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 Nonius Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 Posted: 2011-09-29 13:31 Oh, common guys.  who loves ya baby? Advice for contacting Edward Witten: Don't.
 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 Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 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. Advice for contacting Edward Witten: Don't.
 Nonius Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 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. Advice for contacting Edward Witten: Don't.
 Nonius Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 Posted: 2011-09-30 23:10 Advice for contacting Edward Witten: Don't.
 numbersix Total Posts: 255 Joined: Jan 2007
 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 Joined: Jan 2007
 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 Total Posts: 745 Joined: Feb 2007
 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 Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 Posted: 2011-10-04 00:22 Melt, Jslade, Melt. Advice for contacting Edward Witten: Don't.
 Nonius Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 numbersix Total Posts: 255 Joined: Jan 2007
 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 Joined: Jan 2007
 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 Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 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. Advice for contacting Edward Witten: Don't.
 Martingale NP House Mouse Total Posts: 2580 Joined: Jun 2004
 Posted: 2011-10-04 23:42 It kind of felt like rap battle between two of you, lol
 Nonius Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 Posted: 2011-10-04 23:52 sort of like I'm Jay-Z rapping Takeover?  Ok, I can see that, sort of. Advice for contacting Edward Witten: Don't.
 Tradenator Total Posts: 1178 Joined: Sep 2006
 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 Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 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. Advice for contacting Edward Witten: Don't.
 Tradenator Total Posts: 1178 Joined: Sep 2006
 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 Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 Posted: 2011-10-05 00:13 this is true TRadenator. tis true.  I'm in your city soon, muhahaha. shall we partake in uncut? Advice for contacting Edward Witten: Don't.
 numbersix Total Posts: 255 Joined: Jan 2007
 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 Founding MemberNonius Unbound Total Posts: 11299 Joined: Mar 2004
 Posted: 2011-10-05 00:40 Insolence! Advice for contacting Edward Witten: Don't.
 deeds Total Posts: 236 Joined: Dec 2008
 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 Joined: Jan 2007