## simulating correlated Binomials [another Bernoulli factory]

Posted in Books, Kids, pictures, R, Running, Statistics, University life with tags , , , , , , , on April 21, 2015 by xi'an

This early morning, just before going out for my daily run around The Parc, I checked X validated for new questions and came upon that one. Namely, how to simulate X a Bin(8,2/3) variate and Y a Bin(18,2/3) such that corr(X,Y)=0.5. (No reason or motivation provided for this constraint.) And I thought the following (presumably well-known) resolution, namely to break the two binomials as sums of 8 and 18 Bernoulli variates, respectively, and to use some of those Bernoulli variates as being common to both sums. For this specific set of values (8,18,0.5), since 8×18=12², the solution is 0.5×12=6 common variates. (The probability of success does not matter.) While running, I first thought this was a very artificial problem because of this occurrence of 8×18 being a perfect square, 12², and cor(X,Y)x12 an integer. A wee bit later I realised that all positive values of cor(X,Y) could be achieved by randomisation, i.e., by deciding the identity of a Bernoulli variate in X with a Bernoulli variate in Y with a certain probability ϖ. For negative correlations, one can use the (U,1-U) trick, namely to write both Bernoulli variates as

$X_1=\mathbb{I}(U\le p)\quad Y_1=\mathbb{I}(U\ge 1-p)$

in order to minimise the probability they coincide.

I also checked this result with an R simulation

> z=rbinom(10^8,6,.66)
> y=z+rbinom(10^8,12,.66)
> x=z+rbinom(10^8,2,.66)
cor(x,y)
> cor(x,y)
[1] 0.5000539


Searching on Google gave me immediately a link to Stack Overflow with an earlier solution with the same idea. And a smarter R code.

## 2013 WSC, Hong Kong

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , on August 28, 2013 by xi'an

After an early but taxing morning run overlooking the city, and a recovery breakfast (!), I went from my flat to the nearby Hong Kong Convention Centre where the ISI (2013 WSC) meeting is taking place. I had a few chats with friends and publishers (!), then read a chapter of Rissanen’s book over an iced coffee before attending the Bernoulli session. This was a fairly unusual session with a mix of history of probability, philosophy of probability and statistics, and computational issues (my talk). Edith Sylla gave some arguments as to why Ars Conjectandi (that she translated) was the first probability book ever. Krzys Burdzy defended his perspective on why von Mises and de Finetti were wrong (in their foundational views of statistics). And I gave my talk on a mixture of Bernoulli factory, Russian roulette and ABC  (After my talk, Victor Perez Abreu told me that Jakob Bernoulli had presumably used simulation to evaluate the variance of the empirical mean in the Bernoulli case.) What I found most interesting in the historical talk was that Bernoulli had proven his result in the late 1680’s but he waited to complete his book on moral and commercial issues, waited too long since he died before. This reminded me of Hume using probabilistic arguments a few years later to disprove the existence of miracles. And of Price waiting for Bayes’ theorem to counter Hume. The talk by Krzys was a quick summary of the views exposed in his book, which unsurprisingly did not convince me that von Mises and de Finetti (a) had failed and (b) needed to use a new set of (six) axioms to define probability. I often reflected on the fact that when von Mises and de Finetti state(d) that probability does not exist, they applied the argument to a single event and this does not lead to a paradox in my opinion. Anyway, this talk of Krzys’ induced most of the comments from the floor, my own talk being in fine too technical to fit in this historical session. (And then there was still some time to get to a tea shop in Sheng Wan to buy some Pu Ehr, if not the HK\$3000 variety…!)

## from Jakob Bernoulli to Hong Kong

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , , , , on August 24, 2013 by xi'an

Here are my slides (or at least the current version thereof) for my talk in Hong Kong at the 2013 (59th ISI) World Statistical Congress(I stopped embedding my slideshare links in the posts as they freeze my broswer. I wonder if anyone else experiences the same behaviour.)

This talk will feature in the History I: Jacob Bernoulli’s “Ars Conjectandi” and the emergence of probability invited paper session organised by Adam Jakubowski. While my own research connection with Bernoulli is at most tenuous, besides using the Law of Large Numbers and Bernoulli rv’s…,  I [of course!] borrowed from earlier slides on our vanilla Rao-Blackwellisation paper (if only  because of the Bernoulli factory connection!) and ask Mark Girolami for his Warwick slides on the Russian roulette (another Bernoulli factory connection!), before recycling my Budapest slides on ABC. The other talks in the session are by Edith Dudley Sylla on Ars Conjectandi and by Krzys Burdzy on his book The Search for Certainty. Book that I critically reviewed in Bayesian Analysis. This will be the first time I meet Krzys in person and I am looking forward to the opportunity!

## Abraham De Moivre

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , on March 7, 2012 by xi'an

During my week in Roma, I read David Bellhouse’s book on Abraham De Moivre (at night and in the local transportations and even in Via del Corso waiting for my daughter!)… This is a very scholarly piece of work, with many references to original documents, and it may not completely appeal to the general audience: The Baroque Cycle by Neal Stephenson is covering the same period and the rise of the “scientific man” (or Natural Philosopher) in a much more novelised manner, while centering on Newton as its main character and on the earlier Newton-Leibniz dispute, rather than the later Newton-(De Moivre)-Bernoulli dispute. (De Moivre does not appear in the books, at least under his name.)

Bellhouse’s book should however fascinate most academics in that, beside going with the uttermost detail into De Moivre’s contributions to probability, it uncovers the way (mathematical) research was done in the 17th and 18th century England: De Moivre never got an academic position (although he applied for several ones, incl. in Cambridge), in part because he was an emigrated French huguenot (after the revocation of the Édit de Nantes by Louis XIV), and he got a living by tutoring gentry and aristocracy sons in mathematics and accounting. He also was a consultant on annuities. His interactions with other mathematicians of the time was done through coffee-houses, the newly founded Royal Society, and letters. De Moivre published most of his work in the Philosophical Transactions and in self-edited books that he financed by subscriptions. (As a Frenchman, I personally[and so did Jacob Bernoulli!] found puzzling the fact that De Moivre never wrote anything in french but assimilated very quickly into English society.)

Another fascinating aspect of the book is the way English (incl. De Moivre) and Continental mathematicians fought and bickered on the priority of discoveries. Because their papers were rarely and slowly published, and even more slowly distributed throughout Western Europe, they had to rely on private letters for those priority claims. De Moivre’s main achievement is his book, The Doctrine of Chances, which contains among clever binomial derivations on various chance games an occurrence of the central limit theorem for binomial experiments. And the use of generating functions. De Moivre had a suprisingly long life since he died at 87 in London, still giving private lessons as old as 72. Besides being seen as a leading English mathematician, he eventually got recognised by the French Académie Royale des Sciences, if as a foreign member, a few months prior to his death (as well as by the Berlin Academy of Sciences). There is also a small section in the book on the connections between De Moivre and Thomas Bayes (pp. 200-203), although very little is known of their personal interactions. Bayes was close to one of De Moivre’s former students, Phillip Stanhope, and he worked on several of De Moivre’s papers to get entry to the Royal Society. Some open question is whether or not Bayes was ever tutored by De Moivre, although there is no material proof he did. The book also mentions Bayes’ theorem in connection with an comment on The Doctrine of Chances by Hartley (p.191), as if De Moivre had an hand in it or at least a knowledge of it, but this seems unlikely…

In conclusion, this is a highly pleasant and easily readable book on the career of a major mathematician and of one of the founding fathers of probability theory. David Bellhouse is to be congratulated on the scholarship exhibited by this book and on the painstaking pursuit of all historical documents related with De Moivre’s life.