Chateau Puech-Haut

Posted in pictures, Wines with tags , , , , , , , on September 20, 2016 by xi'an

Bayesian model selection without evidence

Posted in Books, Statistics, University life with tags , , , , , , , on September 20, 2016 by xi'an

“The new method circumvents the challenges associated with accurate evidence calculations by computing posterior odds ratios using Bayesian parameter estimation”

One paper leading to another, I had a look at Hee et al. 2015 paper on Bayes factor estimation. The “novelty” stands in introducing the model index as an extra parameter in a single model encompassing all models under comparison, the “new” parameterisation being in (θ,n) rather than in θ. With the distinction that the parameter θ is now made of the union of all parameters across all models. Which reminds us very much of Carlin and Chib (1995) approach to the problem. (Peter Green in his Biometrika (1995) paper on reversible jump MCMC uses instead a direct sum of parameter spaces.) The authors indeed suggest simulating jointly (θ,n) in an MCMC or nested sampling scheme. Rather than being updated by arbitrary transforms as in Carlin and Chib (1995) the useless parameters from the other models are kept constant… The goal being to estimate P(n|D) the marginal posterior on the model index, aka the posterior probability of model n.

Now, I am quite not certain keeping the other parameter constants is a valid move: given a uniform prior on n and an equally uniform proposal, the acceptance probability simplifies into the regular Metropolis-Hastings ratio for model n. Hence the move is valid within model n. If not, I presume the previous pair (θ⁰,n⁰) is repeated. Wait!, actually, this is slightly more elaborate: if a new value of n, m, is proposed, then the acceptance ratio involves the posteriors for both n⁰ and m, possibly only the likelihoods when the proposal is the prior. So the move will directly depend on the likelihood ratio in this simplified case, which indicates the scheme could be correct after all. Except that this neglects the measure theoretic subtleties that led to reversible jump symmetry and hence makes me wonder. In other words, it follows exactly the same pattern as reversible jump without the constraints of the latter… Free lunch,  anyone?!

Le bayésianisme aujourd’hui

Posted in Books, Statistics with tags , , , , , , , on September 19, 2016 by xi'an

A few years ago, I was asked by Isabelle Drouet to contribute a chapter to a multi-disciplinary book on the Bayesian paradigm, book that is now soon to appear. In French. It has this rather ugly title of Bayesianism today. Not that I had hear of Bayesianism or bayésianime previously. There are chapters on the Bayesian notion(s) of probability, game theory, statistics, on applications, and on the (potentially) Bayesian structure of human intelligence. Most of it is thus outside statistics, but I will certainly read through it when I receive my copy.

snapshots from Nature

Posted in Books, Kids, pictures, University life with tags , , , , , , , , , , on September 19, 2016 by xi'an

Among many interesting things I read from the pile of Nature issues that had accumulated over a month of travelling, with a warning these are mostly “old” news by now!:

  • the very special and untouched case of Cuba in terms of the Zika epidemics, thanks to a long term policy fighting mosquitoes at all levels of the society;
  • an impressive map of the human cortex, which statistical analysis would be fascinating;
  • an excerpt from Nature 13 August 1966 where the Poisson distribution was said to describe the distribution of scores during the 1966 World Cup;
  • an analysis of a genetic experiment on evolution involving 50,000 generations (!) of Escherichia coli;
  • a look back at the great novel Flowers for Algernon, novel I read eons ago;
  • a Nature paper on the first soft robot, or octobot, along with some easier introduction, which did not tell which kind of operations could be accomplished by such a robot;
  • a vignette on a Science paper about the interaction between honey hunters and hunting birds, which I also heard depicted on the French National Radio, with an experiment comparing the actual hunting (human) song, a basic sentence in the local language, and the imitation of the song of another bird. I could not understand why the experiment did not include hunting songs from other hunting groups, as they are highly different but just as effective. It would have helped in understanding how innate the reaction of the bird is;
  • another literary entry at the science behind Mary Shelley’s Frankenstein;
  • a study of the Mathematical Genealogy Project in terms of the few mathematicians who started most genealogies of mathematicians, including d’Alembert, advisor to Laplace of whom I am one of the many descendants, although the finding is not that astounding when considering usual genealogies where most branches die off and the highly hierarchical structure of power in universities of old.

Skeleton Tree

Posted in Statistics with tags , , on September 18, 2016 by xi'an

A new album by Nick Cave and the Bad Seeds. Dark and daunting. And beautiful!

The Magicians [book review]

Posted in Books, Kids, Travel, Wines with tags , , , , , , , , , on September 17, 2016 by xi'an

While in Melbourne, I heard a recommendation for Lev Grossman’s The Magicians and the next day, while checking the Melbourne Writers Festival bookstore, found the book (rather than the Kristoff volume I was seeking), bought it, and read it within a few days.

‘Brakebills will remind readers of Hogwarts, though with more illicit fondling. Grossman has written what could crudely be labeled a Harry Potter for adults.” , NYT

So is this an Harry Potter for adults?! First, I think Harry Potter can be read by adults (if I qualify as adult!). This remark presumably means the book should not be read by young readers, maybe, due to recurrent sex and alcohol consumption, plus some drugs and an overall depressive tone.

Back to Harry Potter, there is the same magical boarding school feeling, even though it is located in upstate New York on the Hudson river.  And not in Scotland. With an equivalent to Quidditch, an evil magician, exams, surly teens, one or two love triangles, &tc. If in a more modern and American way. The difference with Harry Potter is that it also doubles as Narnia! A Narnia eventually turned wrong and sour, but nonetheless a strong similarity of stories and ideas. Of course, this parallel could be seen as an attempt at deconstruction, exhibiting the inconsistencies in the original novels, but it is so subtle it does not feel like it. There are the same encounters with sentient animal creatures, who never reappear after, the same call for Kings and Queens, as in Narnia. This lack of depth at exploring the connections between Harry Potter, Narnia and even some aspects of the Wheel of Time is frustrating in that something great could have come of it. And then… then… comes the worst literary trick in my list, the call to a subterranean quest with endless monsters and accidents! (I obviously exclude Tolkien’ Moria episode from this list!!!) Concluding with the evil character dumping information in the last battle to explain missing bits and pieces in the story.

So, in conclusion, not such a magical book, even though I read it within a few days thanks to my 39 hour trip back to Paris. The Magicians remains too teeny for my taste, hearing self-deprecating depressive monologues occurs way too often to make the main character congenial, and the story has not enough depth or structure to be compelling. A reviewer rightly pointed out it feels like fandom fiction. Rather than a universe on its own. (As for instance Aaronovitch’ Rivers of London series.)

random walk on a torus [riddle]

Posted in Books, Kids, pictures with tags , , , , , , , , , on September 16, 2016 by xi'an

Galgate, Lancastershire, July 19, 2011The Riddler of this week(-end) has a simple riddle to propose, namely given a random walk on the {1,2,…,N} torus with a ⅓ probability of death, what is the probability of death occurring at the starting point?

The question is close to William Feller’s famous Chapter III on random walks. With his equally famous reflection principle. Conditioning on the time n of death, which as we all know is definitely absorbing (!), the event of interest is a passage at zero, or any multiple of N (omitting the torus cancellation), at time n-1 (since death occurs the next time). For a passage in zero, this does not happen if n is even (since n-1 is odd) and else it is a Binomial event with probability

{n \choose \frac{n-1}{2}} 2^{-n}

For a passage in kN, with k different from zero, kN+n must be odd and the probability is then

{n \choose \frac{n-1+kN}{2}} 2^{-n}

which leads to a global probability of

\sum_{n=0}^\infty \dfrac{2^n}{3^{n+1}} \sum_{k=-\lfloor (n-1)/N \rfloor}^{\lfloor (n+1)/N \rfloor} {n \choose \frac{n-1+kN}{2}} 2^{-n}

i.e.

\sum_{n=0}^\infty \dfrac{1}{3^{n+1}} \sum_{k=-\lfloor (n-1)/N \rfloor}^{\lfloor (n+1)/N \rfloor} {n \choose \frac{n-1+kN}{2}}

Since this formula is rather unwieldy I looked for another approach in a métro ride [to downtown Paris to enjoy a drink with Stephen Stiegler]. An easier one is to allocate to each point on the torus a probability p[i] to die at position 1 and to solve the system of equations that is associated with it. For instance, when N=3, the system of equations is reduced to

p_0=1/3+2/3 p_1, \quad p_1=1/3 p_0 + 1/3 p_1

which leads to a probability of ½ to die at position 0 when leaving from 0. When letting N grows to infinity, the torus structure no longer matters and the probability of dying at position 0 implies returning in position 0, which is a special case of the above combinatoric formula, namely

\sum_{m=0}^\infty \dfrac{1}{3^{2m+1}}  {2m \choose m}

which happens to be equal to

\dfrac{1}{3}\,\dfrac{1}{\sqrt{1-4/9}}=\dfrac{1}{\sqrt{5}}\approx 0.4472

as can be [unnecessarily] checked by a direct R simulation. This √5 is actually the most surprising part of the exercise!