Hello Jochen I think you returned back!

I just mention a nice link now:

http://www.stat.ufl.edu/archived/casella/Talks/BayesRefresher.pdf

('Official answer' another time)

I think this is possibly similar to Bayesians' attitude towards the notion of a sample space. Bayesians say: 'Why should I care what might have happened under imaginary data which were never observed?'. Frequentists might say: 'Why should I care about the value of the parameter in imaginary universes I will never have access to?'

Of course, if you form a sample space with elements (e.g. possible worlds) among which the value of a parameter can differ, the mathematical machinery of Kolmogorov's axioms allows you to regard the parameter as a random variable. But anyway you cannot sample from that 'sample' space, so I think that's the difference with the experiment framework.

(You can object to the experiment replications being what the theory calls for, but unlike universe replications, at least you *can* do experiment replications.)

I'm not sure anybody in the world is a frequentist à la Von Mises. I think frequentist statisticians just use Kolmogorov's machinery with 'frequentism' being, so to speak, no more than a semantic layer on top of it in order to translate between math-world and real-world.

Demetris, thanks. So far I can not find anything helpful in the linked presentation.

It repeats that the parameters are fixed. Well known. But again no rationale is given. This is a quite metaphysical assumption and there should be some good reason for such a strange assumption.

I miss this point being addressed:

Frequentists take parameters as fixed and the whole concept is about data that has not been observed. That is, it is a bout data that is still unknown. So evenatually it looks like modelling the uncertainty (lack of knowledge) of unseen data. In contrast, Bayesians are said to take data as fixed and the concept is about unknown parameters (or our knowledge we have about these parameters).

What makes the difference? What makes it impossible (or non-sensical) for a frequentists to model uncertainty about parameters, when he obviousely can do so for data?

There is nothing like variable data. Whatever data can be used is known and it does not change. The frequency aspect used is only required to indicate that we can not (perfectly) predict future/unseen data, and this is precisely not more than an indication of uncertainty. It is thus natural for frequentists to have expectations about the data, but they seem not be allowed to have an expectation about parameters. I don't understand where this prohibition comes from.

The second point in the presentation related to my problem seems to be the sentence that frequentists say that "studies are repeatable". With respect to what? An ideal repetition can not be meant. In every example I know, a "repetition" must include some "variation", and the important thing is the the amount of variation is undeterminable or unknown. If everything was known, there was no variation in the data and no (meaningful) probability. All examples made up on a real identity of conditions are metaphysical. I can watch atoms decay radioactively, and I may not know any difference in conditions between two decays I observe, but assuming that anything that can influence / cause? a decay is therefore absolutely constant is a very strong and strange assumption. Given a deterministic world, two atoms in under identical conditions will both behave identical, e.g. decay at the same time. They don't. So if they were existing under identical conditions, then the decay was an indeterministic process. However, if two atoms were identical, they would be one and the same atom, quantumphysically. Or the other way around: two atoms are quantumphysically different, so what should one even expect that they would behave identically (with respect to whatever, for instance time of decay)?

Pedro, frequentists never sample from the sample space they refer to. The sampling space is purely imaginary. It is what they think it should be, given some information from some available data. And even if it is taken for "real", the meaning only appears for n->Inf, and this is not possible, neither in space nor in time. The ugly problem is that at any finite endpoint the result can be arbitrarily far off the mark.

And iven taken this for granted, it is not clear how I can repeat an experiment. Logically, repeating the same does not add any information. The repetitions are different (and therefore do add information). But what is the difference? If I measure the pH in a solution 5 times I get 5 slighly different results. What should be done to repeat this experiment? How much different are which conditions allowed to be in order to be counted as a repetition? How precisely is "repetation" defined? How can one proof that another measurement is a valid repetition so that its result resembles another value of the infinite sequence leading to the very same (unknown) pH I was trying to estimate?

I don't think that " Kolmogorov's machinery" has anything to do with frequentism. It is an axiomatc system defining a measure. This measure has very much in common with the measure of relative frequency, so most (if not all) mathematical operations that apply to relative frequencies apply to probabilities, too. But this is a coincidence, not a logical or even causal connection. Kolmogorov's machinery is the only one that produces coherence. This is the point. YOu can develop the axioms based on Cox' postulates. There is no frequentitic point of view required.

Fausto, look here:

http://en.wikipedia.org/wiki/Frequentist_probability

http://plato.stanford.edu/entries/probability-interpret/#FreInt

http://goo.gl/VSV6Og

Fausto, right, there is a differece between defiition and interpretation. The axioms are a mathematical definition, they build the fundamet of probability calculus. So it is defined as a measure (sigma-additive Borel set) - that's fine. But what I read is that

P(x) == lim[n->Inf) h(x)/n

is an interpretation (for me personally it is rather a circular assumption; the frequency of an event is defined like ths´when the limiting frequency behaves like this...) and that this interpretation is the reason that P(hypothesis) is an impossible thing because h(hypotheis) makes no sense (I doubt even this; see the "multiple-universe argument").

Correct my when I am wrong:

the theorem expresses a "convergence in probability".

Such a convergence follows from an "almost sure convergence".

The converse does not hold, i.e. from a "convergence in probability" does not follow an "almost sure convergence". Therefore, the relation given in the theorem can not be simplified that lim(h(x)/n) == P(x).

Is this correct?

But then, what if one postulated that there is an "almost sure convergence" of P(x) and h(x)/n? Then the theorem is still valid (and follows from this postulate/assumption). However, it would surely not explain the "almost sure convergence".

I want to make only some remarks, I cannot answer the answer about such a strong and old debate:

Personally I am trying to avoid the adoption of a model based approach and I am looking for methods that can identify a hidden property by using consequences of the definition of that property. I don't know if this kind of thought can be classified and I would not like to join any army camp.

Demetris, thank you for your contribution. This question was not asked to sort people into "army camps" ;) I also do not want to get involved into the battle itself. I try to understand aspects of the "used waepons", so to say.

The aim of "reliability of predictions" is a good point. Just here I am also confused about the way this is usually argued. From an epistemic point of view this cannot be an aim at all, this is simply not in the "domain" of this interpretation (there was a huge discussion about utility measures to still address this, but to my understanding this descussion eventually went nowhere [or in circles]). Reliability is connected to the frequentistic interpretation, since only here can be made an argument that some percentage of predictions must be correct (see the whole concept of confidence intervals). But I miss a way to really check or confirm this. The reliability follows from the assumptions, and the assumptions may not be valid. The only confirmations or "proofs" invoke the use of fixed limited frequencies itself. This seems to be a circular argument. Additionally, the low rate of successful confirmations of study results (even after consideration of publication bias and "bad" scientific practices) is an indication that the "nominal reliability" is far away from the theoretical reliability, what in turn makes the use of "reliabilty" as a measure of "quality" in statistical analyses quite useless.

Fausto, it's perfectly fine, you should correct me whenever you think my thoughts or conclusions are wrong. I understood that you were critczizing the definition/interpretation and not my comments on it. I am looking forward for your contributions.

I understood the circularity of the reliability concept, but in this case we have to define what we should in common sense mean by the term reliability. Probably I didn't use a good term. So, to put it another way, we have to find a way in order to measure (common sense, not math!) the 'goodness' of each practise. How could we do this? Forget the definitions we know till now...

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two critical papers about finite and hypothetical frequentism :

http://joelvelasco.net/teaching/3865/hajek%20-%20mises%20redux%20redux.pdf

(point 6 is quite interesting :" 'Measurement' of the probability is mistaken for the probability itself")

http://philrsss.anu.edu.au/people-defaults/alanh/papers/fifteen.pdf

.

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to be fair with Hajek, he is also harsh on other interpretations of probability ; see "The reference class problem is your problem too"

http://philrsss.anu.edu.au/people-defaults/alanh/papers/rcp_your_problem_too.pdf

.

@Demetris: should we consider that there simply is no such measure?

@Fabrice: I had to read it several times. Hajek seems to have collected all my thoughts (and more) :) I think most of the arguments against finite frequentism are obvious. But I have problems to understand several arguments for the infinite case. Some aspects I simply do not get or I "believe" that either the premises or the conclusions are wrong. Nevertheless there are arguments I can follow and which I think proof that the concept of a limiting relative frequency is not compatible with "probability" (even when moving to NSA and hyperreal field definitions).

Taking this and the fact that the "limiting frequency interpretation" is based on the tautological argument P{|Sn/n-p| < delta} (as Fausto referenced), I face severe problems with the whole "standard practice" in statistics, particularily with the Neymanian concept of the control of error rates and the "meaning" of confidence intervals. If probability is not a limiting relative frequency, there is no such tool to "control long-run error rates". The tools may ensure a coherent (decision) behaviour, but they won't control error-rates (or, to be less strict: they need not control them).

Do I miss something important or are we in trouble as deep as I fear?

@Fabrice: I also found this (it's from Hajek, too):

http://plato.stanford.edu/entries/probability-interpret/

Unless I missed smth. in the previous  discussion. There is no problem assigning probability to a hypothesis. As we, the planetary scientists, often write in our articles:... it is  probable (plausible) that certain microorganisms can survive in the Martian subsurface at a depth of ..." To quantify this probability:  in contrast to interpreting probability as the "frequency"  of some phenomenon, Bayesian probability is assigned to a hypothesis, whereas under the frequentist view, a hypothesis is typically tested without being assigned a probability.