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Nonius
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Nonius Unbound
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Posted: 2011-09-29 13:31
Oh, common guys.  who loves ya baby?Big Smile

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numbersix


Total Posts: 255
Joined: Jan 2007
 
Posted: 2011-09-29 17:36
> Present your own creations or discuss someone else's.

When I say that the risk-neutral probability distribution that you infer from the prices of contingent claims has nothing to do with the real probability distribution, I do not mean the probability distribution of the real world in which you and I live and which extends outside theory; I mean the 'real world' from the point of view of theory itself or at least of its interpretation, as when the risk-neutral probability measure, under which assets are valued as the discounted expectation of their payoffs, is said to be obtained, through a change of measure, from the real probability distribution.

Measure theory in itself is a mathematical formalism that does not require interpretation and totally ignores words like 'real' or 'world'. There, probability is just a measure defined on sets and verifying certain axioms of positivity and additivity. A stochastic variable is defined measure-theoretically as a mapping between a probability space and a measurable space. Stochastic processes are in turn defined as stochastic variables that take whole trajectories t --> X(t) as values. The stochastic integral is defined theoretically under a mean-square convergence hypothesis which itself relates to vanishing variance and, consequently, to the notion of probability as previously axiomatized; stochastic differential equations are defined subsequently, etc.

In this whole un-interpreted episode, we don't know what probability means, apart from what the founding axioms say (it certainly doesn't relate to 'frequencies' or 'statistics' or 'events' or 'experiments' or 'possibilities' or 'randomness' or any of the notions that belong to our semantics); we don't know what a derivative means, apart from a mathematical function of a stochastic variable called the 'underlying'; we don't know what value means (and even less so, price), apart from the value of a mathematical function; as a matter of fact, we don't know what time means, apart from the index of the theoretically defined filtration, etc. And, to repeat, we don't know what 'real probability measure' means. All we know is the definition of equivalence between measures, none of which is distinguished as 'real', since they are all symmetrical in their equivalence, etc.

Interpretation enters into the picture when the stochastic process under scrutiny -- for instance a Wiener process -- is said to take place in the real world. To repeat, this is not yet the real 'real world', but an idealized real world. This is the stage when semantics is produced as an overlay on the theoretical formalism, or when a model is brought to bear on a theory (to speak the language of model theory). Trying to connect with empirical reality is a later stage. At the present stage, interpretation is still theoretical, although no longer formal-theoretical...


The necessary book is subtracted from chance.

numbersix


Total Posts: 255
Joined: Jan 2007
 
Posted: 2011-09-30 11:03
The word 'real' brings to mind the word 'realization', and this, of course, suggests the later notion of statistics and its later usage in empirical reality. Before we get there, however, the trajectory t --> X(t) has to be interpreted as a trajectory; in other words, t has to be interpreted as time and some fixed or constantly re-identifiable entity has to be assumed to lie behind X, in order that X be interpreted as a time-dependent property or manifestation of that entity, typically the movement of a particle in space, or the price of an asset, etc.

Had one particle, or one asset, or generally some entity, not been recognized and constantly re-identified behind X(t), there would have been no way, later, to recognize the sequence of independent and identically distributed (iid) random variables that the increments of the Wiener process will represent as repeated experiments pertaining to the same random phenomenon. In other words, there would have been no notion of statistics. Likewise, had one population not been recognized and constantly re-identified behind the sequence of iid random variables known as the height of an individual, there would have been no notion of the probability of the height of the representative individual of that population being of such or such magnitude. Without this reduction of the multiple to the one, there would have been no notion of a random generator that would later produce the statistics of human heights or the statistics of moves of the Brownian particle or the statistics of returns of the asset price.

The law of large numbers simply states that, given a sequence of iid stochastic variables indexed by n, their average converges in probability to the expected value of (any of) the variables; it does not say that it is the same experiment we are talking about, or the same population we are drawing from, or that the different variables are different measurements, indexed by n, of one and the same magnitude. As a matter of fact, measure-theoretic probability does not know what 'experiment' means. Kolmogorov dedicates a special paragraph, at the opening of his monograph, to emphasize the fact that the 'repeated experiment' idea, and thus frequency and statistics, are only one possible interpretation of his formalism...

The necessary book is subtracted from chance.

Nonius
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Nonius Unbound
Total Posts: 11299
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Posted: 2011-09-30 15:54
what? who? where? when? why?
I step away for a moment and it appears Hamilton has taken over. Ok N6, we'll sort it out. Don't really get why you want to post on this thread, it's not even a POPULAR thread.

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Nonius
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Nonius Unbound
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Posted: 2011-09-30 22:38
anyway, again, for the googolth time. where was I? Yes, indeed, just as Leo ceased from his sonorous maniacal guffaws, there was a loud crash emanating from the room behind a wall lined with dusty books wrapped in henna coloured leather bindings. the wall vibrated and a rumbling, grumbling, snarling reverberating din ensued. Leo waxed silent and his smile waned and was replaced by a half smile, half frown, the smile of Smerdyakov stirring the soup, the smile of a mind-fucking rabble rouser with a several brains. He let out a giggle and put his index finger to his lips, motioning for our collective silence. He then raised his claws and motioned me to follow him into the corridor that was seemingly carved in an askance angle to the wall of book teeth. Kitty obeyed and was quiet, or bored or sleepy. I rose and my heart started beating at a speed that sent tingles down my spine, and I followed him surreptitiously down the dark corridor. He said shhh, but let out a giggle here and there, and I followed but the voices in my mind were telling me to stop, turn around, flip the coin again after I left this lair and hope it would land on tails. Or was it heads? maybe the story was a better bet than The Bot.

The corridor was cold and with each step onward and forward I heard the distinct profound growl that seemed to broadcast its tone to mix with Leo's shuffling feet, his giggles, my breathing and syncopated heart beats in some expressionistic and preposterous polyphony. muhahaha, said he. Leo opened the door at the end slowly and methodically and it squeeked the sound of some strange strange synthetically derived creature that was being fucked for the first time. Squeeeek. Growl. Giggles. Then as the dark room was exposed to us, I could see a large object growling and moving about. Leo turned on a light and there, lo and behold, was a large brown bear encased in a rusty cage. There was a table the middle of the cage with the remains of a laptop strewn about. The bear was pacing back and forth, back and forth, and he did not seem very happy, to say the least.

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Nonius
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Posted: 2011-09-30 23:10

 


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numbersix


Total Posts: 255
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Posted: 2011-10-01 14:48
Let's say I am conducting an new kind of experiment in reading-writing. In the Borgesian library, this thread seems to me as appropriate as any other. It is not popular, you say? Let us just wait and see and fuck the bastards anyway. On the other hand, I am not sure I won't be making up the stuff I will be posting here any less (or is it any more?) that you are your own. Who knows? Could sense emerge from this hybrid thread? I suddenly realized that until now I had been constantly writing and posting under the authority -- and legacy -- of my published book. I wonder what my writing and thinking would be like in this totally divergent, uncorrelated, unrelated, 'unpopular' thread. Maybe I will discover absolute writing, meta-contextual writing, writing with no background. Could art just be the performance of art? There are many ways a forum can be deconstructed, and the one you have adopted in this thread is only the first iteration of the fractal. As my friend Spangberg says: 'We are engaging in choreography remember and that's important shit.'

The necessary book is subtracted from chance.

numbersix


Total Posts: 255
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Posted: 2011-10-01 15:26
So where was I? Oh yes. The whole idea of linking (measure-theoretic) probability with experiments, or events, or statistics, is only an interpretation. There is no randomness in Kolmogorov's probability, because there is no random phenomenon. Randomness presupposes identification of the random 'something'. A sequence of random variables will mathematically always remain of sequence of different variables -- or different things. It is outside the formalism (in interpretation) that you appoint them to a unique entity. Put differently, you need an interpretation of what reality is (and what realization is), in order to make sure that, every time you repeat the experiment and move from one independent stochastic variable in the sequence to the next, the underlying entity does not change!

You need the entity to be the same and you need the probability distribution of the outcome relating to that entity to remain the same. That the outcomes of two successive rolls of a die should be two independent identically distributed variables presupposes that the entity in question (or more generally the whole set-up: the die + the hand that throws it etc.) somehow has and retains this probability distribution. What does that mean? What does it mean, indeed, that the probability distribution should be inherent in the die, i.e. in a material entity, when it used to pertain only to a formal stochastic variable? Doesn’t the whole problem of making sense of physical probability and of saying what it is and where it resides lie in just this question? To repeat, there are no entities -- there is literally no-thing -- in the mathematical formalism and we don't know what 'repetition' means. A sequence of different variables will remain a sequence of different variables.

I should apologize too to drag us back to high school. Or rather not; this is not high school. This is even simpler than high school. These are the metaphysical presuppositions that no one ever discusses: meaning of 'experiment', meaning of 'repetition', re-identification, entity vs. sequence of formal variables, etc., etc.

The necessary book is subtracted from chance.

jslade


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Posted: 2011-10-03 23:17
If I weren't so clean living, I'd love to drop acid with you guys.

"Learning, n. The kind of ignorance distinguishing the studious."

Nonius
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Posted: 2011-10-04 00:22
Melt, Jslade, Melt.

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Nonius
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Posted: 2011-10-04 00:24

Yes, of course, of course.  Voices in my head, at this moment, were echoing the things that I said.  Or it said.  The merging of thoughts/memes/bots/dreams in the neverending fractal pattern of impossible complexity that defies quantification and definition and semantics and Four Tet Fornication of N6 mumbo jumbo was the thought that I thought at the moment that the bear was pacing at 2 knots in a pattern of the unknot next to his cot and he was fraught with the thought of doom.  Everything has been thought of before, according to my polish piimp.  We shall get to that, do not think that I am a wimp.

Or something like that. 

 My knuckles were white.  My teeth were clenched and the dreams of losing teeth were fetched from my finite state machine.  I was scared.  I needed the coin.  I needed an out from this place.  Don’t build The Bot, bro, write the story about building The Bot, thought I.

Leo babbled a bunch of things in Russian (I knew it was Russian because I know some Russian) to the bear, firstly softly and then, in western polysonic symmetry, his decibel levels increased in powers of something, I think 2, with each nanosecond.  Compute that .  He barked out gospadi boza moi to the bear and the bear  growled and rattled the cage and then the bear meandered to the central table and smashed the remaining macroscopic piece of the microscopic technologically infused laptop into bifurcating /2 pieces of plastic metal micro chip silicon grains of sand, which is, er, made of silicon maybe.  Leo then shook his head and hobbled to the hyperbolic edge of the room on the opposite side of the cage and opened a door to what appeared to be a closet filled with an endless supply of new laptops.  He pulled out a new laptop and brandished what looked like a stun-gun, the sight of which sent the poor bear screaming in horror and crawling like a cowardly cub to the antipodal side of the cage. Just then, Kitty sauntered in coyly and let out a rare, which precipitated a noticeable hesitation on the part of Leo, as if he had to wet his hand with his tongue, grab a mirror and make sure that the scant dearth of optimally distributed oases of hair on his homotopically nontrivial head were handsomely combed into a vector field of beauty that should, surely, hopefully, invalidate the hairy ball theorem but which, if true, would imply that Leo was not a sphere.  Proof by contradiction, and fuk the constructionists.  If Leo could comb the hairs on his head without creating a cowlick, then leo is not a sphere.  But who is a sphere anyway?

Leo ranted and entered the cage, averting his eyes away from Kitty, but I sensed he also felt proud to show his bravery to his kitten, for I did sense a few fleeting stolen glances from him towards her.  He plopped the laptop on the table whilst the bear cowered in the corner with the thing that looked like a taser pointed at him by Leo.  Leo then giggled in triumph and backstepped his way out of the cage, took a deep breath, locked the cage door, laid down his gun and then he rested and I sensed he was taking in the splendor, the enjoyment of the feat he had just achieved.  He then rose and explained that the bear was his “partner” and that the bear had proposed that pretty much, everything has been thought of before and that the closed timelike curves (CTC) which could be afforded through careful solutions of the Einstein Field Equations, plus Energy Conditions and experiments and so on, ie, the RHS of the equations that relate the a=a etc nonsense (stress, momentum, energy, blab bla bla) to the (n,1) signature pseudo-riemannian metric, and, well, let’s face it, measure theory is just a drop in the bucket in the complex sequence of ideas leading to something so satisfying and brilliant,  billiard ball brilliant,  appealing as, gravity is geometry but, heh, have fun with the right hand side.  The right hand side.  Anyway, Leo said that the bear said that the lifespan of all thoughts in the universe was finite, that the set of all thoughts/ideas/memes/dreams was finite since the universe was finite, would have finite lifespan, that thoughts were information, that information could be quantized, and that, therefore, due to the Copernican Principle of boring middledom that there was nothing important about the spacetime encapsulation of a thought.   I didn’t quite get it at the time, but I finally did and will go through it with all of you at the opportune time, but it basically boils down to Bayesian Statistics and assumptions about things.  Mental Doomsday, basically.  Basically, the bear said that there was very small probability that they had a new idea/strategy, and, when factoring in the possibility of CTCs (the short looped, highly curved CTCS could propogate into the future via an analytical continuation, and, even accounting for the noise, by basic probabilities, all the future thoughts could be brought back to present, rendering any current thought been there done that-ish), that it made new thoughts even less probable.  Leo laughed and asked why is it that everyone from D to T was recreating, for example, the Anthropic principle?  He was mocking the bear, because he didn’t believe it.  Thoughts were unique creations in Leo’s mind; there was a never ending supply of ideas/thoughts/dreams/memes, The Quixote was still alive, don’t give up, no matter whether there is darkness and dark matter and decaying bodies and brains requiring mental versions of viagra.  Yes, indeed, said Leo, and he then spiraled into a tumultuous torrent of speaking in tongues; he commenced in to repeating over and over again that all swans are white, that this is the same as all non-white things are not swans, that the brain interacts with the future, that we operate in a plane of unknowable, possibly inconsistent logical systems, that QM proved that, that disjunctive syllogisms and ex falso quodlibet should be thrown into the toilet of reasoning, that holes in logic could be measured with topological methods, that the worthiness of a logical system could boil down to the computation of certain topological invariants that could be assigned to CW complexes of categories of theorems/semantics/axioms/NumberSixians and entailments and that, in fact, he agreed with Penrose that maybe GUT would boil down to doing math on objects and arrows, and, well (he was working up quite a froth a this point), that he could just keep putting an endless supply of laptops on the bear’s desk and that finally, come hell or highwater, that the bear would simply would understand, as would kitty, and threads could be cross pollinated and everyone would be happy and books would be purchased, Philosophy lectures given, and meaning could be given to the dullness in the life of your above average survival machine.

Now, the disturbing part about all of this was the following "fact".  when Leo fired up the new laptop on the bear's table and the bear finally stopped whimpering and lazily ambled on his hind legs to the new machine, he pryed open the computer like a clam and I did, I swear I did, see the following information written in gothic font....what the fuck?


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numbersix


Total Posts: 255
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Posted: 2011-10-04 18:10
Now surely, the law of large numbers is empirically verified and we all know that the frequency of appearance of heads or tails in a sequence of coin tosses converges to a limit. But try to prove that without attaching, at some stage, a probability distribution to THE experiment as such (i.e. as uniquely identified yet repeatable), or to the experimental disposition -- what Popper calls the 'propensity' of the coin-throw experiment (or by metonymy, of the coin itself) to produce heads and tails with a given frequency. Only in this way could a sequence of stochastic variables become a sequence of draws from the same probability distribution. Measure-theoretic probability does not know what 'draw' or ‘trial’ means.

Popper did not hesitate to take this step. He claims indeed that his propensity interpretation of probability is what is most faithful to the measure-theoretic notion of probability (as opposed to von Mises’s frequency interpretation). What I think he means by this is that probability is defined first, then the sequence of stochastic variables is considered, then the interpretation is advanced that a property, which is called 'propensity' and which transcends the sequence, is inherent a priori in some entity and just makes it so that 'identically distributed' becomes equivalent to 'one identified distribution'. To make one thing out of a sequence of different things belongs in semantics and in interpretation; it is not part of the formalism.

The reality of that thing, to which attaches the propensity of producing outcomes that are iid stochastic variables, is an interpretation and no part of the formalism. Popper did not formally show that the frequency of heads converges. He assumed that the series of outcomes are iid variables and then he relied on the measure-theoretic law of large numbers. What granted his assumption? The notion of propensity. But then, propensity is not a given fact. It is itself defined in the process. To repeat, probability distributions attach to stochastic variables (in this case, the different outcomes), not to a real thing supposed to generate the outcomes. To posit that thing and to avail oneself to the concept of repetition, thus conflating the sequence of stochastic variables in the random generator -- what Popper calls the 'generating condition of the event' -- is the definition of propensity. It is not its consequence.

Now of course, we all feel that Popper is right and that there is something inherent in the die or in the coin which explains that the successive outcomes are draws from the same probability distribution, or trials of the same experiment. But the whole problem is precisely to say what this something is, this modality which makes the stochastic variables (the outcomes) identically distributed. Of course, we are all able to describe the die, to observe that it is unbiased (or biased and in what way) and symmetrical (or not) and to expect that it will land on one of its faces when thrown. But the extra step, which reasons from the possession, by the die, of a perfectly static and perfectly available physical description to the possession of a subsistent and persistent probability distribution (when probability distributions were supposed to be consummated with the stochastic variables and not to stick to the material entity), is the unwarranted step.

In other words, only if you combine the formal (analytic) law of large numbers and the material (synthetic) notion of propensity can the empirical convergence of the frequency of heads be derived.

The necessary book is subtracted from chance.

deeds


Total Posts: 236
Joined: Dec 2008
 
Posted: 2011-10-04 20:57

Numbersix, 

This thread is the right one...I think you must be creating a fictional author...because I feel confident you have done the basic literature search on your topic of choice and have read the vast array of authors who have worked to explore notions of probability distinct from the straw version you are evoking.  Notions that address the limitations to which you  call attention.

Also, as a courtesy, please include me in your audience by avoiding constructions like "Now of course, we all feel..."

(How easily I am baited, no?)

D

 


numbersix


Total Posts: 255
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Posted: 2011-10-04 22:20
deeds,

I pledged to sink deeper than the layer of the real and the fictional and embrace, for the time being, only the formal and the foundational (is this what you call the straw version of probability?); from which, I promise you, I shall extract riches unheard of as yet and come up with avenues and flights of interpretation -- that is to say, of reality -- quite unexpected.

My author is yet to come.

The necessary book is subtracted from chance.

Nonius
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Posted: 2011-10-04 23:37

From there it was a bit hazy, but I do remember running out of the corridor, offward and downward away from the bear's lair, pausing in the cavernous candlelit drawing room of the topologically nontrivial creature (and I don't mean the usual topology of you and me, he was different for sure and I have a simple language- Nonlinear N- to express that, later), tapping my shoes and chanting that there's no place like home, but that didn't work, so I ran out of the apartment, down the ten or so flights of spiralling stairs, out through the aperiodically black and white tiled floors and into the street, which was rain swept since it was a dark and stormy night.  I hailed a cab, and after figeting with my portable telephone machine I was able to retrieve the address of the scorpion friend who taunted me into this trick.  I wanted a second chance.  I wanted to flip the coin again and, well, maybe have a new lease on life.  The cab screeched down the arterie of the gothic city that was replete with tenebristic towers with black toothed windows,  reaching for the sky and into the clouds, as if trying to grab with grubby concrete fingers  for something more, more more and fuck everyone below in this city of devils, this amalgam of Dutch traders with no moral compass and worse English and followed by other people with nothing to lose, ie, Irish and Italians and Jews and Puerto Ricans and special high schools and so forth.  Cartesian arteries sprinkled with Northern European late Industrial Revolution and Modern sensibility drenched facades and funny faces informed me through light and shadows of their presence, fleeting as it was, whilst the drunk and swarthy cabbie drove at egregiously high speeds, right, then left, right angled manuvering in the process and listening to asymetrically constructed music from, I don't know, Greece, Turkey or some other cross roads of Linear X stuff.  But after the Minoans. 

 


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Martingale
NP House Mouse

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Posted: 2011-10-04 23:42
It kind of felt like rap battle between two of you, lol

Nonius
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Posted: 2011-10-04 23:52
sort of like I'm Jay-Z rapping Takeover?  Ok, I can see that, sort of.

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Tradenator


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Posted: 2011-10-05 00:02
It should be apparent to the reader now that numbersix is actually a character invented by nonius.  I suspect melt had something to do with it.

Nonius
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Posted: 2011-10-05 00:04

He has been a spicy creation, hasn't he?  a bit irreverent and insolent, but he is forgiven. oh, Nonius dammit, I am forgiven.


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Tradenator


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Posted: 2011-10-05 00:08
Next time, wait until you have some good & pure stuff that hasn't been cut with something dodgy before you invent characters, please.  It really stands out in this case.

Nonius
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Posted: 2011-10-05 00:13
this is true TRadenator. tis true.  I'm in your city soon, muhahaha. shall we partake in uncut?

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numbersix


Total Posts: 255
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Posted: 2011-10-05 00:33
I am positive that Nonius's creations are what makes this thread so popular.

The necessary book is subtracted from chance.

Nonius
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Posted: 2011-10-05 00:40
Insolence! Angry

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deeds


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Posted: 2011-10-05 13:15

Numbersix,

(baited again! Last time, Nonius, deepest apologies)

No, what I call the straw version of probability is not only the formal and the foundational.

I am referring to the "Straw Man" version of probability that seems to be a large motivation for your writing.  It asserts that most (all?) people hold the most naive (and unhelpful) beliefs about the relationship between probability and statistics and various systems under study.  It ignores the work already done to address the deficits you suggest.  You promise all sorts of alternative approaches, and then these are only cursorily considered.  I think it diminishes your credibility as an author, but that may be a convenient feature, not a bug, again...I know you are very clever.

Anyway, here I am polluting a perfectly good thread.  Again, apologies for that.

Kind regards

D


numbersix


Total Posts: 255
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Posted: 2011-10-05 14:57
Consider, by contrast, the approach of von Mises, who did not want first to formally define probability and only subsequently to make it inhere, through propensity, in the material entity. Von Mises wanted to define probability empirically, after the empirically observed phenomenon of convergence of the frequency of heads in the coin-tossing experiment, or the convergence of frequency of appearance of the six faces in the die-throwing experiment. There were no stochastic variables, no probability distributions or independent identically distributed variables in von Mises, because there was no notion of probability beforehand. Therefore, there was no law of large numbers. Instead, von Mises had to work with sequences where the frequency of the particular event was assumed to converge.

Having started in empirical reality as an unexplained and improvable given, von Mises had every latitude to consider a repeatable experiment and to assume the notion of trial; however, the probability, which he had defined as the limit of the frequency of the event in the series, could not, as a result, inhere in the experiment (or condition of the experiment) as such because it had no meaning outside the whole series. Either you had the formal law of large numbers but you needed propensity as a given in order to translate the law into empirical reality and to conduct statistics; or you had statistics and empirical convergence of frequencies as a given but then there was nothing more to your definition of probability than a reformulation of the empirical phenomenon you had assumed to be given.

We call the symbol Ω of measure-theoretic probability the 'universe of possibilities', we call the symbol ω the 'elementary event', we even call the average of a sequence of iid stochastic variables a 'statistics', and all these semantically loaded terms are signs that the designers of the formal theory had in mind the heuristic probability that they hoped to cover with their formalism. However, the formalism had its own way, once established, and even took its total independence.

For instance, we all believe that the succession of appearances of the face 'six' in a die-throwing experiment is the repetition of occurrence of the event ω that the stochastic variable known as the die maps into the value 'six'. We all believe that by the law of large numbers the frequency of occurrences of the event 'six' will converge to its probability. However, the measure-theoretic law of large numbers says nothing of the sort. It does not handle a sequence of events but a sequence of stochastic variables. What the formal law says is that the average of the outcomes of throws, now interpreted as iid stochastic variables by the propensity interpretation of probability attaching to the die, is itself a synthetic stochastic variable with a peculiar probability distribution –- a probability distribution such that the measure of the event whereby the average value (i.e. the value in which the synthetic stochastic variable is mapped) would differ from the common expected value of the iid stochastic variables known as the outcomes by more than a given threshold can be made as small as we wish by including sufficiently many stochastic variables (or outcomes) in the sequence under consideration.

At no moment was anything like an actual event, or a series of actual events, mentioned. Nothing was realized. The whole discourse involved only stochastic variables, or complete mappings of elementary events into numerical outcomes. The stochastic variables remained packed with all their possible events. Crucially, the stochastic variables are different yet we are considering their sum (or their average). This means that, on any specific instance, they should all take the same ω, or elementary event, as argument. This does not mean that the outcomes will be the same, of course, since the stochastic variables, or the functions mapping the elementary events into values, are different. The only constraint is that they are identically distributed. My point is that the whole sequence, and consequently the average, is instantiated at once, through the single draw of ω that all the stochastic variables admit as argument at once. It is not the case that the stochastic variables are drawn in a succession.

There is no time and no succession in the formalism of the sequence; the notion of 'trial' or, as Gnedenko defines it, 'the realization of a definite set of conditions as the result of which some elementary event of the space Ω of elementary events can occur', is external to the formalism and belongs to its interpretation (semantics). In the words of Kolmogorov: 'There is assumed a complex of conditions, C, which allows of any number of repetitions. We study a definite set of events which could take place as a result of the establishment of the conditions C.’ Clearly, the formalism totally ignores the notion of 'set of conditions', and consequently of 'trial' or 'repetition of the trial'.

The necessary book is subtracted from chance.
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