## Bayesian News Feeds

### Approximate Bayesian Computation in state space models

**W**hile it took quite a while (!), with several visits by three of us to our respective antipodes, incl. my exciting trip to Melbourne and Monash University two years ago, our paper on ABC for state space models was arXived yesterday! Thanks to my coauthors, Gael Martin, Brendan McCabe, and Worapree Maneesoonthorn, I am very glad of this outcome and of the new perspective on ABC it produces. For one thing, it concentrates on the selection of summary statistics from a more econometrics than usual point of view, defining asymptotic sufficiency in this context and demonstrated that both asymptotic sufficiency and Bayes consistency can be achieved when using maximum likelihood estimators of the parameters of an auxiliary model as summary statistics. In addition, the proximity to (asymptotic) sufficiency yielded by the MLE is replicated by the score vector. Using the score instead of the MLE as a summary statistics allows for huge gains in terms of speed. The method is then applied to a continuous time state space model, using as auxiliary model an augmented unscented Kalman filter. We also found in the various state space models tested therein that the ABC approach based on the marginal [likelihood] score was performing quite well, including wrt Fearnhead’s and Prangle’s (2012) approach… I like the idea of using such a generic object as the unscented Kalman filter for state space models, even when it is not a particularly accurate representation of the true model. Another appealing feature of the paper is in the connections made with indirect inference.

Filed under: Statistics, Travel, University life Tagged: ABC, indirect inference, Kalman filter, marginal likelihood, Melbourne, Monash University, score function, state space model

### ABC model choice via random forests [expanded]

**T**oday, we arXived a second version of our paper on ABC model choice with random forests. Or maybe [A]BC model choice with random forests. Since the random forest is built on a simulation from the prior predictive and no further approximation is used in the process. Except for the computation of the posterior [predictive] error rate. The update wrt the earlier version is that we ran massive simulations throughout the summer, on existing and new datasets. In particular, we have included a Human dataset extracted from the 1000 Genomes Project. Made of 51,250 SNP loci. While this dataset is not used to test new evolution scenarios, we compared six out-of-Africa scenarios, with a possible admixture for Americans of African ancestry. The scenario selected by a random forest procedure posits a single out-of-Africa colonization event with a secondary split into a European and an East Asian population lineages, and a recent genetic admixture between African and European lineages, for Americans of African origin. The procedure reported a high level of confidence since the estimated posterior error rate is equal to zero. The SNP loci were carefully selected using the following criteria: (i) all individuals have a genotype characterized by a quality score (GQ)>10, (ii) polymorphism is present in at least one of the individuals in order to fit the SNP simulation algorithm of Hudson (2002) used in DIYABC V2 (Cornuet et al., 2014), (iii) the minimum distance between two consecutive SNPs is 1 kb in order to minimize linkage disequilibrium between SNP, and (iv) SNP loci showing significant deviation from Hardy-Weinberg equilibrium at a 1% threshold in at least one of the four populations have been removed.

In terms of random forests, we optimised the size of the bootstrap subsamples for all of our datasets. While this optimisation requires extra computing time, it is negligible when compared with the enormous time taken by a logistic regression, which is [yet] the standard ABC model choice approach. Now the data has been gathered, it is only a matter of days before we can send the paper to a journal

Filed under: Statistics, University life Tagged: 1000 Genomes Project, ABC, ABC model choice, admixture, bootstrap, DIYABC, Human evolution, logistic regression, out-of -Africa scenario, posterior predictive, prior predictive, random forests

### The chocolate factory gone up in smoke

**T**here was a major fire near my house yesterday with many fire-engines rushing by and a wet smoke smell lingering by the whole night. As I found out during my early morning run, the nearby chocolate factory had completely burned. Actually, sixteen hours after the beginning of the fire, the building was still smouldering, with a dozen fire-engines yet on site and huge hoses running on adjacent streets. A fireman told me the fire had started from an electric spark and that the entire reserves had been destroyed. This is quite sad, as hitting a local business and a great chocolate maker, Patrick Roger. I do not know whether or not the company will survive this disaster, but if you happen to come by one of the shops in Paris or Brussels, drop in and buy some chocolates! For the taste of it and as a support.

Filed under: Kids, Running Tagged: chocolate, fire, Patrick Roger, Sceaux

### The Unimaginable Mathematics of Borges’ Library of Babel [book review]

**T**his is a book I carried away from JSM in Boston as the Oxford University Press representative kindly provided my with a copy at the end of the meeting. After I asked for it, as I was quite excited to see a book linking Jorge Luis Borges’ great Library of Babel short story with mathematical concepts. Even though many other short stories by Borges have a mathematical flavour and are bound to fascinate mathematicians, the Library of Babel is particularly prone to mathemati-sation as it deals with the notions of infinite, periodicity, permutation, randomness… As it happens, William Goldbloom Bloch [a patronym that would surely have inspired Borges!], professor of mathematics at Wheaton College, Mass., published the unimaginable mathematics of Borges’ Library of Babel in 2008, so this is not a recent publication. But I had managed to miss through the several conferences where I stopped at OUP exhibit booth. (Interestingly William Bloch has also published a mathematical paper on Neil Stephenson’s Cryptonomicon.)

**N**ow, what is unimaginable in the maths behind Borges’ great Library of Babel??? The obvious line of entry to the mathematical aspects of the book is combinatorics: how many different books are there in total? [Ans. 10¹⁸³⁴⁰⁹⁷...] how many hexagons are needed to shelf that many books? [Ans. 10⁶⁸¹⁵³¹...] how long would it take to visit all those hexagons? how many librarians are needed for a Library containing all volumes once and only once? how many different libraries are there [Ans. 1010⁶...] Then the book embarks upon some cohomology, Cavalieri’s infinitesimals (mentioned by Borges in a footnote), Zeno’s paradox, topology (with Klein’s bottle), graph theory (and the important question as to whether or not each hexagon has one or two stairs), information theory, Turing’s machine. The concluding chapters are comments about other mathematical analysis of Borges’ Grand Œuvre and a discussion on how much maths Borges knew.

**S**o a nice escapade through some mathematical landscapes with more or less connection with the original masterpiece. I am not convinced it brings any further dimension or insight about it, or even that one should try to dissect it that way, because it kills the poetry in the story, especially the play around the notion(s) of infinite. The fact that the short story is incomplete [and short on details] makes its beauty: if one starts wondering at the possibility of the Library or at the daily life of the librarians [like, what do they eat? why are they there? where are the readers? what happens when they die? &tc.] the intrusion of realism closes the enchantment! Nonetheless, the unimaginable mathematics of Borges’ Library of Babel provides a pleasant entry into some mathematical concepts and as such may initiate a layperson not too shy of maths formulas to the beauty of mathematics.

Filed under: Books, Statistics, Travel, University life Tagged: book review, Boston, cohomology, combinatorics, infinity, information theory, Jorge Luis Borges, JSM 2014, Library of Babel, Oxford University Press, Turing's machine

### future of computational statistics

I am currently preparing a survey paper on the present state of computational statistics, reflecting on the massive evolution of the field since my early Monte Carlo simulations on an Apple //e, which would take a few days to return a curve of approximate expected squared error losses… It seems to me that MCMC is attracting more attention nowadays than in the past decade, both because of methodological advances linked with better theoretical tools, as for instance in the handling of stochastic processes, and because of new forays in accelerated computing via parallel and cloud computing, The breadth and quality of talks at MCMski IV is testimony to this. A second trend that is not unrelated to the first one is the development of new and the rehabilitation of older techniques to handle complex models by approximations, witness ABC, Expectation-Propagation, variational Bayes, &tc. With a corollary being an healthy questioning of the models themselves. As illustrated for instance in Chris Holmes’ talk last week. While those simplifications are inevitable when faced with hardly imaginable levels of complexity, I still remain confident about the “inevitability” of turning statistics into an “optimize+penalize” tunnel vision… A third characteristic is the emergence of new languages and meta-languages intended to handle complexity both of problems and of solutions towards a wider audience of users. STAN obviously comes to mind. And JAGS. But it may be that another scale of language is now required…

If you have any suggestion of novel directions in computational statistics or instead of dead ends, I would be most interested in hearing them! So please do comment or send emails to my gmail address bayesianstatistics…

Filed under: Books, pictures, R, Statistics, University life Tagged: ABC, Apple II, approximation, BUGS, computational statistics, expectation-propagation, JAGS, MCMC, MCMSki IV, Monte Carlo, optimisation, STAN, statistical computing, sunset, variational Bayes methods

### redshirts

*“For the first nine years of its existence, aside from being appointed the flagship, there was nothing particularly special about it, from a statistical point of view.”*

**A** book I grabbed at the last minute in a bookstore, downtown Birmingham. Maybe I should have waited this extra minute… Or picked the other Scalzi’s on the shelf, * Lock In* that just came out! (I already ordered that one for my incomiing lecture in Gainesville. Along with the

**final volume of Patrick Rothfuss’ masterpiece, The Slow Regard of Silent Things, which will just be out by then! It is only a side story within the same universe, as pointed out by Dan…)**

*not**“What you’re trying to do is impose causality on random events, just like everyone else here has been doing.”*

**W**hat amazes most me is that Scalzi’s *redshirts* got the 2013 Hugo Award. I mean, The Hugo Award?! While I definitely liked the Old Man Wars saga, this novel is more like a light writing experiment and a byproduct of writing a TV series. Enjoyable at a higher conceptual level, but not as a story. Although this is somewhat of a spoiler (!), the title refers to the characters wearing red shirts in Star Trek, who have a statistically significant tendency to die on the next mission. [Not that I knew this when I bought the book! Maybe it would have warned me against the book.] And *redshirts* is about those characters reflecting about how unlikely their fate is (or rather the fate of the characters before them) and rebelling against the series writer. Ensues games with the paradoxes of space travel and doubles. Then games within games. The book is well-written and, once again, enjoyable at some level, with alternative writing styles used in different parts (or coda) of the novel. It still remains a purely intellectual perspective, with no psychological involvement towards those characters. I just cannot relate to the story. Maybe because of the pastiche aspect or of the mostly comic turn. *redshirts* certainly feels very different from those Philip K. Dick stories (e.g., Ubik) where virtual realities abounded without a definitive conclusion on which was which.

Filed under: Books, pictures, Travel Tagged: Birmingham, England, Hugo Awards, John Scalzi, Patrick Rothfuss, redshirts, Star Trek

### métro static

“Mon premier marathon je le fais en courant.” *[I will do my first marathon running.]*

Filed under: pictures, Running Tagged: Badwater Ultramarathon, Florida, Marathon FL, métro, métro static, Paris, sea, sunset

### all models are wrong

*“Using ABC to evaluate competing models has various hazards and comes with recommended precautions (Robert et al. 2011), and unsurprisingly, many if not most researchers have a healthy scepticism as these tools continue to mature.”*

**M**ichael Hickerson just published an open-access letter with the above title in Molecular Ecology. (As in several earlier papers, incl. the (in)famous ones by Templeton, Hickerson confuses running an ABC algorithm with conducting Bayesian model comparison, but this is not the main point of this post.)

*“Rather than using ABC with weighted model averaging to obtain the three corresponding posterior model probabilities while allowing for the handful of model parameters (θ, τ, γ, Μ) to be estimated under each model conditioned on each model’s posterior probability, these three models are sliced up into 143 ‘submodels’ according to various parameter ranges.”*

**T**he letter is in fact a supporting argument for the earlier paper of Pelletier and Carstens (2014, Molecular Ecology) which conducted the above splitting experiment. I could not read this paper so cannot judge of the relevance of splitting this way the parameter range. From what I understand it amounts to using mutually exclusive priors by using different supports.

*“Specifically, they demonstrate that as greater numbers of the 143 sub-models are **evaluated, the inference from their ABC model choice procedure becomes increasingly.”*

**A**n interestingly cut sentence. Increasingly unreliable? mediocre? weak?

*“…with greater numbers of models being compared, the most probable models are assigned diminishing levels of posterior probability. This is an expected result…”*

**T**rue, if the number of models under consideration increases, under a uniform prior over model indices, the posterior probability of a given model mechanically decreases. But the pairwise Bayes factors should not be impacted by the number of models under comparison and the letter by Hickerson states that Pelletier and Carstens found the opposite:

*“…pairwise Bayes factor[s] will always be more conservative except in cases when the posterior probabilities are equal for all models that are less probable than the most probable model.”*

**W**hich means that the “Bayes factor” in this study is computed as the ratio of a marginal likelihood and of a compound (or super-marginal) likelihood, averaged over all models and hence incorporating the prior probabilities of the model indices as well. I had never encountered such a proposal before. Contrary to the letter’s claim:

*“…using the Bayes factor, incorporating all models is perhaps more consistent with the Bayesian approach of incorporating all uncertainty associated with the ABC model choice procedure.”*

**B**esides the needless inclusion of ABC in this sentence, a somewhat confusing sentence, as Bayes factors are not, *stricto sensu*, Bayesian procedures since they remove the prior probabilities from the picture.

“Although the outcome of model comparison with ABC or other similar likelihood-based methods will always be dependent on the composition of the model set, and parameter estimates will only be as good as the models that are used, model-based inference provides a number of benefits.”

**A**ll models are wrong but the very fact that they are models allows for producing pseudo-data from those models and for checking if the pseudo-data is similar enough to the observed data. In components that matters the most for the experimenter. Hence a loss function of sorts…

Filed under: Statistics, University life Tagged: ABC, Bayes factor, Bayesian model choice, George Box, model posterior probabilities, Molecular Ecology, phylogenetic model, phylogeography

### an der schöne blau Donau (#2)

### two, three, five, …, a million standard deviations!

I first spotted Peter Coles’ great post title “Frequentism: the art of probably answering the wrong question” (a very sensible piece by the way!, and mentioning a physicist’s view on the Jeffreys-Lindley paradox I had intended to comment) and from there the following site jumping occured:

*“I confess that in my early in my career as a physicist I was rather cynical about sophisticated statistical tools, being of the opinion that “if any of this makes a difference, just get more data”. That is, if you do enough experiments, the confidence level will be so high that the exact statistical treatment you use to evaluate it is irrelevant.” John Butterworth, Sept. 15, 2014
*

**A**fter Val Johnson‘s suggestion to move the significant level from .05 down to .005, hence roughly from 2σ up to 3σ, John Butterworth, a physicist whose book Smashing Physics just came out, discusses in The Guardian the practice of using 5σ in Physics. It is actually induced by Louis Lyons’ arXival of a recent talk with the following points (discussed below):

- Should we insist on the 5 sigma criterion for discovery claims?
- The probability of A, given B, is not the same as the probability of B, given A.
- The meaning of p-values.
- What is Wilks Theorem and when does it not apply?
- How should we deal with the `Look Elsewhere Effect’?
- Dealing with systematics such as background parametrisation.
- Coverage: What is it and does my method have the correct coverage?
- The use of p0 versus p1 plots.

**B**utterworth’s conclusion is worth reproducing:

*“…there’s a need to be clear-eyed about the limitations and advantages of the statistical treatment, wonder what is the “elsewhere” you are looking at, and accept that your level of certainty may never feasibly be 5σ. In fact, if the claims being made aren’t extraordinary, a one-in-2million chance of a mistake may indeed be overkill, as well being unobtainable. And you have to factor in the consequences of acting, or failing to act, based on the best evidence available – evidence that should include a good statistical treatment of the data.” John Butterworth, Sept. 15, 2014*

esp. the part about the “consequences of acting”, which I interpret as incorporating a loss function in the picture.

**L**ouis’s paper-ised talk 1. [somewhat] argues in favour of the 5σ because 2σ and 3σ are not necessarily significant on larger datasets. I figure the same could be said of 5σ, no?! He also mentions (a) “systematics”, which I do not understand. Even though this is not the first time I encounter the notion in Physics. And (b) “subconscious Bayes factors”, which means that the likelihood ratio [considered here as a transform of the p-value] is moderated by the ratio of the prior probabilities, even when people do not follow a Bayesian procedure. But this does not explain why a fixed deviation from the mean should be adopted. 2. and 3. The following two points are about the common confusion in the use of the p-value, found in most statistics textbooks. Even though the defence of the p-value against the remark that it is wrong half the time (as in Val’s PNAS paper) misses the point. 4. *Wilk’s theorem* is a warning that the χ² approximation only operates under some assumptions. 5. *Looking elsewhere* is the translation of multiple testing or cherry-picking. 6. *Systematics* is explained here as a form of model misspecification. One suggestion is to use a Bayesian modelling of this misspecification, another non-parametrics (why not both together?!). 7. *Coverage* is somewhat disjunct from the other points as it explains the [frequentist] meaning of the coverage of a confidence interval. Which hence does not apply to the actual data. 8. *p0 versus p1* plots is a sketchy part referring to a recent proposal by the author. So in the end a rather anticlimactic coverage of standard remarks, surprisingly giving birth to a sequence of posts (incl. this one!)…

Filed under: Books, Statistics, University life Tagged: Bayesian modeling, five sigma, John Butterworth, likelihood ratio, Louis Lyons, p-values, PNAS, The Guardian, Valen Johnson

### interesting mis-quote

**A**t a recent conference on Big Data, one speaker mentioned this quote from Peter Norvig, the director of research at Google:

*“All models are wrong, and increasingly you can succeed without them.”*

quote that I found rather shocking, esp. when considering the amount of modelling behind Google tools. And coming from someone citing Kernel Methods for Pattern Analysis by Shawe-Taylor and Christianini as one of his favourite books and Bayesian Data Analysis as another one… Or displaying Bayes [or his alleged portrait] and Turing in his book cover. So I went searching on the Web for more information about this surprising quote. And found the explanation, as given by Peter Norvig himself:

*“To set the record straight: That’s a silly statement, I didn’t say it, and I disagree with it.”*

Which means that weird quotes have a high probability of being misquotes. And used by others to (obviously) support their own agenda. In the current case, Chris Anderson and his End of Theory paradigm. Briefly and mildly discussed by Andrew a few years ago.

Filed under: Books, pictures, Statistics, Travel, University life Tagged: Alan Turing, all models are wrong, artificial intelligence, George Box, misquote, Peter Norvig, statistical modelling, The End of Theory, Thomas Bayes

### snapshot from Vienna (#3)

Filed under: pictures, Travel Tagged: Austria, Baroque architecture, Franz Joseph I, Habsburgs, Schönbrunn palace, Unesco World Heritage List, Vienna

### a weird beamer feature…

**A**s I was preparing my slides for my third year undergraduate stat course, I got a weird error that got a search on the Web to unravel:

which was related with a fragile environment

\begin{frame}[fragile] \frametitle{simulation in practice} \begin{itemize} \item For a given distribution $F$, call the corresponding pseudo-random generator in an arbitrary computer language \begin{verbatim} > x=rnorm(10) > x [1] -0.021573 -1.134735 1.359812 -0.887579 [7] -0.749418 0.506298 0.835791 0.472144 \end{verbatim} \item use the sample as a statistician would \begin{verbatim} > mean(x) [1] 0.004892123 > var(x) [1] 0.8034657 \end{verbatim} to approximate quantities related with $F$ \end{itemize} \end{frame}\begin{frame}but not directly the verbatim part: the reason for the bug was that the \end{frame} command did not have a line by itself! Which is one rare occurrence where the carriage return has an impact in LaTeX, as far as I know… (The same bug appears when there is an indentation at the beginning of the line. Weird!) [Another annoying feature is wordpress turning > into > in the sourcecode environment...]

Filed under: Books, Kids, Linux, R, Statistics, University life Tagged: Beamer, bootstrap, course, fragile environment, LaTeX, R, random number generation, rnorm(), slides, Statistics, Université Paris Dauphine, verbatim, \end{frame}

### Statistics second slides

**T**his is the next chapter of my Statistics course, definitely more standard, with some notions on statistical models, limit theorems, and exponential families. In the first class, I recalled the convergence notions with no proof but counterexamples and spend some time on a slide not included here, borrowed from Chris Holmes’ talk last Friday on the linear relation between blood pressure and the log odds ratio of an heart condition. This was a great example, both to illustrate the power of increasing the number of observations and of using a logistic regression model. Students kept asking questions about it.

Filed under: Books, Kids, Statistics, University life Tagged: blood pressure, exponential families, logistic regression, statistical modelling, undergraduates, Université Paris Dauphine

### snapshot from Vienna (#2)

Filed under: pictures, Running, Travel Tagged: Austria, Heldendenkmal der Roten Armee, Red Army, Siege of Vienna, Vienna, WWII

### BAYSM’14 recollection

**W**hen I got invited to BAYSM’14 last December, I was quite excited to be part of the event. (And to have the opportunities to be in Austria, in Wien and on the new WU campus!) And most definitely and a posteriori I have not been disappointed given the high expectations I had for that meeting…! The organisation was seamless, even by Austrian [high] standards, the program diverse and innovative, if somewhat brutal for older Bayesians and the organising committee (Angela Bitto, Gregor Kastner, and Alexandra Posekany) deserves an ISBA recognition award [yet to be created!] for their hard work and dedication. Thanks also to Sylvia Früwirth-Schnatter for hosting the meeting in her university. They set the standard very high for the next BAYSM organising team. (To be hold in Firenze/Florence, on June 19-21, 2016, just prior to the ISBA World meeting *not* taking place in Banff. A great idea to associate with a major meeting, in order to save on travel costs. Maybe the following BAYSM will take place in Edinburgh! Young, local, and interested Bayesians just have to contact the board of BAYS with proposals.)

So, very exciting and diverse. A lot of talks in applied domains, esp. economics and finance in connection with the themes of the guest institution, WU. On the talks most related to my areas of interest, I was pleased to see Matthew Simpson working on interweaving MCMC with Vivek Roy and Jarad Niemi, Madhura Killedar constructing her own kind of experimental ABC on galaxy clusters, Kathrin Plankensteiner using Gaussian processes on accelerated test data, Julyan Arbel explaining modelling by completely random measures for hazard mixtures [and showing his filliation with me by (a) adapting my pun title to his talk, (b) adding an unrelated mountain picture to the title page, (c) including a picture of a famous probabilist, Paul Lévy, to his introduction of Lévy processes and (d) using xkcd strips], Ewan Cameron considering future ABC for malaria modelling, Konstantinos Perrakis working on generic importance functions in data augmentation settings, Markus Hainy presenting his likelihood-free design (that I commented a while ago), Kees Mulder explaining how to work with the circular von Mises distribution. Not to mention the numerous posters I enjoyed over the first evening. And my student Clara Grazian who talked about our joint and current work on Jeffreys priors for mixture of distributions. Whose talk led me to think of several extensions…

Besides my trek through past and current works of mine dealing with mixtures, the plenary sessions for mature Bayesians were given by Mike West and Chris Holmes, who gave very different talks but with the similar message that data was catching up with modelling and with a revenge and that we [or rather young Bayesians] needed to deal with this difficulty. And use approximate or proxy models. Somewhat in connection with my last part on an alternative to Bayes factors, Mike also mentioned a modification of the factor in order to attenuate the absorbing impact of long time series. And Chris re-set Bayesian analysis within decision theory, constructing approximate models by incorporating the loss function as a substitute to the likelihood.

Once again, a terrific meeting in a fantastic place with a highly unusual warm spell. Plus enough time to run around Vienna and its castles and churches. And enjoy local wines (great conference evening at a Heuriger, where we did indeed experience Gemütlichkeit.) And museums. Wunderbar!

Filed under: Books, Kids, pictures, Statistics, Travel, University life, Wines Tagged: ABC, approximate likelihood, architecture, Austria, BAYSM 2014, Donau, econometrics, Heuriger, interweaving, MCMC, Vienna, Wien, WU Wien, young Bayesians

### An der schönen blauen Donau

### new kids on the block

**T**his summer, for the first time, I took three Dauphine undergraduate students into research projects thinking they had had enough R training (with me!) and several stats classes to undertake such projects. In all cases, the concept was pre-defined and “all they had to do” was running a massive flow of simulations in R (or whatever language suited them best!) to check whether or not the idea was sound. Unfortunately, for two projects, by the end of the summer, we had not made any progress in any of the directions I wanted to explore… Despite a fairly regular round of meetings and emails with those students. In one case the student had not even managed to reproduce the (fairly innocuous) method I wanted to improve upon. In the other case, despite programming inputs from me, the outcome was impossible to trust. A mostly failed experiment which makes me wonder why it went that way. Granted that those students had no earlier training in research, either in exploiting the literature or in pushing experiments towards logical extensions. But I gave them entries, discussed with them those possible new pathways, and kept updating schedules and work-charts. And the students were volunteers with no other incentive than discovering research (I even had two more candidates in the queue). So it may be (based on this sample of 3!) that our local training system is missing in this respect. Somewhat failing to promote critical thinking and innovation by imposing too long presence hours and by evaluating the students only through standard formalised tests. I do wonder, as I regularly see [abroad] undergraduate internships and seminars advertised in the stats journals. Or even conferences.

Filed under: Kids, R, Statistics, University life Tagged: academic research, research internships, training of researchers, undergraduates

### snapshot from Vienna

### Le Monde puzzle [#879]

**H**ere is the last week puzzle posted in Le Monde:

*Given an alphabet with 26 symbols, is it possible to create 27 different three-symbol words such that*

*all symbols within a word are different**all triplets of symbols are different**there is no pair of words with a single common symbol*

Since there are

28x27x26/3×2=2925

such three-symbol words, it could be feasible to write an R code that builds the 27-uplet, assuming it exists. However, by breaking those words into primary words [that share no common symbols] and secondary words [that share two symbols with one primary word], it seems to me that there can be a maximum of 26 words under those three rules…

Filed under: Books, Kids Tagged: combinatorics, Le Monde, mathematical puzzle