## Xian's Og

### fine-sliced Poisson [a.k.a. sashimi]

**A**s my student Kévin Guimard had not mailed me his own Poisson slice sampler of a Poisson distribution, I could not tell why the code was not working! My earlier post prompted him to do so and a somewhat optimised version is given below:

**A**s you can easily check by running the code, it does not work. My student actually majored my MCMC class and he spent quite a while pondering why the code was not working. I did ponder as well for a part of a morning in Warwick, removing causes for exponential or factorial overflows (hence the shape of the code), but not eliciting the issue… (This now sounds like lethal fugu sashimi! ) Before reading any further, can you spot the problem?!

**T**he corrected R code is as follows:

**T**he culprit is thus the R function * sample* which simply does not understand Dirac masses and the basics of probability! When running

you can clearly see where the problem stands…! Well-documented issue with * sample* that already caused me woes… Another interesting thing about this slice sampler is that it is awfully slow in exploring the tails. And to converge to the centre from the tails. This is not very pronounced in the above graph with a mean of 6. Moving to 50 makes it more apparent:

**T**his is due to the poor mixing of the chain, as shown by the raw sequence below, which strives to achieve a single cycle out of 10⁵ iterations! In any case, thanks to Kévin for an interesting morning!

Filed under: Books, Kids, pictures, R, Running, Statistics, University life Tagged: convergence assessment, ENSAE, Gibbs sampling, Monte Carlo Statistical Methods, Poisson distribution, R, sample, sampling from an atomic population, slice sampling, slow convergence

### The 8th Rimini Bayesian Econometrics Workshop

Just reporting the announcement for the 8th Rimini Bayesian Econometrics Workshop, June 9-10, 2014, in the very pleasant beach resort of Rimini, workshop that I attended a few years ago:

This Workshop is organized by the Rimini Centre for Economic Analysis (RCEA) and will be run within the RIMINI CONFERENCE in ECONOMICS and FINANCE RCEF-2014

Call for papers: Authors should submit an extended abstract of up to 500 words by Monday 31st of March 2014. Please include with the submission JEL classification codes for the paper, keywords as well as JEL classification codes of the author(s) specialization field(s). Complete papers may be submitted but the extended abstract is required. In case of more than one author, please note the corresponding author. Proposals for sessions, consisting of three papers, are particularly welcome. If you are interested in submitting a session please send the session topic, paper titles and names of authors and arrange for the abstracts to be sent to the addresses provided below.

Filed under: Running, Statistics, Travel, University life Tagged: Bayesian econometrics, Italy, Rimini

### Annual Review of Statistics and its Application, vol. 1

**G**ot this book in my mailbox the other day. It is somewhat a unique object in that it is a high quality book with solid bindings, glossy paper, colours on every page and a nice layout… And even my name (as well as all other authors’) on the cover!

**I** am somewhat surprised there are still books “like that” (private joke for Sempé‘s aficionadi) as I wonder at which niche “they” are aiming at. What is basically unclear to me is the nature and business model of the “non-profit” Annual Reviews. Presumably selling the entire collection of Annual Reviews for $9312 helps in making the scheme “non-profitable”, at $339 for this single volume. If just a few libraries buy them. Especially when authors get no perks other than a single copy of the book. (To be completely fair, the book is available on-line for free for the first year.) So, in retrospect, (a) I should have been more careful in checking the fundamentals of this publication, as I foolishly assumed when contacted that it was a new journal (!), and (b) I should also have taken better advantage of the editing and printing facilities, using margins for highlights and comments, adding markers on important bibliographical references, and including vignettes as those Sam Behesta suggested for my CHANCE book reviews… (It did not come at the best of times, but still I should have tried harder.)

Filed under: Books, Statistics

### sliced Poisson

**O**ne of my students complained that his slice sampler of a Poisson distribution was not working when following the instructions in Monte Carlo Statistical Methods (Exercise 8.5). This puzzled me during my early morning run and I checked on my way back, even before attacking the fresh baguette I had brought from the bakery… The following R code is the check. And it does work! As the comparison above shows…

Filed under: Books, Kids, pictures, R, Running, Statistics, University life Tagged: ENSAE, Monte Carlo Statistical Methods, Poisson distribution, R, slice sampling

### misty dawn [#3]

### Approximate Bayesian model choice

The above is the running head of the arXived paper with full title “Implications of uniformly distributed, empirically informed priors for phylogeographical model selection: A reply to Hickerson et al.” by Oaks, Linkem and Sukuraman. That I (again) read in the plane to Montréal (third one in this series!, and last because I also watched the Japanese psycho-thriller Midsummer’s Equation featuring a physicist turned detective in one of many TV episodes. I just found some common features with The Devotion of Suspect X, only to discover now that the book has been turned into another episode in the series.)

*“Here we demonstrate that the approach of Hickerson et al. (2014) is dangerous in the sense that the empirically-derived priors often exclude from consideration the true values of the models’ parameters. On a more fundamental level, we question the value of adopting an empirical Bayesian stance for this model-choice problem, because it can mislead model posterior probabilities, which are inherently measures of belief in the models after prior knowledge is updated by the data.”*

**T**his paper actually is a reply to Hickerson et al. (2014, *Evolution*), which is itself a reply to an earlier paper by Oaks et al. (2013, Evolution). *[Warning: I did not check those earlier references!]* The authors object to the use of “narrow, empirically informed uniform priors” for the reason reproduced in the above quote. In connection with the msBayes of Huang et al. (2011, *BMC Bioinformatics*). The discussion is less about ABC used for model choice and posterior probabilities of models and more about the impact of vague priors, Oaks et al. (2013) arguing that this leads to a bias towards models with less parameters, a “statistical issue” in their words, while Hickerson et al. (2014) think this is due to msBayes way of selecting models and their parameters at random.

*“…it is difficult to choose a uniformly distributed prior on divergence times that is broad enough to confidently contain the true values of parameters while being narrow enough to avoid spurious support of models with less parameter space.”*

**S**o quite an interesting debate that takes us in fine far away from the usual worries about ABC model choice! We are more at the level empirical versus natural Bayes, seen in the literature of the 80′s. (The meaning of empirical Bayes is not that clear in the early pages as the authors seem to involve any method using the data “twice”.) I actually do not remember reading papers about the formal properties of model choice done through classical empirical Bayes techniques. Except the special case of Aitkin’s (1991,2009) integrated likelihood. Which is essentially the analysis performed on the coin toy example (p.7)

*“…models with more divergence parameters will be forced to integrate over much greater parameter space, all with equal prior density, and much of it with low likelihood.”*

**T**he above argument is an interesting rephrasing of Lindley’s paradox, which I cannot dispute, but of course it does not solve the fundamental issue of how to choose the prior away from vague uniform priors… I also like the quote “the estimated posterior probability of a model is a single value (rather than a distribution) lacking a measure of posterior uncertainty” as this is an issue on which we are currently working. I fully agree with the statement and we think an alternative assessment to posterior probabilities could be more appropriate for model selection in ABC settings (paper soon to come, hopefully!).

Filed under: Books, R, Statistics, Travel, University life Tagged: ABC, ABC model selection, 真夏方程式, Detective Galileo, empirical Bayes methods, integrated likelihood, Jeffreys-Lindley paradox, model posterior probabilities, Montréal, vague priors

### misty dawn [#2]

Filed under: pictures, Running Tagged: air pollution, Château de Sceaux, Colbert, fog, mist, Parc de Sceaux, sunrise

### ISBA 2016 in Banff???

**S**cott Schmidler, Steve Scott and myself just submitted a proposal for holding the next World ISBA Conference in 2016 in Banff, Canada! After enjoying the superb environment of the *Advanced in Scalable Bayesian computation* workshop last week, we thought it would be worth a try as a potential location for the next meeting, esp. when considering the superlative infrastructure of the Banff Centre (meaning we really do not have to be* local* to be *local organisers*!), the very reasonable rates for renting the site and securing two hundred rooms, the potential for a special collaboration with BIRS, the scarcity of alternative proposals (as far as I can fathom) and the ultimate mountain environment… I remember fondly the IMS annual meeting of 2002 there, with a great special lecture by Hans Künsch and, exceptionally, an RSS Read Paper by Steve Brooks, Paulo Guidici and Gareth Roberts. (Not mentioning en exhilarating solo scramble up Mount Temple and another one with Arnaud Guillin up the chimneys of Mount Edith!) Since the deadline was this Saturday, March 15, we should hear pretty soon if we are successful in this bid. (Good luck to our Scottish friends from Edinburgh for their bid for holding ISBA 2018! Moving from the feet of Mount Rundle [above] to the feet of Arthur’s Seat would make for a great transition.)

Filed under: Mountains, pictures, Statistics, Travel, University life Tagged: Alberta, Arthur's Seat, Banff, Banff Centre, Canada, Canadian Rockies, Edinburgh, ISBA, ISBA 2016, ISBA 2018, Mount Rundle, Mount Temple, Scotland

### Clockers [book review]

**T**hroughout my recent trip to Canada, I read bits and pieces of *Clockers* by Richard Price and I finished reading it last Sunday. It is an impressive piece of literature and I am surprised I was not aware of its existence until amazon.com suggested it to me (as I was checking for recent books by another Richard, Richard Morgan!). Guessing from the summary it could be of interest and from comments it was sort of a classic, I ordered it more or less on a whim (given a comfortable balance on my amazon.com account, thanks to ‘Og’s readers!) It took me a few pages to realise the plot was deeply set in the 1990′s, not only because this was the high of the crack epidemics, but also since the characters (drug dealers and policemen) therein are all using beepers, instead of cellphones, and street phone booths).

*“It’s like a math problem. Juan got whacked at point X, he drove away losing blood at the rate of a pint every ninety seconds. He was driving forty-five miles an hour and he bought the farm two miles inside the tunnel (…) So for ten points, [who] in what New Jersey town did Juan?” Clockers (p.272)*

**T**he plot of *Clockers* is vaguely a detective story as an aging and depressed homicide officer, Rosso, hunts the murderer of a drug dealer, being convinced from the start that the self-declared murderer Victor did not do it. In parallel, and somewhat more closely, the book follows the miserable plight and thoughts and desires of Victor’s brother, Strike, who is head of a local crack dealing network, under the domination of the charismatic and berserk Rodney Little… But the resolution of the crime matters very little, much less than the exposure of the deadly economics of the drug traffic in inner cities (years before *Freakonomics*!), of the constant fight of single mothers to bring food and structure to their dysfunctional families, to the widespread recourse to moonlighting, and above all to the almost physical impossibility to escape one’s environment (even for smart and decent kids like Victor and, paradoxically enough, the drug-dealing Strike) by lack of prospect and exposure to anything or anywhere else, as well as social pressure, early pregnancies and gang-related micro-partitioning of cities.

**W**hen I mentioned *Clockers* to Andrew, he told me that he also liked it very much but that the characters were not quite “real”. I somewhat agree in that, while the economics, the sociology and the practice of drug-dealing sound very accurately reproduced (for all I know!), the characters are more caricaturesque or picturesque than natural. The stomach disease of Strike sounds too much like an allegory of both his schizophrenic split between running the drug trade and looking for a definitive quit, while the sacrifice of his brother makes little sense, except as a form either of suicide or of escape from an environment he can no longer stand. What is most surprising is that Richard Price (just like Michael Crichton) is a practised screenwriter (who collaborated to Spike Lee’s 1995 Clockers). So he knows how to run an efficient story with convincing characters and plot(s). Hence my little theory of a picaresque novel… (Here is Jim Shepard’s enthusiastic review of *Clockers*. With the definitely accurate title of “Sympathy for the dealer”.)

Filed under: Books, Travel Tagged: Amazon, Clockers, crack epidemics, drug dealers, Freakonomics, Jim Shepard, New Jersey, Richard Price, Spike Lee, The New York Times

### misty dawn

### truncated t’s [typo]

**L**ast night, I received this email from Piero Foscari (im Hamburg) about my moment derivations for the absolute and the positive *t* distribution:

*There might be two typos in the final second moment formula and its derivation (assuming no silly symmetric mistakes in my validation code): the first ν ought to be -ν, and there should be a corresponding scaling factor also for the boundary μ in Pμ,ν-2 since it arises from a change of variable. Btw in the text reference to Fig. 2 |X| wasn’t updated to X+. I hope that this is of some use.*

and I checked that indeed I had forgotten the scale factor ν/(ν-2) in the t distribution with ν-2 degrees of freedom as well as the sign… So I modified the note and rearXived it. Sorry about this lack of attention to the derivation!

Filed under: pictures, Statistics Tagged: arXiv, Elben, Hamburg, moments, truncated t distribution, typo

### Approximate Integrated Likelihood via ABC methods

**M**y PhD student Clara Grazian just arXived this joint work with Brunero Liseo on using ABC for marginal density estimation. The idea in this paper is to produce an integrated likelihood approximation in intractable problems via the ratio

both terms in the ratio being estimated from simulations,

(with possible closed form for the denominator). Although most of the examples processed in the paper (Poisson means ratio, Neyman-Scott’s problem, g-&-k quantile distribution, semi-parametric regression) rely on summary statistics, hence *de facto* replacing the numerator above with a pseudo-posterior conditional on those summaries, the approximation remains accurate (for those examples). In the g-&-k quantile example, Clara and Brunero compare our ABC-MCMC algorithm with the one of Allingham et al. (2009, Statistics & Computing): the later does better by not replicating values in the Markov chain but instead proposing a new value until it is accepted by the usual Metropolis step. (Although I did not spend much time on this issue, I cannot see how both approaches could be simultaneously correct. Even though the outcomes do not look very different.) As noted by the authors, “the main drawback of the present approach is that it requires the use of proper priors”, unless the marginalisation of the prior can be done analytically. (This is an interesting computational problem: how to provide an efficient approximation to a marginal density of a σ-finite measure, assuming this density exists.)

*Clara will give a talk at CREST-ENSAE today about this work, in the Bayes in Paris seminar: 2pm in room 18.*

Filed under: Books, Statistics, University life Tagged: ABC, Bayesian asymptotics, kernel density estimator, marginalisation, marginalisation paradoxes, nuisance parameters, quantile distribution, semi-parametrics, simulation

### where did the normalising constants go?! [part 2]

**C**oming (swiftly and smoothly) back home after this wonderful and intense week in Banff, I hugged my loved ones, quickly unpacked, ran a washing machine, and then sat down to check where and how my reasoning was wrong. To start with, I experimented with a toy example in R:

and (of course!) it produced the perfect fits reproduced below. Writing the R code acted as a developing bath as it showed why we could do *without* the constants!

“**O**f course”, because the various derivations in the above R code all are clearly *independent* from the normalising constant: (i) when considering a product of kernel density estimators, as in the first version, this is an approximation of

as well as of

since the constant does not matter. (ii) When considering a sample from mi and weighting it by the product of the remaining true or estimated mj‘s, this is a sampling weighting resampling simulation from the density proportional to the product and hence, once again, the constants do not matter. At last, (iii) when mixing the two subsamples, since they both are distributed from the product density, the constants do not matter. As I slowly realised when running this morning (trail-running, not code-runninh!, for the very first time in ten days!), the straight-from-the-box importance sampling version on the mixed samples I considered yesterday (to the point of wondering out loud where did the constants go) is never implemented in the cited papers. Hence, the fact that

is enough to justify handling the target directly as the product of the partial marginals. End of the mystery. Anticlimactic end, sorry…

Filed under: R, Statistics, Travel Tagged: beta distribution, big data, consensus, embarassingly parallel, importance sampling, normalising constant, parallel processing, R

### where did the normalising constants go?! [part 1]

**W**hen listening this week to several talks in Banff handling large datasets or complex likelihoods by parallelisation, splitting the posterior as

and handling each term of this product on a separate processor or thread as proportional to a probability density,

then producing simulations from the mi‘s and attempting at deriving simulations from the original product, I started to wonder where all those normalising constants went. What vaguely bothered me for a while, even prior to the meeting, and then unclicked thanks to Sylvia’s talk yesterday was the handling of the normalising constants ωi by those different approaches… Indeed, it seemed to me that the samples from the mi‘s should be weighted by

rather than just

or than the product of the other posteriors

which makes or should make a significant difference. For instance, a sheer importance sampling argument for the aggregated sample exhibited those weights

Hence processing the samples on an equal footing or as if the proper weight was the product of the other posteriors mj should have produced a bias in the resulting sample. This was however the approach in both Scott et al.‘s and Neiswanger et al.‘s perspectives. As well as Wang and Dunson‘s, who also started from the product of posteriors. (Normalizing constants are considered in, e.g., Theorem 1, but only for the product density and its Weierstrass convolution version.) And in Sylvia’s talk. Such a consensus of high calibre researchers cannot get it wrong! So I must have missed something: what happened is that the constants eventually did not matter, as expanded in the next post…

Filed under: R, Statistics, Travel Tagged: big data, consensus, embarassingly parallel, normalising constant, parallel processing

### snapshot from Banff

### shrinkage-thresholding MALA for Bayesian variable selection

**A**mandine Shreck along with her co-authors Gersende Fort, Sylvain LeCorff, and Eric Moulines, all from Telecom Paristech, has undertaken to revisit the problem of large p small n variable selection. The approach they advocate mixes Langevin algorithms with trans-model moves with shrinkage thresholding. The corresponding Markov sampler is shown to be geometrically ergodic, which may be a première in that area. The paper was arXived in December but I only read it on my flight to Calgary, not overly distracted by the frozen plains of Manitoba and Saskatchewan. Nor by my neighbour watching Hunger Games II.)

**A** shrinkage-thresholding operator is defined as acting on the regressor matrix towards producing sparse versions of this matrix. (I actually had trouble picturing the model until Section 2.2 where the authors define the multivariate regression model, making the regressors a matrix indeed. With a rather unrealistic iid Gaussian noise. And with an unknown number of relevant rows, hence a varying dimension model. Note that this is a strange regression in that the regression coefficients are known and constant across all models.) Because the Langevin algorithm requires a gradient to operate, the log target is divided between a differentiable and a non-differentiable parts, the later accommodating the Dirac masses in the dominating measure. The new MALA moves involve applying the above shrinkage-thresholding operator to a regular Langevin proposal, hence moving to sub-spaces and sparser representations.

**T**he thresholding functions are based on positive part operators, which means that the Markov chain does not visit some neighbourhoods of zero in the embedding and in the sparser spaces. In other words, the proposal operates between models of varying dimensions without further ado because the point null hypotheses are replaced with those neighbourhoods. Hence it is *not* exactly simulating from the “original” posterior, which may be a minor caveat or not. Not if defining the neighbourhoods is driven by an informed or at least spelled-out choice of a neighbourhood of zero where the coefficients are essentially identified with zero. The difficulty is then in defining how close is close enough. Especially since the thresholding functions seem to all depend on a single number which does not depend on the regressor matrix. It would be interesting to see if the g-prior version could be developed as well… Actually, I would have also included a dose of g-prior in the Langevin move, rather than using an homogeneous normal noise.

**T**he paper contains a large experimental part where the performances of the method are evaluated on various simulated datasets. It includes a comparison with reversible jump MCMC, which slightly puzzles me: (a) I cannot see from the paper whether or not the RJMCMC is applied to the modified (thresholded) posterior, as a regular RJMCMC would not aim at the same target, but the appendix does not indicate a change of target; (b) the mean error criterion for which STMALA does better than RJMCMC is not defined, but the decrease of this criterion along iterations seems to indicate that convergence has not yet occured, since it does not completely level up after 3 10⁵ iterations.

**I** must have mentioned it in another earlier post, but I find somewhat ironical to see those thresholding functions making a comeback after seeing the James-Stein and smooth shrinkage estimators taking over the then so-called pre-test versions in the 1970′s (Judge and Bock, 1978) and 1980′s. There are obvious reasons for this return, moving away from quadratic loss being one.

Filed under: Statistics, University life Tagged: Bayesian variable selection, ergodicity, Langevin MCMC algorithm, RJMCMC, spike-and-slab prior, variable dimension models

### les sciences face aux créationnismes [book review]

**I **spotted this small book during my last visit to CBGP in Montpellier, and borrowed it from the local librarian. It is written (in French) by Guillaume Lecointre, who is professor of Biology at the Muséum National d’Histoire Naturelle in Paris, specialised in population evolution and philogenies. The book is published by Editions Quae, a scientific editor supported by four founding French institutes (CIRAD, IFREMER, INRA and IRSTEA), hence no wonder I would spot it in an INRA lab. The theme of the book is not to argue against creationism and intelligent design theories, but rather to analyse how the debates between scientists—interestingly this term *scientist* sounds much more like a cult in English than the French noun *scientifique*— and creationists are conducted and to suggest how they should be conducted. While there are redundancies in the text, I found the overall argumentation quite convincing, with the driving lines that creationists are bypassing the rules of scientific investigation and exchange to bring the debate at a philosophical or ideological level foreign to science definition. Lecointre deconstructs the elements put forward in such debates, from replacing the incompleteness of the scientific knowledge and the temporary nature of scientific theories with a total relativism, to engaging scientific supporters from scientific fields not directly related with the theory of evolution, to confusing methodological materialism with philosophical materialism and more fundamentally to imply that science and scientific theories must have a moral or ideological content, and to posturing as anti-establishment and anti-dogmatic free minds… I also liked the points that (a) what really drives the proponents of intelligent design is a refusal of randomness in the evolution, without any global or cosmic purpose; (b) scientists are very ill-prepared to debate with creationists, because the later do not follow a scientific reasoning; (c) journalists are most often contributing to the confusion by picking out-of-their-field “experts” and encouraging the relativity argument. Hence a reasonable recommendation to abstain from oral debates and to stick to pointing out the complete absence of scientific methodology in creationists’ arguments. (Obviously, readers of Alan Sokal’s Beyond the Hoax will be familiar most of the arguments produced in les sciences face aux créationnismes.)

Filed under: Books Tagged: "intelligent" design, Alan Sokal, creationism, evolution, materialism, Philosophy of Science, Science

### Advances in Scalable Bayesian Computation [group photo]

Filed under: Kids, Mountains, pictures, Statistics, Travel, University life Tagged: 14w5125, Advances in Scalable Bayesian Computation, Banff International Research Station for Mathematical Innovation, Bayesian computation, BIRS, Canada, MCMC, Monte Carlo Statistical Methods, simulation

### BIRS building snapshot

Filed under: Mountains, pictures, Statistics, Travel, University life Tagged: Banff, Banff Centre, Banff International Research Station for Mathematical Innovation, BIRS, Canada

### Advances in scalable Bayesian computation [day #4]

**F**inal day of our workshop Advances in Scalable Bayesian Computation already, since tomorrow morning is an open research time ½ day! Another “perfect day in paradise”, with the Banff Centre campus covered by a fine snow blanket, still falling…, and making work in an office of BIRS a dream-like moment.

**S**till looking for a daily theme, *parallelisation* could be the right candidate, even though other talks this week went into parallelisation issues, incl. Steve’s talk yesterday. Indeed, Anthony Lee gave a talk this morning on interactive sequential Monte Carlo, where he motivated the setting by a formal parallel structure. Then, Darren Wilkinson surveyed the parallelisation issues in Monte Carlo, MCMC, SMC and ABC settings, before arguing in favour of a functional language called Scala. (Neat entries to those topics can be found on Darren’s blog.) And in the afternoon session, Sylvia Frühwirth-Schnatter exposed her approach to the (embarrassingly) parallel problem, in the spirit of Steve’s , David Dunson’s and Scott’s (a paper posted on the day I arrived in Chamonix and hence I missed!). There was plenty to learn from that talk (do not miss the Yin-Yang moment at 25 mn!), but it also helped me to break a difficulty I had with the consensus Bayes representation for two weeks *(more on that later!)*. And, even though Marc Suchard mostly talked about flu and trees in a very pleasant and broad talk, he also had a slide on parallelisation to fit the theme! Although unrelated with parallelism, Nicolas Chopin’s talk was on sequential quasi-Monte Carlo algorithms: while I had heard previous versions of this talk in Chamonix and BigMC, I found it full of exciting stuff. And it clearly got the room truly puzzled by this possibility, in a positive way! Similarly, Alex Lenkoski spoke about extreme rain events in Norway with no trace of parallelism, but the general idea behind the examples was to question the notion of the calibrated Bayesian (with possible connections with the cut models).

**T**his has been a wonderful week and I am sure the participants got as much as I did from the talks and the informal exchanges. Thanks to BIRS for the sponsorship and the superb organisation of the week (and to the Banff Centre for providing such a paradisical environment). I feel very privileged to have benefited from this support, even though I deadly hope to be back in Banff within a few years.

Filed under: Books, Mountains, pictures, R, Statistics, University life Tagged: Art Owen, Banff International Research Station for Mathematical Innovation, Bayesian nonparametrics, Canada, Canadian Rockies, extremes, flu, influenza, low discrepancy sequences, MCMC algorithms, Norway, parallel processing, parallelisation, philogenic trees, quasi-random sequences, vaccine, viruses, Yin-Yang algorithm