Diaconis Persi "Poincaré's Probability"

okay my friends even the best things have to come to an end and so does this point are a celebration and we are approaching the start of the final speech in this conference forward to go well with the celebration of legendary re point array we have here a legendary speaker persi diaconis from stanford known of course for his great abilities of great mathematician and also for his great ability of magician in the past life that maybe is not so not so bad so Percy will tell us about when Kari and probability theory thank you thank you this is a no free lunch theorem Cedric and your job is to make sure that everybody gets one of these pieces of paper there's a one-page handout so it's some kind of revenge or something as a good director he knows how to handle these problems so good good early evening the what I want to tell you about today is is two things the first is I want to try to bring to life for you some of what Poincare a did when he did probability and the second point I want to make and I'll try to make it is that it's still very much worthwhile reading Poincare a there are treasures there there are open problems there are all kinds of interesting ideas and I'll try to make that specific don't expect too much in the following sense Poincare a didn't have probability as his main activity he got interested in the subject when he had to start teaching at the Sorbonne and he he eventually gave a course in probability that became notes that became that became a textbook in 1896 and then so and then he published the second edition of the book in in 1912 the year that he he died and so what we have to work from is not a collection of papers or or a big body of work but this this textbook that he that he wrote and I'm going to talk about three topics that are in the book that are topics that I've worked on also and the first is roulette and I think it's useful to begin with a discussion of real roulette as soon we'll be doing math and okay but there's also real roulette remember that real roulette there's a big wheel and there's a ball that goes ping around the outer wheel and then there's an inner wheel which is spun that goes the opposite way and the ball eventually falls off the outer wheel ends and the inner wheel and it lands on a number and there's a table and you can bet well from that description and as you all know relate as a physical game it's it there's a real object there and when I was a beginning assistant professor in it at Stanford three of us realized you know rule itis a physical game we could clock it and so what we did was make a little gadget which was about the size of this gadget the gadget that is connected to my pocket eventually it'll come out was about about this size and and so we would be in the casino and and the dealer deals the ball around and when the ball comes around the fix point in the room you push a button and when the ball comes around again tap again and now the gadget knows how fast the ball is going and when zero on the inner wheel passes a corner you tap and then tap again when it comes around and the gadget knows how fast the inner wheel is turning and then the gadget does a calculation and and it tells you where the ball will land it's only accurate to within half a wheel but if you're getting thirty five to one on an eighteen to one shot that's like having a vacuum cleaner in the casino and we built such a gadget and went and used it and made money you know it's just thinking right we didn't touch anything now along the way just to put a few sentences of reality in the first thing we did of course is do the equations kind of interesting math to do the equations and then I went and rented a roulette wheel and we tried it out and and all of our equations were completely wrong and we realized that as you usually have to we left friction out in this reasonably serious way and there are three kinds of friction that mattered here there's sliding friction rolling friction and air friction and the type that really mattered was air friction so that when they turned on the air conditioners in the casino all the equations changed it wasn't useless to have done the math in fact the form of the equations we could just leave in their form with some parameters and go into the casino and tune them up and plug in estimate the parameters based on 20 or 30 spins of the wheel and get get quite accurate assessments of the of the wheel and so there is a world of real roulette and it's a physical game is one thought for you so why on earth is rule like random that's a question I mean what what's random about roulette and Poincare a dealt with that question in in in a chapter of his his book and and he phrased it this way as we all do he made a toy model and he said suppose that when the ball is spun it spins around and it it comes to land with a certain probability distribution so let F of I'll call it theta be the probability that the ball lands at theta and so we've taken the roulette wheel and opened it out that's okay this is 0 this is 2 pi and F is some function and of course if you're if the dealer is very accurate maybe F is very peak but anyway this is f and Whangarei talked about the probability of red and so well you can think about around the wheel the reds and blacks alternate and so he took a serration where this is a small radius H and and he reasoned that the the chance of red is the area under the curve over the red squares and so this is some propensity for the ball to land now notice this is simplified an awful awful lot they're not two wheels anymore and there's no bouncing with this is the chance that the ball lands at theta but still those are the assumptions he made and then he proves and we will - that the probability of red if H is small tends to 1/2 as H goes to zero which okay makes of course makes intuitive sense but this is a theorem of panca raised from his probability book and let me give you a version of his argument he said look we may as well just look at the probabilities of H Oh red minus the probability sub H of black and that's the difference between the area under the red squares minus the difference under the black squares and so that's less than or equal to the integral from 0 to 2 pi of f of theta minus f of theta plus h D theta so the the difference between the two areas can be bounded by taking the absolute value inside and then want Caray well says that this is less than or equal to what I'll call Omega sub f of H where Omega sub F of of H is the modulus of continuity it's the maximum over [Music] differences less than or equal to H of f of X minus f of Y how Wiggly the function is and he knew that for continuous functions the modulus of continuity tends to 0 and therefore this tends to 0 and and the probability of red and the probability of black are are equal in the in the limit as as H gets small this is some useful reasoning and it's natural to try to ask how accurate it is or how you know what if H isn't so small how accurate is this if you like calculus problems you might try to prove this the probability of red minus 1/2 I'll do it that way is less than or equal to H over 8 times the integral from 0 to 2 pi of f prime of theta theta you know in order to prove any sort of theorem of this sort somebody could be very accurate and rolling and this density could be very very peak over a certain point so somehow or other you need to say something about about how peak the distribution is and for people who like calculus problems this 8 is sharp okay in the other direction there was literature following up everything that Wonka I wrote about has follow-up literature and pranker a assumed continuity and people said well you could do it with less in fact nowadays it's easy to see that this theorem this theorem is true for measurable functions you don't need any assumptions because translation is continuous in l1 ok if you like that kind of argument it does it does tend to 0 I like this kind of argument which gives me some sort of rate ok so that method got abstracted and I'll tell you one or two sentences more about it let me let me see if I have okay I do have one yeah I do have a lot of things one image of randomness that we talk about all the time is this feel of us and you know why does coin tossing which is a basic image of random phenomena why is it random and Jo Keller wrote a very nice paper in which he said listen you know when we flip a coin if I knew how fast it was going up and I knew how many times a second that was turning Newton tells me whether it's going to come down or soon it lands without bouncing whether it's going to come down heads or tails I know how long it stays up I know how many times a second that's turning I know what it's gonna do right the coin tossing is physics it's not random so you can do the physics and I in the handout there's a there's a picture I should explain well I don't have one but it's ok I know what's on the and out thank you kind person so first this is the picture from Wonka Ray's book if you if you want to see it you can see his version of the argument this is the phase space of I know this picture this is the phase space of coin tossing so let me just tell you what it's like this is velocity how fast the coin is going up and this is omega how many times a second the coin is spinning and so every time you flip a coin in this version in which it's sort of just flipping around an axis through its center of gravity a dot a flip corresponds to a dot on this picture how fast is the coin going up and how many times a second is it turning and so for example a dot here means it's going up with a lot of velocity but very little spin so it's going up like a pizza yeah so you can imagine that there's a region and it's bounded by a hyperbola when you do it where points below this region the coin doesn't turn over at all for example out here but also up here the coin could be going with a tremendous amount of spin but not very much velocity so it doesn't turn over and then there's a region in the next region where it turns over just once and then twice etcetera and and the picture on the right is as a picture of that face space if you look at the picture what you'll see is that as you move away from 0 the the regions get closer together and so intuitively small changes in initial conditions make for the difference between heads and tails and everybody knows that that's that's where randomness comes from I was interested in when we really flipped real coins where are we on that nice picture and so I did a lot of experiments over the years in one of them we got a tunable strobe light and you know painted half the coin black and half the coin white and flipped it and try to adjust the strobe and figure it out you know figured out where we are on this picture and in the units of this picture well one two three four five this in velocity the velocity coins the way you flip them normally go up about five miles an hour in my units and and in the units of this picture the velocity is a fifth well this is you know five this is one a fifth is pretty close to zero but coin spin something like forty revolutions per second and a typical coin flip takes about half a second and so in the units of this picture the velocity variable is 40 units up so actually the picture says nothing about real coin tossing but the math behind the picture tells you how serious region is and of course if you want to know about heads and tails what you have is a distribution of pics here and if the regions get close close together about half of the area lies over heads and half over tails and so one can make a version of plonker A's argument these kinds of arguments for this kind of setup and then the if you wanted to learn more about that in fact real coins do something more complicated when you flip a real coin in fact they process and the paper I reference with Susan Holmes and Richard Montgomery we did the three-dimensional case and so in three dimensions it's actually you know in mechanic's it's twelve dimensions right it's you know it's twelve dimensions and so you have some Syrian of twelve dimensional space and you want to know what what what portion of it lies over heads and what portion over tails and there were versions of this theorem for four for that kind of situation so that's point Carre's rillette argument it's often nowadays the method of arbitrary functions because Huang karere was interested in the fact that for an arbitrary probability density if H is small enough you know the chance of red is is 1/2 and the same is true in in in these other settings I said that there were things to do and one of the things to do which as far as I know hasn't been done is to make the right abstraction of these theorems that as I just told you a couple of special cases yeah you know that one two dimensional three dimensional what's the right setting what's the right abstraction of that theorem so you have a scale I don't know actually so okay how do you know nobody's nobody's come back and done that if you're interested on Kure at the end of his probability book has all kinds of things he's talking about which relate to that okay so that's my first topic of what's in Whangarei the method of arbitrary functions one more sentence this method was brilliantly developed by Eberhard Hoff who began a classification of low water differential equations and if you have uncertainty about the initial conditions or the coefficients of them how does that propagate into uncertainty and hopf ooh I think called it method of arbitrary functions and if you want to see references to the literature there's a section on it in the paper that I wrote with Susan and Richard Montgomery and for what it's worth I put all my papers on my home page if you type Percy into Google you'll find me and if you spell it correctly and and so it's easy easy to find okay so that's my first topic panca ray on the method of arbitrary functions this was a very very innovative idea what he was trying to establish was that there were certain situations in which the exact details of the probability distribution didn't matter he had three examples in the book and this is one of them the second example I'm gonna talk about is a little less widely discussed but hope that will change after this talk and so again remember my job is to try to bring what's in punker a alive to you and so I'm gonna tell you the way we think about it now and then I'll tell you what he did same way I just did so I call this subject well Bayesian numerical analysis and just to start suppose I had a function on zero one say on zero one I have a function f of X you know I'm gonna tell you what it is it's e to the cosh of x squared plus cosine X plus X to the fifth plus cinch X okay that's my function and so there it is my function and now suppose I also had a water pistol pistol dough how do you say water pistol and Fleance thank you and I have my water pistol and I say Cedric what do you think the integral of that function is between zero and one and he says and I go to sad drink so there's a question what does it mean to know a function just seeing a formula for a function doesn't mean you know very much about it you don't know about the maximum you don't know about its integral you don't know lots and lots of things about it well one thing you could do and we're going to do is to own up to that fact and say hey you know just seeing a formula I know something I don't really know all about the function so let me try this so since I don't know I'll put quotes around know the function I'll own up to that fact and I'll put a probability distribution a prior distribution on functions now I'm not a fancy french probabilists so I only know one probability distribution on functions Brownian motion so say I think or I assume for a moment that f of X is I don't know what it is exactly I have this formula sum a plus B times B of X where B of X is Brownian motion that's crazy but bear with me for a second now suppose I observe the function at a few points we see why I equals f of X I 1 less than equal to I less than equal to n so I observed the function at some points so now by calculus I have an apostille redistribution I assumed it was like Brownian motion I see where it is it a few points I have a new distribution which is Brownian motion constrained to go through those points then you could ask the question well if that's correct what's the optimal quadrature rule what's the best guess at the integral with this information so the yes under squared error is lost and I'll call it in the goal of f of X the X is f hat I'll call I had and and what it is is the expected value the average value of the integral of this B of X DX given that given what you're given the conditional distribution it's just Bayes theorem in this setting and what this is is the trapezoid rule the trapezoid rule that is I have my function I know it at some points I connect the points up by straight lines and I use the area under that straight line approximation as that's what this calculation gives you well seeing a classically used rule the trapezoid rule is very often used comes out of this crazy set of assumptions makes me want to try to use this board properly so you might say well look that function that you wrote down there horrible as it may seem it's nothing like Brownian motion it's much smoother well you might therefore say Brownian motion isn't the right prior you could try some smoother prior you might try say f of X is distributed is the integral from 0 to X of B of P DT or some other some other smoothing procedure but let's take once integrated Brownian motion that's a measure on the space of all functions and if you use this prior and go through that calculation you'll find that the Bayes estimate the optimal estimate is the cubic spline interpolant of your function if you if you integrate K times you get 2 K plus 1 ordered splines so seeing standard numerical analysis procedures coming out of this crazy assumption maybe makes you want to think about them a little bit more you can ask going backwards is there some measure on the space of all continuous functions say which gives you even order splines we don't know open open problem let me ask the fact that I was using the quadrature rule that I got out of an assumption and something as as some information about the quality of my prior assumption leads to the math problem is Brownian motion the only measure on functions which gives you the trapezoid rule well the answer is no so let me write a canvas the question is Brownian motion the only measure on functions giving the trapezoid rule and the answer's no any independent increments process of Poisson process gives you the same rule any the increments are well any independent increments gives the same but independent increments processes have jumps and if you want to say I have a measure not on functions but on continuous functions then that gets rid of that problem the second issue is is well twice B of T twice Brownian motion predicts the same way as Brownian motion and there's nothing special about two two could be Sigma and Sigma could be random and independent of Brownian motion but there's a beautiful theorem of David Williams which says that's it that is if you have a measure on continuous functions which predicts in the same way as Brownian motion that is gives the trapezoid rule as its quadrature rule then it is a rescale version of Brownian motion and so that's an example of how little math problems come come about now nobody needs me or us to tell them how to integrate one-dimensional functions but if you're integrating in higher dimensions this idea of working with our priori distributions and seeing what their bays rules gives is a very very healthy way of doing various tasks in numerical analysis and so this little subject is called Bayesian numerical and alysus by me now who did it first well this is a talk on Kure so I'll tell you what Wonka ray did Wonka ray did it first and more or less exactly in this way so here's what Wonka rated he has a chapter in his book which is called interpolation and what's he doing well he says suppose you have have a function f of X on say 0 1 over N or or say and and I don't know what don't know it but we observe it at some points observe Y is equal to f of X I 1 less than equal to I less than equal to n and he wants to know how to what's the best guess at the function at another point so the best guess and f of some X tor some other point and he says I'm going to do this problem by the method of causes the method of causes was 1900s language for Bayes theorem and he puts a prior distribution on functions by assuming that f of X had a power series expansion I equals 0 to infinity of some coefficients or prolem AI times X to the I where according to Poincare a a I was assumed to be Gaussian independent from i to i normal with mean zero say and variance Sigma I squared variances were would fall to zero and then he says well if if if that's true then this unknown function is itself a Gaussian variable and I'm we're being told in linear functions of Gaussian of a Gaussian process and we're being asked to calculate it you know it's mean at another point nowadays that's a standard problem when Whangarei was working there were no reproducing kernel Hilbert spaces he just does it anyway and he figures out what the Bayes rule is and I'll tell you what what he what his estimate is pointer a he made assumptions on Sigma I squared falling off properly so that this exists for all punker a looks at let me call it V of X which is the sum of Sigma I times X to the I equals 0 to infinity this this is a an ordinary function these are just the the variances and says that F hat X the best guess of f at at a given point is is given by the the mean of a posterior doing the same calculations I was doing but he figures out what that is and it's the sum from I equals 1 to N of epsilon I fee of X star times X I this is feet these are the excise that you observe that an epsilon I or parameters that are chosen so to enforce to enforce this this this condition so epsilon I you have to solve a linear system and he works it works out what this is in the case in which this fee is a polynomial in which case he's getting polynomial interpolation and so that's quite a quite an original thing to have done in in 1890s and it's an original thing now and if you haven't seen it before it may seem strange but it's all the rage in certain parts of numerical average case analysis of this type is being used in all kinds of high-powered computations and I put one reference on the references which is a survey article by Andy Stewart and if you want to see you know serious numerical analysts using this kind of Bayesian approach in order to sell very complicated problems and these article is a wonderful place to start looking but many many serious numerical analysts are doing computations of this way in this way so that's my second example of of panca ray doing interesting things and if you read that chapter there are other parts of that chapter and as far as I know nobody's ever read that chapter so no he didn't he doesn't do any examples he does work out exactly what this is in the case of of a certain choice of the Sigma i's vanishing but that's as far as he goes right right and I wrote a that I wrote a survey article a long time ago called Bayesian numerical analysis which traces other people who have done who have done this kind of thing and you can do it for any task in numerical analysis you can try this as a way to go okay so that's a second that's the second example and so the the last example I want to talk about is point Gray's work on vataj descartes and shuffling cards and so shuffling and that was an an example at Wong Kar a used in several articles for the public where he was trying to explain the onset of randomness in it and how things that start out not random could wind up being random and and I want to take you through a little bit of what he did and didn't do so so let me let me start where he started in anything he wrote until the last edition the second edition of his book he said well it's too complicated to do even with three cards let's do it with two cards okay well bear with bear with me for a second so let me work with the deck which has N equals two and so there's two arrangements there cards can be in order one two or they can be in the reverse order two one and what we're going to do is to repeatedly mix them and the mixing is we're either gonna do nothing with probability P one just this is the identity permutation or we're gonna switch the two cards with probability P two and of course P 1 plus P 2 is equal to 1 and punker a says let's consider the following bet we're gonna shuffle K times and if if the cards are in their original order 1 2 you get a dollar franc and if they're in the reverse order you have to pay me a dollar okay or Cedric and and so let's think about that so shuffle K times if after Kay shuffles it's in one two you get you get say one unit if it's to one you give one unit okay so let's do a few calculations little baby calculations suppose you shuffle once if K is one what can happen well if you don't shuffle and the cards are in the original order you get the dollar if you happen to switch them then the cards are out of order and you lose the unit and so the average value is p1 minus p2 right if if you happen to leave them alone you got a unit if you switch them once you lose a unit let's take K equals 2 now you shuffle twice we have 2 what's the average the average pay off well if if so you've done two shuffles you might do nothing twice the average is p1 squared or you might switch them and then switch them again in that case you also get a dollar p2 squared but you might not do anything and switch them or switch them and not do anything in both of those cases you lose a unit so minus 2 P 1 P 2 that's the average pay off to you and of course this is p1 minus p2 squared and for general K the average is easy to show is P 1 minus P 2 to the K power and and Whangarei says you can see that that goes to 0 so that the game is fair if you shuffle a lot if K is large and so that's that's font gray trying to get people to understand that if you shuffle a lot you mix it up you mix up the cards okay Oh beautiful math has to go but it will one Caray must have felt a little guilty for having written four or five times the case of general K is too hard and so in the second edition of his book which was completed in the last year of his life he adds a section and it's quite a long section maybe fifteen or twenty pages and that of that length in which he analyzes the problem of shuffling cards for general deck sizes and I want to at least tell you a little bit about about where he got to and and how he did it the kind of math that that he used in order to solve this problem so he treats the problem so for general deck sizes he says well there are the different ways of arranging a deck of cards and let's label them Sigma 1 Sigma 2 the different orders Sigma and factorial all the different possible arrangements and our shuffling scheme will be starting at the cards all in order starting at the identity I'll have a certain probability distribution over the allowable permutations so we worked with repeated shuffles I'll call it P I is the probability of Sigma I some some way of shuffling and and he says in order to to work with this problem I need to use some algebra and this is no 1912 and but still so he doesn't quite have the language but or care of the language he works in what we now would call the group algebra which is he calls it the generalized complex numbers but which is the set of all linear combinations of permutations Michael's one took in factorial so my probability assignment I can associate to an element of you know and it might only be it might be that P I is is one for transposition and it's a half for a transposition and a half for an end cycle and it's zero for all the other cases so P I is some probability distribution on on on these guys and he understands shows that if you square this element so if probability of Sigma after two shuffles after two shuffles well of course it's equal to the probability of Sigma I times the probability of sigma x sigma i inverse over all i the chance of winding up at sigma after two shuffles you had to have done something your first shuffle and then chosen the permutation that gets you to Sigma after your second shuffle is the coefficient of Sigma in P squared so if you take this element and the group algebra and square it and look at the coefficient of Sigma that's the chance of Sigma after two shuffles and so on and he says what we want to prove and he says theorem under assumptions that I'll make specific in a second P to the K power if you shuffle K times this tends to 1 over n factorial times the sum over all Sigma so and that says that if you shuffle the cards a lot they get all mixed up I mean all Arrangements become equally likely that's what he wants to prove now that theorem the way I've just said it isn't isn't true because you're shuffling could be you just switch the top two cards and that's it well you're not going to mix them up see there need to be some assumptions the method that Wonka ray uses to prove this theorem is methods that Frobenius and carton had derived about algebras and so pangkor a uses the following the idea of the characteristic vector or eigenvalue idea that so p is an element of the group algebra x is another element of the group algebra and carton had shown that we nowadays we understand that there are eigen vectors in algebra x' and this is an eigenvalue and this is the eigenvector so he's gonna use these ideas and he proves proves that that this convergence holds if and only if the support of P I is not in a co set of a sub group so you need the way that you're shuffling to be able to get to all possible arrangements and you need to avoid parity problems and he's aware of that and and in these cases he shows that any eigenvalue is less than or equal to one and that the vector this element of group algebra is an eigenvector with eigenvalue 1 he knows that and he proves that all of the under this assumption that all of the other eigenvalues are strictly less than 1 and he therefore concludes that this converges to two that he's aware and has to deal with the fact that this operator this element P might not be diagonalizable so you need something like you know the Jordan form of an operator and and he worries about that and does Indians with it and does manage to prove this theorem and it's quite an elaborate argument with a lot of details in it and obviously somebody was bugging him about it maybe he was bugging himself about it but for Pangkor a there's an awful lot of detail there some things he doesn't do doesn't one is a mentioned Markov that's not so surprising but Markov 1906 at any rate had developed the theory of Markov chains and if you want an interesting answer to a good question ask me went during questions why did Markov develop Markov chains it's an interesting story Markov developed Markov chains for some interesting reason and he had two examples one of which was rhyming patterns in Eugene Onegin and he went blind gathering the data actually and the second was shuffling cards and so Markov has an elaborate description of shuffling cards and a much more general setup poincare's argument really uses the fact that he's in the group algebra and it wouldn't work for poor for more general Markov chains the second thing he doesn't do is anything quantitative anything quantitative and let me just tell you two quantitative results so you can see what I mean there's the usual way that people shuffle cards you have a deck of cards you cut them about in half and you go like that you riffle shuffle them together and they fire and I showed that three-halves log to the base 2 of n plus c get you e to the minus c close so if K and so this is says you know about seven shuffles required to mix up fifty-two cards if you riffle shuffle on the other hand if you do the other method of shuffling you know this method of shuffling you know what I mean by that you have a deck of cards in clump clump clump clump clump clump it's N squared log n versus N squared log in for overhand shuffle and so when n is 52 n squared is 2,500 log 52 is about 5'4 and a half so it's 10,000 so it takes 10,000 of these verses seven of the others right so there's an interesting world of of quantitative theory point Caray was interested in the fact that for any method of shuffling as long as it wasn't stupid you would converge at infinity and that's what that's what he was set out to doing what he did and see Tempus Fugit so i won't i won't i won't say more about shuffling i promise you i could the last thing to say is something for all of us but in particular for me the math problems that I'm working on right now or trying to understand what I call smashing cards that is that's this method of shuffling you know what I mean people put cards on the table and go like that right and how long does it take touch motion right I'm not going to give a talk about that but the way I'm doing it is using fluid dynamics treating the treating the cards as a fluid and and if you look at the last 30 pages of Poincare raised appendix the last chapter it's all about mixing in fluids and there's just an awful lot of math as far as I know nobody's ever looked at that I'm looking at it pretty hard he doesn't get very far but he you know he's punker a and he he gets someplace what I tried to tell you in this talk is that first of all there's a lot of life in want Caray and and the second thing is that there's still gold in them there hills it's worth picking them up in any area and trying to read them thank you [Applause] okay any questions questions comments yes here microphone please you have two that we have to get a microphone it's coming so hello again Percy do use are you preliminary results show that smooshing is a little between the first ship Lingam it is the preliminary results show that it's an amazingly fast way of mixing just to say the first thing I did was get some data so I got some undergraduate and had her smush cards for a minute and then record the permutation and do that a hundred times so I had 100 permutations now if if you didn't smush enough why wouldn't they be random well you might think there were too many cards which were originally together that are still together for example we found we made 10 test statistics of that sort pushing for a minute passed all tests pushing for half a minute passed all tests 15 seconds started to detect it's a pretty efficient I was surprised it's a pretty efficient way of doing it it's a standard way of shuffling in minor Carlow it's not just me who's interested you might be interested too it's the standard way they shuffle in poker tournaments so it's not you know okay so it seems to be more effective than then I think and we have some preliminary results but there's a-there miles to go so I first have a comment and then a question my comment is that all to the credit of quantity he had an added disadvantage that in France we have this beautiful card game called billet where we don't actually shuffle so he actually did that without actually working with shuffling so much and I have a question why did Mark off and then mark where did you find this question I will answer that because it's a nice story so around the turn of the century but actually it lasted quite a long time in the Soviet Union a kind of mathematical like saenko ism took over the mathematical community it was it had a religious overtones it was one of the main themes of it was freewill and the Russian analysts and the cities to which this was going on were sort of instructed to not work on smooth functions because they didn't have enough free will but that's somehow where the Soviet school of more general functions arose you may think this is crazy nobody can tell us what to work on if they break your windows you know beat you up you know fire you from your job and it all happened all of a sudden you know this is a problem so there was some crazy people who were espousing this freewill is an interesting book about it there was a czar of probability and he was called nekross off like Microsoft but and he declared that the only things that were fit to work on for probably a sums of independent random variables because only they had enough free will in order to so that the basic laws of probability would hold Markov couldn't stand this story and he invented Markov chains as a political statement Markov chains are dependent and he proved the law of large numbers in the central limit theorem for Markov chains and the last line of his paper is thus free will is not necessary to do probability so they were invented as a political example not because what's it some wonderful story and there's a there's beautiful article about it by Eugene Sunita and question there about them wait for the microphone please I may be wrong but I thought that Frank Harris book was a text book actually and it was part of lectures I was wondering why he was asked to give a lecture about probability and if there is anything known about the reaction to his course and who the students were and if they did something with it right so I meant to say this at the beginning and those of you who have my handout had I done it I should have I would have written it down but there's a very very nice survey article about Poincare a and randomness by M a Zhi Li a K and it was given just recently homas yeah thank you good and and he traces very carefully the the answer to that question I'll give it briefly prawn curry didn't have a fancy job a chair of probability in physics came open and he that was the job he wanted it was at the Sorbonne and his first couple of years he didn't teach any probability whatsoever during that time he was teaching thermodynamics and he got involved in controversies with the English school if Frank Frank or I'll say wasn't a believer in it's complicated who knows what he believed in but Frank Ray wasn't a believer in statistical mechanics which many people didn't believe in before the turn of the century after all Boltzmann killed himself and he in the middle of that controversy decided maybe I better figure out about probability and he started to give these lectures they were first you know written down in longhand and then they were used as used as a textbook and so I think the main the main people who read it were students but it did become the standard textbook in probability for about 15 20 years following book his advisor Burt Ron and I don't you know I don't quite know it's not so easy to read punk Rey although this is pretty friendly as as as those things go I was told by somebody in this room that plonker a was a criminally bad lecturer and so maybe they read the books very carefully but I think I find interesting I'd like an answer from you after this is over the probability that punk or a put in his book is the kind of thing I was talking about here their concrete real-world problems what we would nowadays call applied probability given that he was the major force in mathematics the major probabilists of his time how did French probability wonderful healthy French probability not knocking it at all how did it go in such an opposite direction of what punk or a set out to do I really don't understand that it's just completely different subject and it's an interesting question to me how that happened good question well it's not a question it's almost one word and his grave whose yes that he it figures very prominently it was the end of the process of acquitting Dreyfus so when Caray was involved in the Dreyfus scandal and did did write a position paper making a terrible fund deserving fun of Burt Burt not Bertino the the fingerprint guy not the fingerprint kinda anyway the and if you look at the document that there it that there's left of that it's a very sketchy document it's three or four pages and I wasn't so impressed by panca Ray's work on that case now he might have done a lot of work in private or something like that and he did very forcefully defend Dreyfus but but I anyway it was real applied probability and he certainly did did did work at that so I I no but I read the document conquer ace it's on the web if you just type in point grain Dreyfus you'll find it and it was pretty sketchy and it just it was a lot of name-calling and rhetoric there are a few calculations but didn't seem like like he he decided I think to try to argue from on high what like to me there is from what I understood there is a longer and published document that is still at the family point carry about this problem in which he computed probabilities did study a lot the written the handwriting and so on so that would be something that one would have to look at he does have some numbers so it wasn't just you know a quick hatchet job but but it it didn't yeah it didn't I I wasn't I wasn't I was expecting a more thorough investigation given what a big deal it was I mean was a big deal other question okay I have one PC could you I didn't follow when you were explaining about difference between the two kinds of suffering over end and the other what what are the two different ones okay so the riffle shuffle I meant to bring cards I didn't the riffle shuffle is you have a deck of cards you cut it about in half according to the binomial distribution you start dropping the cards with your thumbs according to a rule for example if you have a cards here and B cards here the chance that you drop your next card from here is a over a plus B so you drop cards probability proportional to packet size and they riffle together okay that's one shuffle and then you can ask how many repetitions of that shuffle does it take to get you you know within epsilon close to random and some metric on probabilities and I could say it as carefully as you want but in order to get e to the minus C close you have to do this many shuffles the second way of shuffling I won't do it but you know like I can't but you know if you went like this you kind of took random packets right you know they do that in casinos they they do strip shuffling but we with cards here's a facedown deck of cards here's that you clump clump clump clump clump clump clump and so a mathematical model for that is in between each pair of cards you put a coin with probability theta say 1/2 and then you drop off all until the first head and all until the second head and so you drop little packets and in that model it's a theorem of robbing P mammals that it takes N squared log n repetitions and that so another question in the first part of the talk you talked about guessing the integral for a random function like an approximation yes you threatened me and you didn't talk about rates of convergence about how typically what is the rest of approximation I didn't but but people do in fact [Music] house-door and and and Bono wrote about that kind of thing and you can find in appendix by house-door Flynn bollocks book about putting measures on l1 and rates of convergence there has been some work but by and large this was something that punk RA did that I noticed in the 1980s was an interesting subject fewer students developed it but then it went away again and now it's come back with a vengeance in in numerical analysis so I think there's an awful lot to do there and that so there's a lot of nice things to do and they haven't been done I think so it's a good question but something to do for all of us ok let's thank you again and then I'll make a few announcements so thank you again so first Percy as all the lecturer gets Poincare a middle this is your middle Percy for those of the lecturers who don't didn't get it just just come to see me I will I would give you on one side there is punk array on the other some representation of some of the Poincare a great actions great deeds there we have 20 area middles we have Galois middles that we can provide for those who want them of course if you didn't lecture you need to pay a bit for the middle that's how life goes now about Poincare a about Galois so last year we celebrated Galois this was the 200th anniversary of his of his birth now this year is the 100th anniversary of Poincare II we are trying each year to organize some commemoration of celebration that has some wide interest like this ones it's great that we can see here there's a lot of people we would like to have more young people in this kind of of organization and so on please advertise for young people to participate in these kind of meetings that are not so specialized I think we think a density Poincare a and many of us I think think too that it is very important to open your mind to be exposed not only to the talks in your particular specialty now all the talks in this conference will be available we have been recorded and will be available some at some point not far from now on our web page next year probably will there are two natural things to celebrate one thing is 2013 is the mathematics for planet Earth a year in which all around the world mathematics Institute will be hosting and discussing about things relating mathematics to the earth either environment or human populations or astronomy many things like this will have two of our trimesters will be devoted to this kind of things about ecosystems and there will be one about astronomy and 2013 will also be the 200th anniversary of the death of Lacan so of course there will be something about lag haunch as you know the gorgeous cream both by French and by Italian people and if you go to to ha you can see the house where I can't was born you can see the statue of raccoons and so on so we'll definitely celebrate like harsh and the last thing I will advertise for is that tomorrow is the end of this week so we started the last Saturday with the broad audience celebration at Sorbonne for young people for generations and so on necked was the conference for mathematicians that was concluded by the beautiful talk of Percy at tomorrow there will be the Poincare seminar with ODG daddy girl Allen Sean sinner today this was the to cut problem tomorrow it would be the three-body and so on there will be low Hamas Jack there will be from so a bigger it will be concluded by a movie by Philip films made for all kinds of of audience and this is also the opportunity to advertise for the Poincare seminar which has been doing a great job for years now to produce thematic days that are also intended for prodigals for young people this is great way to get hold on some subject I encourage you to participate in the one of tomorrow if you are available that's all thank you very much for coming and we are always happy to host you in SD to prank arranging thank you you

Channel: Institut Henri Poincaré

Views: 6,270

Rating: 5 out of 5

Keywords: Institut Henri Poincaré, Cédric Villani, Quentin Lazzarotto, Mathématiques

Id: 3tMor37XpE0

Length: 70min 15sec (4215 seconds)

Published: Tue May 28 2013