On the myth that probability depends on knowledge

[A. Neumaier]

You are the one who started telling Varon (on the interpretations poll thread I think) about how the position of a particle does exist, but is not well-defined (you used the term fuzzy) until a measurement is made. What do you use to describe the existence of the particle position prior to the measurement if you don't use |psi|^2?

You misunderstood what I said. Saying that a particle has a fuzzu position means that it actually _has_ this position independent of any measurement, but that its value is meaningful only up to an accuracy determined by the uncertainty relation. The position is given not by |psi|^2 but by xbar=psi^*x psi, with an absolute uncertainty of sqrt(psi^*(x-xbar)^2 psi).

Measuring the position gives a value statistically consistent with this and the measuring accuracy, but does not change the fact that the position remains fuzzy. You cannot read from your meter that the position is at exactly x.

 

[A. Neumaier]

You are presented with a specific bridge over a ravine. [...]
As the Engineer you are asked
Will the bridge collapse if I drive my lorry over it?

Whether you answer ''with 75% probability'' or ''with 10% probability'', nobody can verify whether your answer was correct when the bridge collapsed, or didin't collapse, upon driving the lorry over it.
And if you answer ''with 99% probability'' and you conclude that you better not drive, the answer can again not be checked.

This makes it clear that your answer is not about this bridge collapsing when you drive over it now,
but with the ensemble of all possible lorries and bridges matching the characteristics of your model as derived from your input data.

This represents a one off unique situation and you have to make an assessment ie a subjective decision to allow for the fact that all the facts are not ( and probably cannot be ) known.

As far as it is applied to a particular situation, you always have a subjective probability, which is not verifiable by checking against reality.

You did not read my post correctly either.
Are you not familiar with the difference between analysis and the more difficult process of synthesis (or design)?

I am familiar with it. But the bridge example is one of analysis, not of design. And though I know about limit state design, I was not directly involved in that. Thus I deliberately changed the wording. However, it is not _so_ different from limit state analysis, as it involves the latter as a constraining design condition. So it is part of the total optimization problem to be solved. I have been involved in the design of devices facing uncertainty by other methods; see, e.g., p.81ff of my slides http://arnold-neumaier.at/ms/robslides.pdf

 

[A. Neumaier]

One of the direct consequences of this statement, if true, has deep philosophical implications because it implies determinism.
That is that any point in time the future is completely determined with a probability of either 1 or 0.

It doesn't imply determinism, since no dynamical law is involved in it. It only implies (or assumes, depending on what you regard as given) that after something happened, it is a fact, independent of the future.

 

[A. Neumaier]

No reason, except that the deterministic case is off topic and obvious.

It is not off-topic since it serves to clarify the issue, and it is as obvious in the probabilisitc case as in the deterministc case, hence there is no reason to emphasize it in the latter case. It doesn't add any useful insight into the nature of probability.

Yes, but your definition is not the only valid and accepted definition of probability. Your claim is only true if you require probabilities to be defined only over ensembles. In that case I agree that the posterior probability does not depend on the prior so in that case you are indeed correct that probability does not depend on knowledge. Under the more general definition of probability the posterior can depend on the prior in any case where you do not have a sufficiently large number of observations.

But in that case, the probability is subjective, and not checkable by anyone.

Thus according to the customary criteria, it is not part of science.

 

[A. Neumaier]

This is different from the fixed-prior case. Here, instead of having a fixed prior you have a family of priors with some hyper-parameters which are uniquely specified by available information. Note that in this case the probabilities are objective (user independent), but they do depend on knowledge.

They do depend on the selected parameters, which is part of the specification of the ensemble.

Of course, the model reflects knowledge, prejudice, assumptions, the authorities trusted, assessment errors, and all that, but that's the same as in _all_ modeling. Hence it is not a special characteristics of probability.

 

[Studiot]

As far as it is applied to a particular situation, you always have a subjective probability, QUOTE]

Loud applause all round.

That is the point everyone has been trying to make to you. Subjective probability has a place in physical science.

Further there exist a range of probabilities, useful in science, between the values 0 and 1.

which is not verifiable by checking against reality.[/

You test your assessment by driving over the bridge.

My specific examples separately addressed two different points. (1) Uncertainty and (2)objective v subjective.

Limit State theory (analysis or design) is a real world example of applied science attempts to allow for inevitable uncertainty in an objective way. There is no subjectivism whatsoever in this theory. It has been highly successful in increasing design eficiency.

Bridge assessment contains a specific subjective component as a formal part of the process. An extra factor is introduced called the condition factor. This is a subjective derating factor, not present in normal limit state or other analysis methods. (Assessment does not necessarily use limit state theory.)

 

[A. Neumaier]

Subjective probability has a place in physical science.

No, since it is not testable.

You test your assessment by driving over the bridge.

Whether the assessment was ''with 75% probability'' or ''with 10% probability'', nobody can verify whether the statement was correct after you tried to drive over the bridge. Thus it cannot be regarded as a test.

 

[Studiot]

OK, so we have laid one ghost.

You have not disgreed that there is room, even a necessity, for a subjective component to probability in applied science.


Now for the second one.

You mentioned several times that a probability value exists for something whether the observer knows this value or not.

I agree.

Similarly a probability value exists whether the observer tests, or can test or not.

 

[Fra]

You test your assessment by driving over the bridge.

Yes, exactly.

This is also the gaming analogy. When driving over the bridge, you are placing best, you are taking risks. But this is how nature works. All you ever do, is place your bets and play the game. Along the game you shall then learn and revise your expectations as feedback is arrived.

However, sometimes fatal things happens. Driving over the bridge can be fatal. But this is also part of the game.

The predictions from this game is that only the players that are rational and good guessers and gamers, will survive. So the systems we observer in nature, are then likely to comply to these rationality constraints. But they are not FORCED to them. In fact evolution depends on mistakes and variation.

So subjective probabilites are not tested in the descriptive sense. But they don't need to. Their sole purpose are in evaluating the most rational action (think some action principle). But these "inference systems" that are somewhat subjective are subject to evolution and selection, and anywhere near equilibrium conditions this may yield predictions of expected behaviour (actions) of subsystems in nature; just assuming rationality in their way of placing bets based upon subjective probabilitis.

I think if you take the "rationality constraints" to be exact, and forcing, then the difference to this view and neumaiers "objective constraints" is almost nil.

But the problem is that even the effectively objective constraints are observer dependent and in particular scale dependent. So the only consistent stance as far as I am concerned, is to allow for evolution and selection here and understand that the subjective perspective is what is needed to understand how the effective objective has emerged. Without that, it just is what it is. An ad hoc choice for not particular reason.

The evolutionary picture has a power the deductive way hasn't - to provide a mechanism to understand effective objectivity from a democratic system of subjective views as they interact (equilibrate).

/Fredrik

 

[A. Neumaier]

You have not disgreed that there is room, even a necessity, for a subjective component to probability in applied science.

In the art of using science, not in science itself. Subjective probability is a guide to action in single instances, but not a scientific (testable) concept.

You mentioned several times that a probability value exists for something whether the observer knows this value or not.

Similarly a probability value exists whether the observer tests, or can test or not.

The latter sort of existence is meaningless. In the same sense, ghosts exist (subjectively) no matter whether it can be tested.

 

[A. Neumaier]

Umm... I'd say physics (and natural science in general) is ALL about us learning ABOUT nature, what we can say about nature.

''us learning'' is the subject of psychology, not of physics. The subject of physics is the objective description of the kinematics and dynamics of systems of Nature.

 

[Fra]

''us learning'' is the subject of psychology, not of physics.

In the case of and observer = human scientist, that's of course correct. I agree.

But like I've argued, the subjective interpretation would make no sense if it was all about human observers. Science is FAPP objective in terms of human-human comparasions.

All human scientists will agree upon the description of nature in the sense physicists talk about. We agree there.

But THE physics is about how one subsytems of the universe, "learns" about the states and behaviour of the other subsystems. It's about how the state of a proton, encodes and infers expectations of it's environment (fellow observers, such as other neutrons, electrons etc), and how the action of the proton follows from rationality constraints in this game.

This will have testable predictions for human science, and it may help understand how interactions are scaled as the observer scales down from human laboratory device to a proton which is then a proper inside observer (except WE humans, observe this inside observer form the outside (the lab)).

So the physics analogy, is that the action of a proton is similarly a game. The action of the proton is based upong it's own subjective expectations of it's envionment. It tests this by acting ("driving over the bridge"). A stable proton in equilibrium will have a holographically encoded picture corresponding to external reality. But a system not in equilibrium or in agreement with heavly evolve and changes it's state, sometimes it even decomposes and is destroyed.

This is the "learning" I'm talking about. But it's actually analogous to how science works. So the analogies is still good, but the real thing is one subsystem of the universe makes inferences about it's physical environment. We humans are like very MASSIVE observing system that observes these inside observers interacting. So human science IS like a DESCRIPTION of the inside game. BUT as we also consider cosmological models, this assymmetry does not hold, and we are forced to consider that human scientists are indeed also inside observers playing a game not JUST descriptive scientists. Except of course on a cosmo scale clearly all EARTHBASED human scientists will still indeed agree upon science.

So nothing of what I say threatens the integrity and soundness of science. On the contrary does it deepen in.

/Fredrik

 

[lalbatros]

Originally Posted by Studiot
Subjective probability has a place in physical science.

No, since it is not testable.

It is testable: humans are testable!

 

[A. Neumaier]

It is testable: humans are testable!

There is a difference between testing a human and testing the assertion that a particular bridge will collapse with 75% probability when a particular truck crosses it at a particular time. The latter is impossible and proves that the statement has no scientific content.

 

[A. Neumaier]

In the case of and observer = human scientist, that's of course correct. I agree.

In the case of a machine, it is a matter of artificial intelligence, not of physics.

Physics is about interpreting experiments in an observer-independent way.

 

[Studiot]

There is a difference between testing a human and testing the assertion that a particular bridge will collapse with 75% probability when a particular truck crosses it at a particular time. The latter is impossible and proves that the statement has no scientific content.

Actually that is where you are wrong.

You introduced the example of radioative decay, which is exactly parallel.

I take particular exception to the notion that my statements 'have no scientific content'.

That is a highly coloured value judgement sir!

 

[A. Neumaier]

Actually that is where you are wrong.

You haven't proven me wrong. You haven't provided a way to test the statement, thus making it amenable to the scientific method.

You introduced the example of radioative decay, which is exactly parallel.

No, it isn't. Radioactive decay is a mass phenomenon and the probability for decay applies (as I had explicitly argued) _only_ to the ensemble of all isotopes of a particular kind, and not to any single decay. The latter is completely unpredictable and a probability statement about it is - like any statement assigning a probability dofferent from 0 or 1 to a single event - completely uncheckable.

Thus applying the probability for the decay of an anonymous atom to a particular atom has as much scientific content as claiming that a ghost has appeared on my desk.

I take particular exception to the notion that my statements 'have no scientific content'.

The statement that I called devoid of scientific content, namely ''that a particular bridge will collapse with 75% probability when a particular truck crosses it at a particular time'' was mine, not yours.

 

[A. Neumaier]

I take particular exception to the notion that my statements 'have no scientific content'.

The statement that I called devoid of scientific content, namely ''that a particular bridge will collapse with 75% probability when a particular truck crosses it at a particular time'' was mine, not yours.

Whereas the statement that you actually made in this context, namely

You test your assessment by driving over the bridge.

is plain wrong.

How can a nontestable statement have scientific content?

 

[Studiot]

No, it isn't. Radioactive decay is a mass phenomenon and the probability for decay applies (as I had explicitly argued) _only_ to the ensemble of all isotopes of a particular kind, and not to any single decay. The latter is completely unpredictable and a probability statement about it is - like any statement assigning a probability dofferent from 0 or 1 to a single event - completely uncheckable.

I grow weary of this verbal fencing - it achieves nothing.

Instead of constantly flatly refuting everyone else's comments you might gain something if you asked for more information about why such and such statement was made.

Radioactive decay, for instance, is actually a function of time, not mass.
The objective measure is the fraction ( a pure number) decaying within a certain time period.

So it is with lorry journeys and bridges.

Again I repeat this is a quantum mechanics forum.

In QM there are at least two ways of interpreting probability, since there are at least two independent variables.

So it is with lorry journeys and bridges.

 

[Dale]

Of course, the model reflects knowledge, prejudice, assumptions, the authorities trusted, assessment errors, and all that, but that's the same as in _all_ modeling. Hence it is not a special characteristics of probability.

If the model depends on knowledge and the result of the model is a probability then how can you claim that probability does not depend on knowledge? And I agree that it is not peculiar to probability.

I think you are confusing your concept of "subjective" with knowledge. With a specified family of priors and an algorithm for determining the hyper parameters from the available knowledge then the probability depends on the knowledge objectively. I believe that you are really just saying that scientists shouldn't just use subjective "gut feeling" priors.

 

[SpectraCat]

There is a difference between testing a human and testing the assertion that a particular bridge will collapse with 75% probability when a particular truck crosses it at a particular time. The latter is impossible and proves that the statement has no scientific content.

Ok .. let's work this through:

There is a bridge, trucks drive over it. Each time a truck drives over it, one of two things will happen .. it will collapse or it won't. Objectively, for each trial (i.e. truck journey) there is no way to say with certainty which outcome will be obtained until either the truck crosses safely, or the bridge collapses. Ok so far?

Now consider two bridges, a wooden bridge designed for pedestrian traffic, and a steel bridge designed for truck traffic. You are the truck driver ... which bridge do you take? I guess that is what you are calling a subjective probability judgment? It seems to me that there is a higher objective probability that the wooden bridge will collapse when the truck is driven across it. Do you agree with that? If you do agree, then can you explain how you measure the difference between the cases? Or is the difference unmeasurable?

Note the exactly the same analogy can be drawn for radioactive decay lifetimes of different isotopes: given two atoms of different isotopes, one with a half-life of 5 seconds, the other with a half-life of 5 years, which is more likely to decay in a given time interval? It seems that there is a clear, objective difference between the probabilities of the two events. What is wrong with that analysis?

 

[Studiot]

Well SC you seem to have caught the essence of it.

The bridge assessment question is faced by some bridge engineers every working day of their lives.

You may have heard of AILs - Abnormal Indivisible Loads.

When a load larger than the legally allowable the max gross weight needs to be transported the transport company approaches the bridge authority for any bridge they propose to pass over to ask under what conditions they can cross the bridge.

A real world example might be a train company transporting a 250 tonne locomotive to another location along roads and across bridges where the max gross weight is 38 tonnes.

 

[lalbatros]

There is a difference between testing a human and testing the assertion that a particular bridge will collapse with 75% probability when a particular truck crosses it at a particular time. The latter is impossible and proves that the statement has no scientific content.

Is that not a bad news for engineers, specially in the nuclear safety field?
Reliability theory and practice is completely build on the assumption that probabilities (even very small) have a meaning even though they often cannot be measured.

The book "The Black Swan" by Taleb illustrated very well the risk of blindly using probabilities.

This discussion is interresting.
Jaynes has clearly shown that the concept of probability needs to be analysed more deeply.
I have no doubt that probabilities are -in way- subjective and that this explains conceptual difficulties in physics, specially quantum mechanics.

The "frequentist" interpretation is conveniently used to hide difficulties in quantum mechanics, but these difficulties remain even if they are hidden.

 

[Studiot]

Reliability theory and practice is completely build on the assumption that probabilities (even very small) have a meaning even though they often cannot be measured.

The engineering answer to a quantity that cannot be calculated or measured exactly is to bracket it between upper and lower bounds to prove that is lies within acceptable limits.

The whole idea of limit state is that the probability of failure is quantifiable in this way and acceptably low.

 

[lalbatros]

The engineering answer to a quantity that cannot be calculated or measured exactly is to bracket it between upper and lower bounds to prove that is lies within acceptable limits.

The whole idea of limit state is that the probability of failure is quantifiable in this way and acceptably low.

I agree, but the evaluations cannot be tested ... and humans often do mistakes!
The black swarn approach would be to ban Nuclear Power plants: never trust Gaussian assumptions if your life is at stake.
Or in extended form: never trust any assumption if your life is at stake.

 

[Studiot]

Mistakes can usually be caught and corrected if proper procedures are followed.
That is what the independent check is all about for instance.

Deliberate mis-evaluation is more difficult to cope with.

 

[A. Neumaier]

Instead of constantly flatly refuting everyone else's comments.

I only refute what doesn't hold water.

Radioactive decay, for instance, is actually a function of time, not mass.
The objective measure is the fraction ( a pure number) decaying within a certain time period.

a ''mass phenomenon'' does not refer to masses measured in kg, but to masses measured in large numbers. I could have written as well ''ensemble phenomenon''.

So it is with lorry journeys and bridges.

Again I repeat this is a quantum mechanics forum.

I don't see the connection of lorries and bridges with quantum mechanics.

 

[A. Neumaier]

There is a bridge, trucks drive over it. Each time a truck drives over it, one of two things will happen .. it will collapse or it won't. Objectively, for each trial (i.e. truck journey) there is no way to say with certainty which outcome will be obtained until either the truck crosses safely, or the bridge collapses. Ok so far?

Yes, and since you say ''each'' time, you acknowledge that it is a matter of ensembles, not of driving across this bridge now. The single instance is not a matter of probability, but what happens each time someone does something is. That's the whole point.

Note the exactly the same analogy can be drawn for radioactive decay lifetimes of different isotopes: given two atoms of different isotopes, one with a half-life of 5 seconds, the other with a half-life of 5 years, which is more likely to decay in a given time interval? It seems that there is a clear, objective difference between the probabilities of the two events. What is wrong with that analysis?

That you equate objective probabilities for ''each time'' with subjective probabilities for
a single instance. Applying the probability is admissible only if you regard the single instance as member of the observed ensemble, and then it refers to the ensemble and not to the single instance. This becomes obvious if you ask for the reason why the subjectve probability was assigned. invariably there will be an explanation involving
''each time''.

Suppose a second person would assign different probabilities based on ignorance, and a third person would assign different probabilities based on better knowledge unknown to the driver. Since all are subjective probabilities, all are as valid as any other. Now the driver picks one of the roads and drives - with or without success. Who of the three were correct or wrong? Being subjective probabilities, all were right. Thus the scientific method is impotent to distinguish between these probability assignments - although they would be mutually conflicting if they were saying something about the bridge rather than the subject defining them. This clearly shows that subjective probabilities are properties of the subject and not properties of the bridge.

 

[A. Neumaier]

When a load larger than the legally allowable the max gross weight needs to be transported the transport company approaches the bridge authority for any bridge they propose to pass over to ask under what conditions they can cross the bridge.

But this is a matter of law, not of science.

 

[Studiot]

But this is a matter of law, not of science.

Are you seriously suggesting that the weight carrying capacity (ie whether it is physically possible to support a statd weight) of a structure is a matter of human legislature, not of science?

''mass phenomenon'' does not refer to masses measured in kg, but to masses measured in large numbers

Are you suggesting that the probability of atomic decay (chain reactions apart) is a function of the number of atoms present.
And I always thought that the measure was the probability that a certain % would decay in a specific time, regardless of quantity.