On the myth that probability depends on knowledge

[A. Neumaier]

I agree.

And since, according to you, nothing is objective, the word has lost its descriptive meaning. It can be applied nowhere. This shows that your notion of objectivity is not the standard one. Mine is.

 

[A. Neumaier]

The question is what the ensemble IS (ie. how the mathematical abstractions map to physical states)

They are mapped in the usual informal way.

I think there is no other choice but to assume that they are representative as you say. The limited instances we have is IMO more fundamental than the ensemble, since the true ensemble is never available for decision making anyway.

In practice, the mathematical ensemble is fundamental, once the situation is a bit complex. We make simplifying modeling assumptions all the time, and these determine the ensemble. Whereas the data we have to fit the parameters of the ensemble changes in amount and value, and hence cannot be taken to be fundamental.

At least that's how science proceeds.

 

[Studiot]

The probability of encountering the black state is objectively determined by whatever ensemble you assume. It has nothing to do wiith knowledge.

You are making flat assertions without substantiation, mathematical or otherwise sir!

One could 'assume' anything one liked.

But making assumptions casts serious doubt on the validity of any (derived?) result.

There were no assumptions in my scenario.

 

[Fra]

And since, according to you, nothing is objective, the word has lost its descriptive meaning. It can be applied nowhere. This shows that your notion of objectivity is not the standard one. Mine is.

We can still communicate right? So there IS indeed an "effective objectivity". But the difference is that in my perspective, this is emergent and evolving. In particular it's a result of negotiating interactions between subjective views.

Objectivity is the result of interactions. So there sort of is effective objectivity.

I think the difference is that you think of it as a forcing, timeless constraint, in which all subjective views are related as per some relativity principle. I instead thing that the symmetry transformations are emergent.

I think the difference is if you see observer invariance as "forcing constraints", or if you see it in terms of observer democracy, where the constraints are instead emergent.

All we admittedly DO need, is some form of FAPP type objectivity, it does not NEED to be rigid. This objectivity I never denied.

I also agree that your notion of objectivity, is indeed more common than mine.

/Fredrik

 

[A. Neumaier]

You are making flat assertions without substantiation, mathematical or otherwise sir!

One could 'assume' anything one liked.

But making assumptions casts serious doubt on the validity of any (derived?) result.

There were no assumptions in my scenario.

Indeed. It is impossible to derive anything without making assumptioons. But
the quality of one's predictions depend crucially on which assumptions are made.

If you have a die in which the six is replaced by another copy of 1, the ignorance assumption of equal probabilities for 1:6 will simply be faulty, although it is generally considered to be the right choice.
Independent of our knowledge, in this particular case, only the probability distribution 2:1:1:1:1:0 is correct, and hence objectively true.

In general, only correct assumptions lead to good predictions.

The purpose of scientific theories is therefore precisely to discover and inform about the assumptions that lead to the best predictions, and are in this sense objective. In particular, we learn which ensembles are most appropriate for which kind of physical situation.

 

[A. Neumaier]

We can still communicate right? So there IS indeed an "effective objectivity".

Others simply call objective what you call ''effectively objective''. There is no need to eliminate the uses for a word and then to substitute a more complex version for the previous usage.

But the difference is that in my perspective, this is emergent and evolving. In particular it's a result of negotiating interactions between subjective views.

One can consider objectivity to be dependent on social agreements (and hence subject to potential change), without having to change the terminology. On the other hand, if Aristotle would visit the Earth today, I think he would agree with that much of our science is objective while some of what he thought is correct wasn't.

Thus I believe that objectivity doesn't change but only the degree to whioch we come close to objectivity, and how convinced we are of something to be objective. Real objectivity is not a time-dependent thing.

I also agree that your notion of objectivity, is indeed more common than mine.

In the interest of easy communication, one should strive to use the most common terminology rather create one's own.

 

[Fra]

Others simply call objective what you call ''effectively objective''. There is no need to eliminate the uses for a word and then to substitute a more complex version for the previous usage.

I think the distinction is still important, because when the objectivity is in fact "just" effective, it limits the applicability of extrapolations. You can no longer made deductions at arbitrary length unless you also show that the assumption of "effectiveness" still holds.

This is I think less trivial when you consider extending QM, talking about QG and unification of forces. Here effective notions, does need non-trivial renormalisations when you change observer or observer scale.

/Fredrik

 

[Studiot]

It is impossible to derive anything without making assumptioons

I disagree.

I am pointing out that the black state is unique because in any other state there is only one possible answer to the question Is a signal getting through?

The black state may occur either because no signal is getting therough or because a valid (black) signal is getting through.

No other state is subject to this restriction and can be 'deduced' to be valid (valid = one of the possible states caused by a valid signal) what ever other colour is displayed.

 

[A. Neumaier]

I disagree.

I am pointing out that the black state is unique because in any other state there is only one possible answer to the question Is a signal getting through?

But this is already an assumption (or alleged knowledge) Someone else seeing your setup for the first time will treat the black state not different from all the others.

 

[Studiot]

What do you mean this is an assumption?

Why is any observer entitled to assume this?

 

[A. Neumaier]

What do you mean this is an assumption?

Why is any observer entitled to assume this?

If you have a screen and don't know anything about it, the black color has no special significance.

Thus giving the black color a special status has the character of an assumption. The assumption may be wrong (perhaps the white color should have a special status?) or correct. It is likely correct if taken from a manual for using the screen, or if one has a lot of experience with doing controlled experiment with various inputs and observing the output.

But back to your original post #5. Your first question had the answer zero since the entropy of any realization of an ensemble is zero. The second question was announced but not formulated. What is the relevance of your scenario in the context of the theme of the thread?

 

[Fra]

But in physics, the assumptions are part of the scientific consensus, and hence there is no choice. To describe a thermodyneamic equilibrium state of a chemical system, say, you _have_ to use the grand canonical ensemble, otherwise you don't get an equilibrium state.

Therefore in physics, probabilities are objective while in gambling they aren't.

Yes there is a choice. Each individual still has the choice to accept majority consensus, or not. Consensus or not, it's still a choice and a game. When acting as per consensus expectations, we are placing our bets.

After all, diversity is necessariy for development. It's healthy for individuals to question the consensus. This happens also in science. Either consensus changes, or it just gets stronger!

The bets are only objective in the sense of "everybody in my neighbourhood" agrees with me, thus it's quite RATIONAL to act as if this was objective truth, because it's all we know. But as we know consensus can both change and be relative. It's quite possible that there are two research groups reaching different consensus becauase they both have a biased perspective.

My only point is that the analogy to gambling is stronger that I think you say.

/Fredrik

 

[A. Neumaier]

Yes there is a choice. Each individual still has the choice to accept majority consensus, or not. [...]
My only point is that the analogy to gambling is stronger that I think you say./QUOTE]

Physics has nothing in common with gambling.

Someone who knows that there is a highly predictive system and chooses an inferior one to serve ther same task is a fool.

 

[Dale]

Probabilities are never assigned to a single event but always to the sigma-algebra of all events - in physics language: to the ensemble.

You should practice what you preach.

In the interest of easy communication, one should strive to use the most common terminology rather create one's own.

It is indeed common and accepted terminology to talk about the probability of a single event.

This is purely a semantic debate. If you use the frequentist definition of probability then probability does not depend on knowledge and can only be defined on an ensemble. If you use the Bayesian definition of probabilty then probability does depend on knowledge and can be defined on ensembles or on single events. Both definitions are common and well-accepted so to call one definition or the other a "myth" is rather absurd.

 

[Fra]

Physics has nothing in common with gambling.

Someone who knows that there is a highly predictive system and chooses an inferior one to serve ther same task is a fool.

I don't know what you refer to, but I think you must confuse general inference and gambling with something else. This can certainly be highly predictive; it is however not deductive.

What I think you refer to as deductions, is really just rational inductive style inference that are so confident that they are "effectively deductive".

Gambling and inductive inference does in no way contradict predictive power. On the contrary would it rather have superior learning capability. Deductions are either right or wrong. Inductive reasoning can be adaptive. The adaptions of a deductive framework are completely meaning changes in the axiomatic systems is treated completely outside the system. This is unsatisfactory.

Yes, deductive logic is more strict and precise. But inductive logic is more flexible and more appropriate for realistic situations. The only fatal failure is failing to adapt and learn. Making false predictions along the road is completely and fully part of the game. It's the learning curve that is relevant. Here I think deductive logic is way too rigid and unfit.

/Fredrik

 

[A. Neumaier]

You should practice what you preach.It is indeed common and accepted terminology to talk about the probability of a single event.

This is purely a semantic debate. If you use the frequentist definition of probability then probability does not depend on knowledge and can only be defined on an ensemble. If you use the Bayesian definition of probabilty then probability does depend on knowledge and can be defined on ensembles or on single events. Both definitions are common and well-accepted so to call one definition or the other a "myth" is rather absurd.

The objective probability of a single event is 1 if it happens and 0 otherwise.

There may be also a subjective probability in the Bayesian sense, but such probabilities are physcally meaningless. And this is a discussion in a physics forum.

 

[SpectraCat]

The objective probability of a single event is 1 if it happens and 0 otherwise.

There may be also a subjective probability in the Bayesian sense, but such probabilities are physcally meaningless. And this is a discussion in a physics forum.

What about |psi|^2? That can certainly be considered as a probability for a single event ... if you choose a point in space, |psi|^2 tells you the probability that the particle will be observed at that position. Is that physically meaningless as well? That would seem to be at odds with statements you have made on other threads ...

 

[A. Neumaier]

What about |psi|^2? That can certainly be considered as a probability for a single event ... if you choose a point in space, |psi|^2 tells you the probability that the particle will be observed at that position. Is that physically meaningless as well?

Yes, it is meaningless. For either the particle will be observed, or it won't. Thus the probability must be one or zero, but |psi|^2 typically isn't.

|psi|^2 is the probability for observing the position in the ensemble of _all_ particles prepared in the same state psi, but says nothing about any particular such particle.

That would seem to be at odds with statements you have made on other threads ...

Please give a more precise context for this claim.

 

[Physics Monkey]

Applying probability theory to single instances is foolish.

How is this a scientific argument? You're just baldly asserting your point of view and calling those who don't agree with you fools.

 

[A. Neumaier]

How is this a scientific argument?

Not every statement in a scientific discussion must be a scientific argument. And if you look at the context, you see that here ''applying probability theory'' meant ''deducing from a single case a probability'', which simply doesn't make sense.

You're just baldly asserting your point of view and calling those who don't agree with you fools.

Hardly. doing something that I consider foolish and being a fool are worlds apart.
I sometimes do foolish things, but don't think that this makes me a fool. And those who don't agree with me won't take my statement that ''Applying probability theory to single instances is foolish'' seriously anyway. Thus the statement is harmless.

 

[Physics Monkey]

Yes, if the gas is deterministic, and hence determined by the initial condition.

Should I read the subtext here to say that you don't believe classical gases are deterministic?

I am not that interested in money to accept your hypotheses. You may think of probabilities of single cases - these are very subjective, though. They have nothing to do with the probabilities used in physics.

Again, you're just making an assertion without any evidence. I claim the probabilities used in physics are highly subjective. They contain our prejudices about beauty and symmetry. They include our limited access to experimental data and our subjective assumptions about the relevant degrees of freedom, sources of error, etc. We even use them to help determine what are the interesting questions in physics. In short, they are always constrained and defined by our own limited experience and knowledge. I have no interest in forbidding you from talking about "objective probabilities" as some platonic notion, but real physics is done with subjective probabilities.

For example, the Boltzmann distribution is certainly subjective. It assigns non-zero weight to states that the system will never access, and indeed, many distributions will give precisely the same answers for macroscopic physical observables. Thus choosing Boltzmann is a subjective assignment.

In any case, since I don't know the properties of your 6-sided die, assigning probabilities is completely arbitrary. Unless I assume that the die is just like one of the many I have seen before, in which case I assign equal probabilities to each outcome, because I substitute ensemble probabilities for ignorance.
But if your die had painted 1 on each side, my choice of 2:6 based on my assumption would be 100% wrong.

Thus probabilities are based on _assumptions_, not on _knowledge_.

Assumptions are based on knowledge. You assign probabilities to the die rolls based on your knowledge and experience with other die. You want to make the best guess you can based on your limited knowledge. It's ok to be 100% wrong so long as you made a good guess. If you get to roll the die many times then you can improve your guess. Of course, it could really roll a classical many times in exactly the same then you would always get the same answer, thus the probabilities one assigns to die rolls are actually only even relevant because one has limited knowledge of the conditions of the throw. Another manifestation of subjectivity in physics.

 

[Physics Monkey]

Not every statement in a scientific discussion must be a scientific argument. And if you look at the context, you see that here ''applying probability theory'' meant ''deducing from a single case a probability'', which simply doesn't make sense.

Hardly. doing something that I consider foolish and being a fool are worlds apart.
I sometimes do foolish things, but don't think that this makes me a fool. And those who don't agree with me won't take my statement that ''Applying probability theory to single instances is foolish'' seriously anyway. Thus the statement is harmless.

I disagree, we were talking about assigning probability to a single event. Your phrase "deducing from a single case a probability" presupposes the notion that there is some abstract correct probability to be obtained.

I further disagree that your statement is harmless. It can discourage participation in the discussion and it can sway opinions based on rhetoric rather than sound scientific argument. I imagine you would agree those are both negative outcomes.

Finally, I agree there is a distinction between being a fool and acting foolish. I misquoted you. Nevertheless, I think you're totally missing the point. Telling someone they're doing something foolish still has no place in a scientific discussion.

 

[A. Neumaier]

I claim the probabilities used in physics are highly subjective. ?

Let us be specific. The probability of decay of a radium atom in the next 10 minutes is a constant independent of anyone's knowledge. It had that value even before there were physicists knowing about the existence of radium. No amount of subjectivity in the views about beauty and symmetry, relevant degrees of freedom, sources of error changes this fact.

Assumptions are based on knowledge. ?

They may be based on knowledge. They may also be based on ignorance or false information, unchecked belief, etc.. But all this is irrelevant for physics. Once your assumptions specified the ensemble in question, the probabilities are objectively determined. No matter whether you can calculate them, or whether you have any knowledge about the system so defined.

You assign probabilities to the die rolls based on your knowledge and experience with other die. You want to make the best guess you can based on your limited knowledge. It's ok to be 100% wrong so long as you made a good guess. If you get to roll the die many times then you can improve your guess. ?

This only implies that the guesses made depend on your knowledge. But the probabilities are not dependent on whether you guess them well or poorly. Nature doesn't care about our knowledge, it doesn't change its behavior when we get to know something new. And physics is about the properties of Nature, not about the psychology of human knowledge.

 

[A. Neumaier]

Telling someone they're doing something foolish still has no place in a scientific discussion.

I wasn't telling someone they're doing something foolish. I was telling something about my standards of judging, not meaning anyone in particular. If you felt offended, I apologize.

 

[Fra]

But the probabilities are not dependent on whether you guess them well or poorly. Nature doesn't care about our knowledge, it doesn't change its behavior when we get to know something new. And physics is about the properties of Nature, not about the psychology of human knowledge.

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

So whatever nature is, or probabilities are, the PROBLEM is how to INFER it. THIS is the primary problem of the scientific method. The problem is not really what nature is or isn't. The problem, is how to, by means of experiments and interactions make rational inferences, that lead to rational and sound beliefs (scientific knowledge).

To me, physics is how to make rational inferences and produce rational expectations ABOUT nature, from past interaction history. And I even think that all physical interactions obey this structure, that two interacting atoms are in fact making inferences about each other. This is why I probably consider my self the complete opposite to your very strong structural realist position.

I agree it's not about psychology or human mind. But none that sees the inference perspective seriously makes that confusion. Observations, information states, expectations etc are thought to be encoded in any physical system. No brains are needed.

/Fredrik

 

[Studiot]

@A. Neumaier

I find discussion in this thread very difficult.

This is partly because I agree with much of what you say and partly because the thread appears to be a compartmentalised set of bilateral conversations, rather than a group discussion.

I am also inviting you to look a little further about probability.

Take for instance limit state design.
Or bridge strength assessment.
Or diversity as applied to electrical installation design
Or the error term as applied to many mathematical calculations.

You state that single event probability is either 1 or zero.
In the case of my bridge example this implies that a bridge either collapses or it doesn't.
In reality the bridge may suffer a partial collapse, indeed some bridges may suffer a small partial collapse (=degradation) on every use until finally that last straw walks over it.

 

[Stephen Tashi]

The objective probability of a single event is 1 if it happens and 0 otherwise.

"if it happens"? That's a nice conditional for a statement about a probability. You've made a Bayesian utterance.

By that reasoning, all "single events" have probabilities that are 0 or 1. So now we must look at non-single events. But what are "non-single" events? - collections of single events? Collections of events, each of which has probability 0 or 1 ? This sounds like the old Von Mises approach to probability theory using "collectives".


Are there any actual consequences to the theory of "objective probabilities"? Can it make any testable predictions that disagree with Bayesian predictions?

 

[SpectraCat]

Yes, it is meaningless. For either the particle will be observed, or it won't. Thus the probability must be one or zero, but |psi|^2 typically isn't.

|psi|^2 is the probability for observing the position in the ensemble of _all_ particles prepared in the same state psi, but says nothing about any particular such particle.

No, |psi|^2 defines the probability density .. it applies equally well to the probability of single measurements (before they are made obviously), as it does to ensembles of measurements. Of course *after* the measurement the particle position will be a delta function (for theoretically infinite precision), but that is not really a probability at all .. it is a result. Furthermore, if you consider the space of all possible results, the particle will always be observed somewhere, so the probability then is always 1. That seems a lot more meaningless than |psi|^2 to me ...

Please give a more precise context for this claim.

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?

 

[A. Neumaier]

Take for instance limit state design.
Or bridge strength assessment.
Or diversity as applied to electrical installation design
Or the error term as applied to many mathematical calculations.

You state that single event probability is either 1 or zero.
In the case of my bridge example this implies that a bridge either collapses or it doesn't.
In reality the bridge may suffer a partial collapse, indeed some bridges may suffer a small partial collapse (=degradation) on every use until finally that last straw walks over it.

I have been doing a lot of practical work in uncertainty analysis (including FORM, SORM and various other engineering techniques). I even did research in advanced methods of uncertainty estimation in complex settings; see http://arnold-neumaier.at/clouds.html

Thus I make my assertions based on thorough and quite diverse experience.

Predicting a partial collapse is different from predicting a probability of collapse.
The correct modeling would try predict the expected amount of collapse or degradation, not a probability of collapse. Bringing this into play only confuses issues, and I'll disregard it in the following.

Saying that there is a 60% chance that it will rain tomorrow may sound like a probability statement about the single event tomorrow, but it isn't - this statement cannot be verified, whether or not it actually rains, and hence is empty. Instead it is a statement about the known preconditions of the weather tomorrow - namely that they belong to an ensemble described by a stochastic model in which the probability of raining is 60%.

Essentially the same holds for all other of the many engineering uses of probability I have met during my career.

A lot of knowledge (but also prejudice, or more or less justified assumptions) goes into the creation of an appropriate stochastic model for defining the ensemble. In this (and only this) sense, probabilities are knowledge-dependent. But this knowledge-dependence is of the same character as that of anything we say or believe, and hence is not something worth emphasizing.

On the other hand, once the ensemble is fixed, probabilites are objective. Of course, the language assigns probabilities to single events, but (as in the case of tomorrow's weather), these are not properties of these events but of an associated theoretical ensemble chosen
such that averaged over many actual events the predictions are maximally useful.

Thus if two people assign different probabilities to the same event, it means that they have different ensembles in mind for modeling the same situation.

Now suppose that we have a real ensemble, such as whether or not it rains at Vienna airport each day of the next two years, or whether or not some of the bridges in Europe crash in the next two years Then there are objective probabilities associated with them, namely the relative frequencies of the actual events. Again, these are completely independent of the knowledge of any observer or analyst. They are unknown now, but can be determiend in two years time, hence they are objective.

On the other hand, the probabilities we assign to them based on a particular model for predictions are approximations, whose quality depends on the knowledge (but also prejudice, or more or less justified assumptions) of the modeler.

But again, this is nothing surprising, and nothing special for probabilities - the quality of the _description_ of any property of anything depends on the describer's knowledge, although the properties themselves are objectively fixed (if they deserve the name ''property'').

Thus knowledge plays in probability no role different from that it plays everywhere - at least not in those aspects of probability that can be checked in reality.

Subjective probability are a different matter. They are not verifiable or falsifiable, hence do not fall under the above analysis. But because of that, they should have no place in science or engineering.

 

[Dale]

There may be also a subjective probability in the Bayesian sense, but such probabilities are physcally meaningless. And this is a discussion in a physics forum.

On what basis do you make the claim that Bayesian probabilities are physically meaningless? You can use them to make predictions, test hypotheses, and all of the other things that you would expect to be able to do with probabilities in physics. Your claim seems to represent simply a personal distaste for Bayesian reasoning rather than an informed understanding of how it can be used in science.