QBism - Is it an extension of "The Monty Hall Problem"?

[atyy]

In contrast, the methods of Bayesian statistics are indifferent as to the number of trials. You can get information from a single trial. You can get more information from 1000 trials, but there is no magic number of trials.

The results of Bayesian statistics are dependent on the number of trials. Regardless of the prior, even if it places an infinitesimally small probability over the true probability, as long as the prior is non-zero over the true hypothesis, the Bayesian will converge to the true probability.

Bayesian statistics is guaranteed to work if one knows in advance all possible hypotheses. Which is why it is beautiful, and also impractical - because if we did, we would already have a candidate non-perturbative definitions of all of string theory.
http://en.wikipedia.org/wiki/Bernstein–von_Mises_theorem
http://www.encyclopediaofmath.org/index.php/Bernstein-von_Mises_theorem

The other important theorem is the de Finetti representation theorem that allows Bayesians to be "effectively frequentist".
http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture1.pdf
http://www.stats.ox.ac.uk/~steffen/teaching/grad/definetti.pdf

 

[bhobba]

I would think that justifying that choice would be very difficult for the frequentist. The frequentist would say that there is no probability of theory A versus theory B. Either one or the other is correct, even if we don't know which.

A Bayesian could say the same thing as well - its just our knowledge of which is correct is subjective.

A Frequentest would say in a large number similar situations a certain proportion would be wrong. But, in that application it seems a strange way to view it - for hypothesis testing I think Bayesian is better.

But take Stochastic Modelling and say a queue at a bank. It's quite natural to think of if we did a large number of trials we would find a certain proportion with this or that number. Although of course one could view it the Bayesian way and think of a each number as simply subjective likelihood - but for me such isn't that visual.

As far as QM goes, for exactly the same reason I find the ensemble view more appealing - its concrete thinking of repetitions of the same observation - exactly as Vanhees says.

You can view it Bayesian and get something like Copenhagen, but like the queue it sort of seems unnatural.

One thing for sure, Bayesian is the obvious correct way to view Many Worlds. We know we must be in a world but which one. However I don't want to revive that long thread we had about it.

Thanks
Bill

 

[microsansfil]

When I did my degree
But never in all my studies have those terms been used.

I will not give you my CV.

There are different languages and cultures

Ontic := Objectif
Epistemic : Subjectif

Many different interpretation
http://en.wikipedia.org/wiki/Frequentist_probability
http://en.wikipedia.org/wiki/Probabilistic_logic
http://en.wikipedia.org/wiki/Propensity_probability
http://en.wikipedia.org/wiki/Bayesian_probability
...

The first thing I need to ask - how do they diverge from the Kolmogorov axioms?
l

Proof theory
Model theory
Gödel's completeness theorem

Kolmogorov is included in this theoretical framework Measure (mathematics).

In QM there are, inter alia, Quantum probability Theory :

http://arxiv.org/abs/quant-ph/0601158 That isn't Kolmogorov axioms (mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933)As in general relativity there are Differential geometry.However physical is not as mathematical

Patrick

 

[bhobba]

There are different languages and cultures

I think that's obvious

Ontic := Objectif
Epistemic : Subjectif

Its good to know what you mean.

I did a scan on the book you linked to and it did not mention either of those terms.

Many different interpretations

I am aware there are a number of different views. That's not my issue. My issue is since they all are based on the Kolmogorov Axioms they all must give the same results.

My background is math mate - I am well aware of what constitutes a valid proof.

I am well aware of Godel's theorem, but its relevance here has me beat.

I am well aware of Model theory. Its application to non standard analysis is one of the most beautiful pieces of math I have ever seen - and one of the most difficult - its decidedly non-trivial. Again its relevance here has me beat.

Kolmogorov is included in this theoretical framework Measure (mathematics)

Mate - didn't I just do a post on a book of rigorous probability theory I studied a similar one of. Exactly what do you think it's about?

That isn't Kolmogorov axioms (mathematics of classical probability theory was subsumed into classical measure theory by Kolmogorov in 1933)

By definition the Kolmogorov axioms are a measure space with total measure one and conversely a measure space of total measure one obeys the Kolmogorv axioms.

I am now starting to suspect your knowledge of rigorous probability theory is rather rudimentary.

As in general relativity there is Differential geometry.

I have studied GR. The situation is exactly the same as Feller wrote for probability.

However physical is not as mathematical

Nobody ever said it was. Again I repeat what Feller said and highlight the key point:
'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of the mass and energy or the geometer discusses the nature of a point. Instead we shall prove theorem's and show how they are applied'

Now exactly what is your issue?

Thanks
Bill

 

[stevendaryl]

The results of Bayesian statistics are dependent on the number of trials. Regardless of the prior, even if it places an infinitesimally small probability over the true probability, as long as the prior is non-zero over the true hypothesis, the Bayesian will converge to the true probability.

Bayesian statistics is guaranteed to work if one knows in advance all possible hypotheses. Which is why it is beautiful, and also impractical - because if we did, we would already have a candidate non-perturbative definitions of all of string theory.
http://en.wikipedia.org/wiki/Bernstein–von_Mises_theorem
http://www.encyclopediaofmath.org/index.php/Bernstein-von_Mises_theorem

The other important theorem is the de Finetti representation theorem that allows Bayesians to be "effectively frequentist".
http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture1.pdf
http://www.stats.ox.ac.uk/~steffen/teaching/grad/definetti.pdf

I once sketched out a Bayesian "theory of everything". Theoretically (not in practice, because it's computationally intractable, or maybe even noncomputable), you would never need any other theory.

Let T_1, T_2, ... be an enumeration of all possible theories. Let H_1, H_2, ... be an enumeration of all possible histories of observations. (It might be necessary to do some coarse-graining to make a discrete set of possibilities.)

Let P(T_i) be the a-priori probability that theory T_i is true.
Let P(H_j | T_i) be the (pretend it's computable) probability of getting history H_j if theory T_i were true. Then we compute the probability of T_i given H_j has been observed via Bayes' rule:

P(H_j) = \sum_i P(T_i) P(H_j | T_i)
P(T_i | H_j) = P(H_j | T_i) P(T_i)/P(H_j)

So this gives us an a posteriori probability that any theory T_i is true.

How can we enumerate all possible theories? Well, we can just think of a theory as an algorithm for computing probabilities of future histories given past histories. Computability theory shows us a way to enumerate all such algorithms.

 

[microsansfil]

Nobody ever said it was. Again I repeat what Feller said and highlight the key point:
'We shall no more attempt to explain the true meaning of probability than the modern physicist dwells on the real meaning of the mass and energy or the geometer discusses the nature of a point. Instead we shall prove theorem's and show how they are applied'

Now exactly what is your issue?

Your rhetoric does not interest me.
Physics is not limited to citations

Once again with the same mathematical axiomatic we can biuld different model/semantic.

Patrick

 

[atyy]

I once sketched out a Bayesian "theory of everything". Theoretically (not in practice, because it's computationally intractable, or maybe even noncomputable), you would never need any other theory.

Let T_1, T_2, ... be an enumeration of all possible theories. Let H_1, H_2, ... be an enumeration of all possible histories of observations. (It might be necessary to do some coarse-graining to make a discrete set of possibilities.)

Let P(T_i) be the a-priori probability that theory T_i is true.
Let P(H_j | T_i) be the (pretend it's computable) probability of getting history H_j if theory T_i were true. Then we compute the probability of T_i given H_j has been observed via Bayes' rule:

P(H_j) = \sum_i P(T_i) P(H_j | T_i)
P(T_i | H_j) = P(H_j | T_i) P(T_i)/P(H_j)

So this gives us an a posteriori probability that any theory T_i is true.

How can we enumerate all possible theories? Well, we can just think of a theory as an algorithm for computing probabilities of future histories given past histories. Computability theory shows us a way to enumerate all such algorithms.

But does this include that the same formal theory can correspond to different semantics (models - both mathematically and physically)?

 

[stevendaryl]

But does this include that the same formal theory can correspond to different semantics (models - both mathematically and physically)?

Well, for the practical purposes of science (trying to build rockets and lasers and computers and so forth), the semantics aren't important. The only thing of importance is how to relate past observations to future observations.

Of course, that viewpoint completely ignores the reason people are drawn to science--not for practical purposes, but to understand. It also ignores the fact that the connection between past observations and future observations is enormously complex, and from a purely computational standpoint, having a semantic understanding of what's going on is tremendously powerful in creating the connection. If you're just tinkering with algorithms without being guided by physical insight, it's hopeless, in practice. But in principle...

 

[bhobba]

Your rhetoric does not interest me. Physics is not limited to citations

No its not. But precisely what makes you think when someone 'show's how they are applied' that does not include a mapping from the abstract things in axioms to what you apply it to? For example in probability you map this abstract thing called probability to outcomes. One then applies the law of large numbers, and a few reasonableness assumptions, to show that abstract thing is the proportion of outcomes of a large number of trials. My suspicion is you don't have much experience in applying math. How its done is usually so obvious its not even spelt out - simply assumed.

Once again with the same mathematical axiomatic we can biuld different model/semantic.

Translation - the same axioms can be applied to different situations. Why you want to make a point out of something utterly trivial has me beat.

But its obvious you come from an entirely different background to me - and I suspect its philosophy - not applied math or physics.

I have discussed this sort of thing with philosophy types before - we talk past each other.

Thanks
Bill

 

[stevendaryl]

Your rhetoric does not interest me.
Physics is not limited to citations

Once again with the same mathematical axiomatic we can biuld different model/semantic.

Patrick

I think you're barking up the wrong tree in arguing with Bill. There's no disagreement between you two about the fact that the same mathematical theory can have different, non-isomorphic models. If you disagree with Bill, it would be helpful to try to pinpoint what the disagreement really is. I can assure you that it is not about model theory or Godel's theorem.

 

[microsansfil]

But its obvious you come from an entirely different background to me - and I suspect its philosophy - not applied math or physics.

That's why the truth lies in the axioms.

No comments.

I had the full range of your Fallacy. You did not interest me, I move on.

Patrick

 

[atyy]

Well, for the practical purposes of science (trying to build rockets and lasers and computers and so forth), the semantics aren't important. The only thing of importance is how to relate past observations to future observations.

Of course, that viewpoint completely ignores the reason people are drawn to science--not for practical purposes, but to understand. It also ignores the fact that the connection between past observations and future observations is enormously complex, and from a purely computational standpoint, having a semantic understanding of what's going on is tremendously powerful in creating the connection. If you're just tinkering with algorithms without being guided by physical insight, it's hopeless, in practice. But in principle...

That's true, but I don't mean mathematical semantics as much as physical semantics. For example, Euclid's points can model either physical lines or to physical points, so the formal object can have more than one valid physical correspondence, and I'm not sure if you can list all conceivable physical correspondences to a given formal theory.

 

[naima]

No comments.

I had the full range of your Fallacy. You did not interest me, I move on.

Patrick

Sorry, all french are not like that.

 

[bhobba]

Sorry, all french are not like that.

Its not a French thing I am sure.

I think, for reasons best known to him, he was simply being contrary.

I looked at his background. He is evidently a research engineer and should have understood many of the fundamental issues he bought up and how they are resolved in practice. I thought his background was philosophy because some (fortunately very few) can carry on like that - but evidently it isn't - which leads me to believe he was simply being contrary.

Thanks
Bill

 

[bhobba]

That's true, but I don't mean mathematical semantics as much as physical semantics. For example, Euclid's points can model either physical lines or to physical points, so the formal object can have more than one valid physical correspondence, and I'm not sure if you can list all conceivable physical correspondences to a given formal theory.

Of course that is true.

But when one evokes axioms in an applied context its usually utterly obvious from the context what you are mapping to what.

When it is said the truth lies in the axioms, and people like Feller say we don't attempt to define what the basic objects are; what is meant is the modern mathematical method. We prove theorems without referencing the meaning of the objects the axioms apply to; so when applied we have all these consequences without any further ado. You can apply the same axioms to many different areas with great economy of thought.

In relation to Frequentest vs Bayesian its simply a matter of interpretation of that undefined thing called probability in the Kolmogogrov axioms. You interpret it as plausibility ie a belief we as human being have and you get a Bayesian view - although that's usually done by the so called Cox axioms - but they are logically equivalent to Kolmogorov's axioms. You can leave it undefined and simply show via the law of large numbers (and yes some other assumptions such as assuming a very small probability is FAPP zero is required as well - but as is usual in applied math its not explicitly stated - you glean it with a bit of experience) that undefined thing is the proportion of a large number of trials. Also you can assume probability is a propensity and arrive at exactly the same thing. Kolomgorv's axioms guarantee it.

The only caveat with all of this is if you decide to get really tricky and map the same axioms to the same physical situation in different ways as mentioned about Euclidean geometry. Then you are in for a whole world of hurt in saying what's true and what isn't - of course you can do it - but great care would be required in keeping each separate.

It goes without saying that's not what is going on here so regardless of what view you take of probability you must get the same results.

Thanks
Bill

 

[microsansfil]

Sorry, all french are not like that.

This is the only argument you have. it is rather pathetic.

Is not a question of nationality because it is E.T Jaynes points of View. E.T Jaynes is not French isn't he ?

In purely mathematical framework (axiomatic) bayesian, frequentist are interpretation. They don't change mathematical framework. This is an evidence since it is outside the mathematical framework.

But mathematics isn't physics and neither the inverse. We can not reduce the QM in a single axiom purely syntactic.

I can only advise you to read E.T Jaynes. He provides concrete examples on different physical results obtained.

I have nothing to sell, no proselytizing in my case. I am just an Amanuensis

Patrick

 

[microsansfil]

« for all practical purposes »

E.T Jaynes :
http://bayes.wustl.edu/etj/articles/confidence.pdf

Confidence Interval (Frequentist) vs. Credible Interval (Bayesian)
AEC Graduate Course - Statistics :
http://www.lhep.unibe.ch/schumann/docs/nirkko_tufanli_intervals.pdf

To express uncertainty in our knowledge after an experiment :

– Frequentist approach uses a "confidence interval"
– Bayesian approach uses a “credible interval”

Example - Cookie jar

See results Confidence vs. credible interval slide 20

my position : Just Amanuensis

Patrick

 

[bhobba]

But mathematics isn't physics and neither the inverse. We can not reduce the QM in a single axiom purely syntactic.

This is the issue that leaves me scratching my head.

I can't find anything in this thread that says otherwise. My quote from Feller states we determine the meaning of axiomatic systems by seeing how they are applied. When someone says the truth is in the axioms, obviously it doesn't mean the axioms are the truth - because such are neither true or false - what it means is when applied the results from the axioms are so well known it becomes a testable theory.

Maybe it's because English is not your first language - I simply do not know.

So let's go back to what you said right at the beginning:

prob = 1/2 is not a property of the coin.
prob = 1/2 is not a joint property of coin and tossing mechanism.
Any probability assignment starts from a prior probability.

As far as I can see that is a philosophical position you are taking. There is zero reason you can't map probability as per the Kolmogerov axioms to the coin. In fact that's exactly what is done in basic courses of probability.

So can you please explain what's wrong with that? Why have textbooks like Feller got it wrong?

Thanks
Bill

 

[bhobba]

Just Amanuensis

Yes - but the issue is you are the 'Amanuensis' for a controversial position that is far from accepted eg:
http://stats.stackexchange.com/ques...ian-credible-intervals-are-obviously-inferior

'So essentially, it is a matter of correctly specifying the question and properly intepreting the answer. If you want to ask question (a) then use a Bayesian credible interval, if you want to ask question (b) then use a frequentist confidence interval.'

While I haven't read Janes book, from my knowledge of Bayesian inference the above looks a lot closer to the truth of the matter than the frequentest view is wrong.

Its simply a matter of what is the most natural way to view things making problems easier. That's nothing new, and as I have posted previously in this thread as far as Bayesian hypothesis testing is concerned the frequentest view is rather unnatural - but wrong - that is another matter.

If you argue non-standard controversial position, you really should be able to justify it - not just fall-back on - all I am doing is repeating.

Thanks
Bill

 

[Drakkith]

Thread closed for the moment, pending possible moderation.

 

[Drakkith]

Since this thread has drifted from the original topic and has devolved into arguing, it shall remain locked.