to produce this.  If there are two eigenvalues and the Born probabilities are 0.5 and 0.5 then it works fine.  But it the Born probabilities are 0.501 and 0.499 then there must be a thousand new worlds,  yet the Schroedinger equation still only predicts two outcomes.

The SWE predicts two fist person outcomes, OK. But the “number” of worlds, or of histories, depends on the metaphysical assumptions.

With mechanism it is a bit hard to not see the physical multiverse as a confirmation of the many-computations (many = 2^aleph_0 at least!) theorem in (meta)-arithmetic (that is not an interpretation). 

Bruno

Brent

The same occur in arithmetic, where the subjective (first person) probabilities are objective (they obey objective, sharable, laws).

Naïve many-worlds view are not sustainable, but there is no problem with consistent histories, and 0 worlds.

Bruno

Bruce wrote:

It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I consider probabilities to be intrinsic properties -- not further analysable. In other words, I favour a propensity interpretation. Relative frequencies are the way we generally measure probabilities, but they do not define them.

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[Lawrence Crowell]

On Sunday, May 17, 2020 at 9:06:12 PM UTC-5, Bruce wrote:

On Mon, May 18, 2020 at 11:20 AM Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Sunday, May 17, 2020 at 1:57:19 AM UTC-5, Bruce wrote:

On Sat, May 16, 2020 at 11:04 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

There is nothing wrong formally with what you argue. I would though say this is not entirely the Born rule. The Born rule connects eigenvalues with the probabilities of a wave function. For quantum state amplitudes a_i in a superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an observable O obeys

⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.

Your argument has a tight fit with this for O_i = ρ_{ii}.

The difficulty in part stems from the fact we keep using standard ideas of probability to understand quantum physics, which is more fundamentally about amplitudes which give probabilities, but are not probabilities. Your argument is very frequentist.

I can see why you might think this, but it is actually not the case. My main point is to reject subjectivist notions of probability:  probabilities in QM are clearly objective -- there is an objective decay rate (or half-life) for any radioactive nucleus; there is a clearly objective probability for that spin to be measured up rather than down in a Stern-Gerlach magnet; and so on.

Objective probabilities are frequentism.

Rubbish. Popper's original propensity ideas may have had frequentist overtones, but we can certainly move beyond Popper's outdated thinking. An objective probability is one that is an intrinsic property of an object, such as a radio-active nucleus. One can use relative frequencies or Bayesian updating to estimate these intrinsic probabilities experimentally. But neither relative frequencies nor Bayesian updating of subjective beliefs actually define what the probabilities are in quantum mechanics.

Probability and statistics are in part an empirical subject. This is not pure mathematics, and it is one reason why there is no single foundation. It would be as if linear algebra or any area of mathematics had two competing axiomatic foundation. There are also subdivisions in the frequentist and subjectivist camps, particularly the first of these. 

Quantum mechanics computes probabilities not according to some idea of incomplete knowledge, but according to amplitudes. There is no lack of information from the perspective of the observer or analyst, but it is intrinsic. From what I can see this means it is irrelevant whether one adopts a Bayesian or frequentist perspective. The division between these two approaches to probability correlate somewhat with interpretations of QM that are ψ-ontic (aligned with frequentist) and ψ-epistemic (aligned with Bayesian). The divisions between the two camps amongst statisticians is amazingly sharp and almost hostile. I think the issues is somewhat irrelevant. 

LC

[Bruce Kellett]

On Mon, May 18, 2020 at 9:27 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Sunday, May 17, 2020 at 9:06:12 PM UTC-5, Bruce wrote:

On Mon, May 18, 2020 at 11:20 AM Lawrence Crowell <goldenfield...@gmail.com> wrote:

Objective probabilities are frequentism.

Rubbish. Popper's original propensity ideas may have had frequentist overtones, but we can certainly move beyond Popper's outdated thinking. An objective probability is one that is an intrinsic property of an object, such as a radio-active nucleus. One can use relative frequencies or Bayesian updating to estimate these intrinsic probabilities experimentally. But neither relative frequencies nor Bayesian updating of subjective beliefs actually define what the probabilities are in quantum mechanics.

Probability and statistics are in part an empirical subject. This is not pure mathematics, and it is one reason why there is no single foundation. It would be as if linear algebra or any area of mathematics had two competing axiomatic foundation. There are also subdivisions in the frequentist and subjectivist camps, particularly the first of these. 

Quantum mechanics computes probabilities not according to some idea of incomplete knowledge, but according to amplitudes. There is no lack of information from the perspective of the observer or analyst, but it is intrinsic. From what I can see this means it is irrelevant whether one adopts a Bayesian or frequentist perspective. The division between these two approaches to probability correlate somewhat with interpretations of QM that are ψ-ontic (aligned with frequentist) and ψ-epistemic (aligned with Bayesian). The divisions between the two camps amongst statisticians is amazingly sharp and almost hostile. I think the issues is somewhat irrelevant. 

OK. So stick with instrumentalism then.

Bruce

[Philip Thrift]

On Monday, May 18, 2020 at 6:27:34 AM UTC-5, Lawrence Crowell wrote:

Probability and statistics are in part an empirical subject. This is not pure mathematics, and it is one reason why there is no single foundation. It would be as if linear algebra or any area of mathematics had two competing axiomatic foundation. There are also subdivisions in the frequentist and subjectivist camps, particularly the first of these. 

Quantum mechanics computes probabilities not according to some idea of incomplete knowledge, but according to amplitudes. There is no lack of information from the perspective of the observer or analyst, but it is intrinsic. From what I can see this means it is irrelevant whether one adopts a Bayesian or frequentist perspective. The division between these two approaches to probability correlate somewhat with interpretations of QM that are ψ-ontic (aligned with frequentist) and ψ-epistemic (aligned with Bayesian). The divisions between the two camps amongst statisticians is amazingly sharp and almost hostile. I think the issues is somewhat irrelevant. 

LC

 

All of this is an opinion, and its painful to read.

Probability Theory  can be just as much a subject of Pure mathematics as Group Theory or Real Analysis.

Quantum mechanics computes probabilities not according to some idea of incomplete knowledge

That some physicists think this is their problem.

@philipthrift

[Lawrence Crowell]

On Monday, May 18, 2020 at 12:12:28 AM UTC-5, Brent wrote:

On 5/17/2020 6:20 PM, Lawrence Crowell wrote:

On Sunday, May 17, 2020 at 1:57:19 AM UTC-5, Bruce wrote:

On Sat, May 16, 2020 at 11:04 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

There is nothing wrong formally with what you argue. I would though say this is not entirely the Born rule. The Born rule connects eigenvalues with the probabilities of a wave function. For quantum state amplitudes a_i in a superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an observable O obeys

⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.

Your argument has a tight fit with this for O_i = ρ_{ii}.

The difficulty in part stems from the fact we keep using standard ideas of probability to understand quantum physics, which is more fundamentally about amplitudes which give probabilities, but are not probabilities. Your argument is very frequentist.

I can see why you might think this, but it is actually not the case. My main point is to reject subjectivist notions of probability:  probabilities in QM are clearly objective -- there is an objective decay rate (or half-life) for any radioactive nucleus; there is a clearly objective probability for that spin to be measured up rather than down in a Stern-Gerlach magnet; and so on.

Objective probabilities are frequentism.

No necessarily.  Objective probabilities may be based on symmetries and the principle of insufficient reason.  I agree with Bruce; just because you measure a probability with frequency, that doesn't imply it must be based on frequentism.

That is not what I meant. Bruce does sound as if he is appealing to an objective basis for probability based on the frequency of occurrences of events. I am not arguing this isy wrong, but rather that this is an interpretation of probability. 

LC

 

The idea from a probability perspective is that one has a sample space with a known distribution. Your argument, which I agree was my first impression when I encountered the Bayesian approach to QM by Fuchs and Schack, who I have had occasions to talk to. My impression right way was entirely the same; we have operators with outcomes and they have a distribution etc according to Born rule. However, we have a bit of a sticking point; is Born's rule really provable? We most often think in a sample space frequentist manner with regards to the Born rule. However, it is at least plausible to think of the problem from a Bayesian perspective, and where the probabilities have become known is when the Bayesian updates have become very precise. 

However, all this talk of probability theory may itself be wrong. Quantum mechanics derives probabilities or distributions or spectra, but it really is a theory of amplitudes or the density matrix. The probabilities come with modulus square or the trace over the density matrix. Framing QM around an interpretation of probability may be wrong headed to begin with.

But if it's not just mathematics, there has to be some way to make contact with experiment...which for probabilistic predictions usually means frequencies.

Brent

 

The argument by Carroll and Sebens, using a concept of the wave function as an update mechanism, is somewhat Bayesian.

It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I consider probabilities to be intrinsic properties -- not further analysable. In other words, I favour a propensity interpretation. Relative frequencies are the way we generally measure probabilities, but they do not define them.

I could I suppose talk to Fuchs about this. He regards QM as having this uncertainty principle, which we can only infer probabilities with a large number of experiments where upon we update Bayesian priors. Of course a frequentists, or a system based on relative frequencies, would say we just make lots of measurements and use that as a sample space. In the end either way works because QM appears to be a pure system that derives probabilities. In other words, since outcome occur acausally or spontaneously there are not meddlesome issues of incomplete knowledge. Because of this, and I have pointed it out, the two perspective end up being largely equivalent.

 

This is curious since Fuchs developed QuBism as a sort of ultra-ψ-epistemic interpretation, and Carroll and Sebens are appealing to the wave function as a similar device for a ψ-ontological interpretation.

I do though agree if there is a proof for the Born rule that is may not depend on some particular quantum interpretation. If the Born rule is some unprovable postulate then it would seem plausible that any sufficiently strong quantum interpretation may prove the Born rule or provide the ancillary axiomatic structure necessary for such a proof. In other words maybe quantum interpretations are essentially unprovable physical axioms that if sufficiently string provide a proof of the Born rule.

I would agree that the Born rule is unlikely to be provable within some model of quantum mechanics -- particularly if that model is deterministic, as is many-worlds. The mistake that advocates of many-worlds are making is to try and graft probabilities, and the Born rule, on to a non-probabilistic model. That endeavour is bound to fail. (In fact, many have given up on trying to incorporate any idea of 'uncertainty' into their model -- this is what is known as the "fission program".) One of the major problems people like Deutsch, Carroll, and Wallace encounter is trying to reconcile Everett with David Lewis's "Principal Principle", which is the rule that one should align one's personal subjective degrees of belief with the objective probabilities. When these people essentially deny the existence of objective probabilities, they have trouble reconciling subjective beliefs with anything at all.

Bruce

Maybe a part of the issue here is the role of counter-factual statements. I think that may be more of an issue here than with the objective vs subjective perspective on QM. Most interpretations of QM are not counter-factual definite. In fact the only one I think is CF definite is deBroglie-Bohm. The Lewis idea was that CF statements are modal logical and thus there exists these alternative worlds. I am not sure to what degree MWI upholds that idea. However, that is also not necessarily an argument against the idea of these branching worlds.

LC

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[scerir]

<<Quantum mechanics computes probabilities not according to some idea of incomplete knowledge>>

“One may call these uncertainties [i.e. the Born probabilities] objective, in that they are simply a consequence of the fact that we describe the experiment in terms of classical physics; they do not depend in detail on the observer. One may call them subjective, in that they reflect our incomplete knowledge of the world.” (Heisenberg)

[Brent Meeker]

On 5/18/2020 3:29 AM, Bruno Marchal wrote:

On 17 May 2020, at 20:59, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/17/2020 3:31 AM, Bruno Marchal wrote:

On 17 May 2020, at 11:39, 'scerir' via Everything List <everyth...@googlegroups.com> wrote:

I vaguely remember that von Weizsaecker wrote (in 'Zeit und Wissen') that probability is 'the expectation value of the relative frequency'.

That is the frequency approach to probability. Strictly speaking it is false, as it gives the wrong results for the “non normal history” (normal in the sense of Gauss). But it works retire well in the normal world (sorry for being tautological).

At its antipode, there is the bayesian “subjective probabilities”, which makes sense when complete information is available . So it does not make sense in many practical situation.

Remark: the expression “subjective probabilities” is used technically for this Bayesian approach, and is quite different from the first person indeterminacy that Everett call “subjective probabilities”. The “subjective probabilities” of Everett are “objective probabilities”, and can be defined trough a frequency operator in the limit.

That's questionable.  For the frequencies to be correct the splitting must the uneven.  But there's nothing in the Schoedinger evolution to produce this.  If there are two eigenvalues and the Born probabilities are 0.5 and 0.5 then it works fine.  But it the Born probabilities are 0.501 and 0.499 then there must be a thousand new worlds,  yet the Schroedinger equation still only predicts two outcomes.

The SWE predicts two fist person outcomes, OK. But the “number” of worlds, or of histories, depends on the metaphysical assumptions.

With mechanism it is a bit hard to not see the physical multiverse

Nobody sees the physical multiverse.  It's as much a theoretical construct as arithmetic is.

Brent

as a confirmation of the many-computations (many = 2^aleph_0 at least!) theorem in (meta)-arithmetic (that is not an interpretation). 

Bruno

Brent

The same occur in arithmetic, where the subjective (first person) probabilities are objective (they obey objective, sharable, laws).

Naïve many-worlds view are not sustainable, but there is no problem with consistent histories, and 0 worlds.

Bruno

Bruce wrote:

It is this subjectivity, and appeal to Bayesianism, that I reject for QM. I consider probabilities to be intrinsic properties -- not further analysable. In other words, I favour a propensity interpretation. Relative frequencies are the way we generally measure probabilities, but they do not define them.

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[Brent Meeker]

On 5/18/2020 4:27 AM, Lawrence Crowell wrote:

Probability and statistics are in part an empirical subject. This is not pure mathematics, and it is one reason why there is no single foundation. It would be as if linear algebra or any area of mathematics had two competing axiomatic foundation. There are also subdivisions in the frequentist and subjectivist camps, particularly the first of these. 

Quantum mechanics computes probabilities not according to some idea of incomplete knowledge, but according to amplitudes. There is no lack of information from the perspective of the observer or analyst, but it is intrinsic. From what I can see this means it is irrelevant whether one adopts a Bayesian or frequentist perspective. The division between these two approaches to probability correlate somewhat with interpretations of QM that are ψ-ontic (aligned with frequentist) and ψ-epistemic (aligned with Bayesian). The divisions between the two camps amongst statisticians is amazingly sharp and almost hostile. I think the issues is somewhat irrelevant. 

Probability has several different meanings and people argue over them as if one must settle on the real meaning.  But this is a mistake.  Just like “cost” or “energy”, “probability” is useful precisely because the same value has different interpretations.  There are four interpretations that commonly come up. 

1. It has a mathematical definition that lets us manipulate it and draw inferences.
2. It has a physical interpretation as a symmetry. 
3. It quantifies a degree of belief that tells us whether to act on it. 
4. It has an empirical meaning that lets us measure it.  

The usefulness of probability is that we can start with one of these, we can then manipulate it mathematically, and then interpret the result in one of the other ways.

Brent

[Bruce Kellett]

On Mon, May 18, 2020 at 10:57 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

On Monday, May 18, 2020 at 12:12:28 AM UTC-5, Brent wrote:

On 5/17/2020 6:20 PM, Lawrence Crowell wrote:

On Sunday, May 17, 2020 at 1:57:19 AM UTC-5, Bruce wrote:

On Sat, May 16, 2020 at 11:04 PM Lawrence Crowell <goldenfield...@gmail.com> wrote:

There is nothing wrong formally with what you argue. I would though say this is not entirely the Born rule. The Born rule connects eigenvalues with the probabilities of a wave function. For quantum state amplitudes a_i in a superposition ψ = sum_ia_iφ_i with φ*_jφ_i = δ_{ij} the spectrum of an observable O obeys

⟨O⟩ = sum_iO_ip_i = sum_iO_i a*_ia_i.

Your argument has a tight fit with this for O_i = ρ_{ii}.

The difficulty in part stems from the fact we keep using standard ideas of probability to understand quantum physics, which is more fundamentally about amplitudes which give probabilities, but are not probabilities. Your argument is very frequentist.

I can see why you might think this, but it is actually not the case. My main point is to reject subjectivist notions of probability:  probabilities in QM are clearly objective -- there is an objective decay rate (or half-life) for any radioactive nucleus; there is a clearly objective probability for that spin to be measured up rather than down in a Stern-Gerlach magnet; and so on.

Objective probabilities are frequentism.

No necessarily.  Objective probabilities may be based on symmetries and the principle of insufficient reason.  I agree with Bruce; just because you measure a probability with frequency, that doesn't imply it must be based on frequentism.

That is not what I meant. Bruce does sound as if he is appealing to an objective basis for probability based on the frequency of occurrences of events. I am not arguing this isy wrong, but rather that this is an interpretation of probability. 

I am sorry if I have given the impression that I thought that objective probabilities were possible only with frequentism. I thought I had made it clear that frequentism fails as a basis for the meaning of probability. There are many places where this is argued, and the consensus is that long-run relative frequencies cannot be used as a  definition of probability.

I was appealing to the propensity interpretation, which says that probabilities are intrinsic properties of some things.; such as decay rates; i.e., that probability is an intrinsic property of radio-active nuclei. But I agree with Brent, probabilities can be taken to be anything that satisfies the basic axioms of probability theory -- such as non-negative, normalisable, and additive. So subjective degrees of belief can form the basis for probabilities, as can certain symmetry properties, relative frequencies, and so on.

The point is that while these things can be understood as probabilities in ordinary usage, they don't actually define what probability is. One can use frequency counts to estimate many of these probabilities, and one can use Bayes's theorem to update estimates of probability based on new evidence. But Bayes's theorem is merely an updating method -- it is not a definition of probability. People who consider themselves to be Bayesians usually have a basically subjective idea about probability, considering it essentially quantifies personal degrees of belief. But that understanding is not inherent in Bayes' theorem itself.

As Brent says, these different approaches to probability have their uses in everyday life, but most of them are not suitable for fundamental physics. I consider objective probabilities based on intrinsic properties, or propensities, to be essential for a proper understanding of radio-active decay, and the probability of getting spin-up on a spin measurement, and so on. These things are properties of the way the world is, not matters of personal belief, or nothing more than relative frequencies. Probabilities may well be built into the fabric of the quantum wave-function via the amplitudes, but the probabilistic interpretation of these amplitudes has to be imposed via the Born rule:  Just as with any mathematical theory -- one needs correspondence rules to say how the mathematical elements relate to physical observables. From that point of view, attempts to derive the Born rule from within the theory are doomed to failure -- contrary to the many-worlders' dream, the theory does not contain its own interpretation.

Bruce

[Alan Grayson]

"Propensity" seems pretty vague. Hard to imagine finding an objective principle underlying probabilities. What precisely does Born's rule mean? AG 

[Bruce Kellett]

"Propensity" is just a word, the usage originates with Karl Popper, who used it to convey the idea that probability may be a primitive concept, like mass or charge, that is not analysable in terms of anything more fundamental. So you cannot expect to explain the concept in terms of anything else.

The Born rule is really just the statement that quantum mechanics is a theory that predicts probabilities, and those probabilities are given by the mod-squared amplitudes. In other words, it is an interpretative rule, connecting the theory with observation. Seen in this light, it does not make sense to attempt to derive Born's rule from within the theory itself. I think the analysis that I gave at the beginning of this thread is probably the best that one can do -- one shows that the mod-squared amplitudes play the role of probabilities, and that those are the probabilities needed to connect the theory to experiment. The probabilities are objective in that they are already part of the theory:  the amplitudes are objective aspects of the theory.

Bruce

[Brent Meeker]

But even if you're right (and I think you are) does that affect the MWI.  In an Everett+Born theory there will still be other worlds and the interpretation will still avoid the question, "When and where is a measurement?"...answer "Whenever decoherence has made one state orthogonal to all other states."   Of course we could then as the question, "When and where has the wave function collapsed?" and give the same answer.  Which would be CI+Zurek.

Brent

[Bruce Kellett]

Does this analysis affect the MWI? I think it does, because if probabilities are intrinsic, and the Born rule is merely a rule that connects the theory with experiment;  not something that can be derived from within the theory, then we are left with the tension that I mentioned a while ago between many-worlds and the probability interpretation. Everett says that every outcome happens, but separate outcomes are in separate "relative states", or separate, orthogonal, branches. This can only mean that the probability is given by branch counting, and the probabilities are necessarily 50/50 for the two-state case. Then the probability for each branch on repeated measurements of an ensemble of N similarly prepared states is 1/2^N, and those probabilities are independent of the amplitudes. This is inconsistent with the analysis that gives the probability for each branch of the repeated trials by the normal binomial probability, with p equal to the mod-squared amplitude. (This is the analysis with which I began this thread.) The binomial probabilities are experimentally observed, so MWI is, on this account, inconsistent with experiment (even if not actually incoherent).

The question of the preferred basis and the nature of measurement is answered by decoherence, and the collapse of the wave function is another interpretative move. On this view, the wave function is epistemic rather that ontological. The ontology is the particles or fields that the wave function describes, and these live on ordinary 3-space. This is not necessarily CI+Zurek, because Zurek still thinks he can derive probabilities from within the theory, and CI does not encompass decoherence and the quantum nature of everything. Nor is it Qbism, since that idea does not really encompass objective probabilities.

Bruce

[smitra]

It's implausible that a fundamental theory can have a macroscopic
concepts like "environment", "experiment" etc. in it build in. That's
what motivated the MWI in the first place. Now, the MWI may not be
exactly correct as QM may itself only be an approximation to a more
fundamental theory. But the approach of trying to address
interpretational problems by relying on macroscopic concepts is a priori
doomed to fail.

Saibal

[Alan Grayson]

I haven't followed every detail of this discussion, but enough to get this important point; now I have no idea what the Born's rule means.  Whereas it affirms something about "probabilities", it doesn't tell us what this means! To quote our Chief Asshole (DJT), Sad! AG  

[Bruce Kellett]

Where are these concepts "built in" to the theory as I have outlined it? One mentions things like experiments and the environment because the theory has, after all, to explain things like that. There is no question that experiments, apparatuses, observers, and the environment are all ultimately quantum, and must be encompassed within the quantum theory;  that is something that Everett's approach taught us. But that insight has not been violated by anything that I have said. As Bell stressed, the fundamental theory should not refer to "measurement" or "observers". But we have to relate the theory to observation, or else the theory is useless.

That's what motivated the MWI in the first place. Now, the MWI may not be
exactly correct as QM may itself only be an approximation to a more
fundamental theory. But the approach of trying to address
interpretational problems by relying on macroscopic concepts is a priori
doomed to fail

Not necessarily doomed to fail -- such an approach may not be entirely useless. The Copenhagen Interpretation did work well for nearly 100 years, and it still works in practise. The CI may not satisfy ones fundamentalist realist leanings, but that is another matter.

Bruce

[Lawrence Crowell]

I will not say that this is wrong, but it strikes me as more philosophical. The game of probability and statistics is heavily laden with this. If this works for you then fine. I tend to think according to what most effectively answers a question or problem.

LC 

[Bruno Marchal]

On 18 May 2020, at 21:38, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/18/2020 3:29 AM, Bruno Marchal wrote:

On 17 May 2020, at 20:59, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 5/17/2020 3:31 AM, Bruno Marchal wrote:

On 17 May 2020, at 11:39, 'scerir' via Everything List <everyth...@googlegroups.com> wrote:

I vaguely remember that von Weizsaecker wrote (in 'Zeit und Wissen') that probability is 'the expectation value of the relative frequency'.

That is the frequency approach to probability. Strictly speaking it is false, as it gives the wrong results for the “non normal history” (normal in the sense of Gauss). But it works retire well in the normal world (sorry for being tautological).

At its antipode, there is the bayesian “subjective probabilities”, which makes sense when complete information is available . So it does not make sense in many practical situation.

Remark: the expression “subjective probabilities” is used technically for this Bayesian approach, and is quite different from the first person indeterminacy that Everett call “subjective probabilities”. The “subjective probabilities” of Everett are “objective probabilities”, and can be defined trough a frequency operator in the limit.

That's questionable.  For the frequencies to be correct the splitting must the uneven.  But there's nothing in the Schoedinger evolution to produce this.  If there are two eigenvalues and the Born probabilities are 0.5 and 0.5 then it works fine.  But it the Born probabilities are 0.501 and 0.499 then there must be a thousand new worlds,  yet the Schroedinger equation still only predicts two outcomes.

The SWE predicts two fist person outcomes, OK. But the “number” of worlds, or of histories, depends on the metaphysical assumptions.

With mechanism it is a bit hard to not see the physical multiverse

Nobody sees the physical multiverse.  It's as much a theoretical construct as arithmetic is.

Nobody sees the moon. It is a theoretical construct, even if a large part of the construction is programmed by millions years of evolution. In that sense, we don’t see anything. But when I said that with mechanism it is hard to not see the multiverse, “see” was used with the meaning of “logically conclude from Mechanism”.

Bruno

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[Brent Meeker]

On 6/25/2020 11:10 PM, Bruno Marchal wrote:
>>> With mechanism it is a bit hard to not see the physical multiverse
>>
>> Nobody sees the physical multiverse.  It's as much a theoretical
>> construct as arithmetic is.
>
> Nobody sees the moon. It is a theoretical construct, even if a large
> part of the construction is programmed by millions years of evolution.
> In that sense, we don’t see anything. But when I said that with
> mechanism it is hard to not see the multiverse, “see” was used with
> the meaning of “logically conclude from Mechanism”.
>

Nobody logically concludes the Moon either.

Brent

[Bruno Marchal]

Exactly. That is why I doubt less on the “many-computations” than on a moon made of some primary material object.

The question is to be clear about what we need to assume (which is then primary, per definition) and what we derive as being observable, believable, knowable, etc.

The goal is not to predict, but to explain. Yet, prediction remains the main verification tool, but that can be used only to refute the explanation, and today, it is not refuted, and I think it (Mechanism) is the only theory explaining both quanta and qualia (with the mathematical details needed to build the experimental devices capable of refuting it). Nobody knows the truth as such. About Reality, we can only make test, and mesure what is more plausible.

Bruno



>
> Brent

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