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Proving $x^{\log n} = n^{\log x}$
A number with 6 distinct digits which get multiplied by 5 if we move the last digit to front

Am I right to assume that *under the frequentist interpretation of probability*,* the set of allowed probabilities isn’t $$\left[0,1\right],$$

but rather is

$$\lim_{m\to\infty}\left.\left\{\frac{n}{m}\,\right|\; n=0,\ldots,m\right\}$$

or perhaps

$$\bigcup_{m=1}^\infty\left.\left\{\frac{n}{m}\,\right|\; n=0,\ldots,m\right\},$$

or $$\left[0,1\right]\cap\Bbb Q,$$

all implying, I assume, e.g., i) that the set of allowed probabilities is countable, and ii) that $\frac1\pi$ isn’t an allowed probability?

*NB: Besides a straight answer, a little background or some references would be highly appreciated.*

*Perhaps, or perhaps not, including that “[a] controversial claim of the frequentist approach is that in the “long run,” [*sic*] as the number of trials approaches infinity, the relative frequency will converge *exactly* to the true probability […]”.

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- Exactly who popularized the modern definition of domain and codomain of functions?
- Is math built on assumptions?

The essence of frequentism are that probabilities are “out there” in the world rather than aspects of our knowledge of the world. A frequentist will not say that there’s a 40% chance that there was life on Mars a billion years ago, unless the frequentist believes that in 40% of all instances of the universe, there was life on Mars a billion years ago. A frequentist may say that there is a 40% chance of contracting X disease if you’re exposed to Whatever, and that will mean that 40% of people exposed to Whatever contract X disease. A frequentist may also say that it is *estimated* that there is a 40% chance of contracting X disease if you’re exposed to Whatever, but the frequentist is then uncertain what the probability is.

To a Bayesian, probabilities are epistemic. They express uncertainties. A Bayesian may say the conditional probability of life on Mars a billion years ago, given the Bayesian’s current knowledge is 40%. That is not an *estimate* that 40% of anything out there in the world is thus-and-so. The Bayesian need not be uncertain about whether it’s 40% or 41%, but the number expresses the Bayesian’s uncertainty about something else.

There is nothing in either the frequentist position or the Bayesian position that requires probabilities to be anything but numbers in $[0,1]$.

Probabilities are rational in a limited probability universe where random variables are restricted to a bounded number of uniform random choices from finite sets. The bound can depend on the variable, so that one comes from throwing 50 or fewer dice, another from up to 307 random bits, and a third from 5 bits and between 3 and 10 dice.

If unlimited throwing of rational coins (stopping “with probability 1”) is allowed, such as throwing coins indefinitely until the sequence HTTTTTH appears, then all real probabilities in $[0,1]$ can be achieved. If number of coins is limited, the issue is whether coins with irrational probability are allowed as elements in the theory.

I will comment soon if you want references on this topic:

Please see Alan Hajek’s papers “Fifteen Arguments Against Hypothetical Frequentism”

And by the same Author, []:[“15 arguments against Finite Frequentism”]

[]2

I also noticed this wmbriggs.com/post/12651/

Remember that there are many brands of probability interpretation, including but not restricted to, further -denominations within, frequent-ism and Bayesian-ism.

It it ironic that the proponents of these two interpretations seem to come into conflict most with another.

Despite the fact that in my view (subjective Bayesian-ism) is essentially, based on exchange-ability, and betting/breaking even arguments.

Sometimes called,the betting-interpretation; as well as due to the bayesian updating, based on frequencies.

That is, it ultimately parasitic on Frequentism, at least in some cases.

Whilst arguably, Objective Bayesianism on the other hand, is just an admixture of the Classico-logical account on the one hand through principals of indifference on priors/exchange-ability, and frequent-ism, through the frequency data on which exchange-ability and updating is based.

They cannot say what they mean by probability. Not that anyone really knows what ‘probability means’, if indeed the concept in general, has any coherent meaning at all) .

But at least some accounts attempt to give reductive/semi-reduction analysis.

They cannot have their cup and eat it by branding their account the one true account by using the words logical, or objective rational credence.

The Principal Principle cannot be derived or resolved,

**Simply by defining one’s account, just to be** ‘**that, **account**, **the likes of which delivers the ONE true objective rational credence’s****’.

See also

Those arguments that ‘frequentism is circular’ because of $(1)$ and $(2)$ seem to me to be moribund (do not work).

*von Mises, R.*, Comments on Donald Williams’ paper, Philos. Phenomenol. Res. 6, 45-46 (1945). ZBL0063.04020.

*Jeffrey, Richard C.*, **Probable knowledge**, ZBL06702506.

*Eagle, Antony*, **Randomness is unpredictability**, Br. J. Philos. Sci. 56, No. 4, 749-790 (2005). ZBL1098.03508.

As well as “random-ness and the right reference class by H Kyburg

https://www.jstor.org/stable/2025794?seq=1#page_scan_tab_contents

“direct Inference” by Isaac Levi”https://www.jstor.org/stable/187801?seq=1#page_scan_tab_contents

And see https://www.jstor.org/stable/187801?seq=1#page_scan_tab_contents

Perhaps there is hope, and someone will actually develop a model and analysis, of single case propensity, that is reductive and individuates it from other accounts

$(1)$ “It uses the Strong Law of Large Numbers to define Probability in terms of Probability”

$(2) “Or that it is moribund because it only speaks about the Long run”

Not that, frequent-ism, is not circular in other ways. For example in hypothetical frequent-ism, when in analyzing, the closest possible worlds, which have the same laws ‘probabilities/frequencies; what is meant by law-

If its Hum-ean in one sense to begin with?

Or, due to reference class issue, which may render it at least both circular, and somewhat pragmatically circular.

See for more on this issue *Hajek, Alan*, **The reference class problem is your problem too**, Synthese 156, No. 3, 563-585 (2007). ZBL1125.03303.

*Hajek, Alan*, The reference class problem is your problem too!, Brown, Bryson (ed.) et al., Truth and probability. Essays in honour of Hugues Leblanc. London: College Publications (ISBN 1-904987-19-2/pbk). Tributes 3, 86-110 (2005). ZBL1269.03028.

Where-as in the single case propensity account, the reference class issue does not render it definition-ally circular. But at least pragmatically circular, or non-useful at least for other reasons.

It depends on what kind of frequent-ist one is;it can be shown that they can allow for irrational probability values. But i wonder if the non-standard- hyper-hyper frequentist could make sense of such irrational fractions. They could not re-order the countably infinite sequence (that is , if they are) to get such fractions.

That(maximally specific reference class), (and Humphreys paradox) of course, is the least of the single case propensity interpretation’s issues.

*Eagle, Antony*, **Twenty-one arguments against propensity analyses of probability**, Erkenntnis 60, No. 3, 371-416 (2004). ZBL1093.03504.)

The single case propensity account, is (arguably) the only account that in some sense, avoiding all **standard objections**; there is hope for it, but not that this is a positive thing.

Its by way of it suffering, a worse objection’ “there is nothing to object to”** We do not even though what ‘single case propensity” even means. Its called the occult power, “alea-tive virtue, or the **no-theory theory** objection

Sometimes subjective Bayesian-ism is attacked by this objection as well. See for this note on bayesianism:’Alan Hajek, Conditional Probability is the very guide to life, chapter 9′ in, See https://books.google.com.au/books?id=nGQRLD6MtIEC&pg=PA346&lpg=PA346&dq=hajek+thalos&source=bl&ots=_KTH2UAMDh&sig=YG3RKfXfeyjyeVA8ogiayVolwxo&hl=en&sa=X&ved=0ahUKEwig45K5yuTTAhXEH5QKHcJTCY0Q6AEIOzAE#v=onepage&q=hajek%20thalos&f=false

That is, whilst I am not a frequent-ist, I can least say this;

$(A)$ Frequent-ism does not even have a need for the ‘strong law of large numbers. and

And why would they want to use it, when they make a stronger claim.as part, or as their semantics of chance.

Why would they want prove their definition and in particular in terms of a weaker claim.

(2) At least they do,(or allege) to get frequencies right in ‘long run’. Other accounts, cannot even get the long run right (it is other accounts which cite the meaning-less with probability one, or almost surely, in the limit…). So what hope do they have for the short run?.

At least the Frequent-ist says something certain, even in hypothetical and justifiable.

Again I stress that I am not a proponent of their account.

I think that the entire concept needs to be re-worked; with a much stronger tie with logic (much much stronger than the non existent connection in the objective bayesian and/or classical/logical views).

There are many others, in some sense including Kolmogovov’s own account, logico-classical views, Propensity and long run propensity view.

Many people who have researched these matters do not subscribe to any account, and it would be incorrect to call it a debate, as if we know that one side is correct and we are trying to figure out which.

Nigh one anyone, who has given some though into this, knows that every position thus adduced, is dead wrong; and not even infinitesimally close to the mark.

And that is, even if there is a mark, to get close to!.

See https://philpapers.org/rec/JEFMR for the original papers by R Jeffreys attacking frequentism; as well I Levi’s, Confirmation and conditionalization, and his direct inference which you can find on the same page.

It depends on what kind of frequent-ist one is;it can be shown that they can allow for irrational probability values. But i wonder if the non-standard- hyper-hyper frequentist could make sense of such irrational fractions. They could not re-order the countably infinite sequence (that is , if they are) to get such fractions.

That(maximally specific reference class), (and Humphreys paradox) of course, is the least of the single case propensity interpretation’s issues.

*Eagle, Antony*, **Twenty-one arguments against propensity analyses of probability**, Erkenntnis 60, No. 3, 371-416 (2004). ZBL1093.03504.)

The single case propensity account, is (arguably) the only account that in some sense, avoiding all **standard objections**; there is hope for it, but not that this is a positive thing.

Its by way of it suffering, a worse objection’ “there is nothing to object to”** We do not even though what ‘single case propensity” even means. Its called the occult power, “alea-tive virtue, or the **no-theory theory** objection

Sometimes subjective Bayesian-ism is attacked by this objection as well. See for this note on bayesianism:’Alan Hajek, Conditional Probability is the very guide to life, chapter 9′ in, See https://books.google.com.au/books?id=nGQRLD6MtIEC&pg=PA346&lpg=PA346&dq=hajek+thalos&source=bl&ots=_KTH2UAMDh&sig=YG3RKfXfeyjyeVA8ogiayVolwxo&hl=en&sa=X&ved=0ahUKEwig45K5yuTTAhXEH5QKHcJTCY0Q6AEIOzAE#v=onepage&q=hajek%20thalos&f=false

That is, whilst I am not a frequent-ist, I can least say this;

$(A)$ Frequent-ism does not even have a need for the ‘strong law of large numbers. and

And why would they want to use it, when they make a stronger claim.as part, or as their semantics of chance.

Why would they want prove their definition and in particular in terms of a weaker claim.

(2) At least they do,(or allege) to get frequencies right in ‘long run’. Other accounts, cannot even get the long run right (it is other accounts which cite the meaning-less with probability one, or almost surely, in the limit…). So what hope do they have for the short run?.

At least the Frequent-ist says something certain, even in hypothetical and justifiable.

Again I stress that I am not a proponent of their account.

I think that the entire concept needs to be re-worked; with a much stronger tie with logic (much much stronger than the non existent connection in the objective bayesian and/or classical/logical views).

There are many others, in some sense including Kolmogovov’s own account, logico-classical views, Propensity and long run propensity view.

Many people who have researched these matters do not subscribe to any account, and it would be incorrect to call it a debate, as if we know that one side is correct and we are trying to figure out which.

Nigh one anyone, who has given some though into this, knows that every position thus adduced, is dead wrong; and not even infinitesimally close to the mark.

And that is, even if there is a mark, to get close to!.

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