Bayesian News Feeds

someone to watch over me [Horfðu á mig]

Xian's Og - Fri, 2014-09-19 17:14

And yet another roman noir taking place in Iceland! My bedside read over the past two months was “Someone to watch over me” by Yrsa Sigurðardóttir. (It took that long because I was mostly away in July and August, not because the book was boring me to sleep every night!) It is a fairly unusual book in several respects: the setting is an institution for mentally handicapped patients that was set on fire, killing five of the patients as a result, the investigator is an Icelandic lawyer, Þóra Guðmundsdóttir, along with her German unemployed-banker boyfriend, the action takes place at the height [or bottom!] of the Icelandic [and beyond!] economic crisis, when most divorce settlements are about splitting the debts of the household, and when replacing a computer becomes an issue, some of the protagonists, including the main suspects, are mentally ill, and the police and justice are strangely absent from most of the story. The the book tells a lot about the Icelandic society, where a hit-and-run is so unheard of that the police is clueless. Or seems to be. And where people see ghosts. Or think they do, as the author plays (heavily?) on the uncertainty about those ghosts. (At least, there are no elves. Nor trolls.) Definitely more in tune with the “true” Iceland than Available dark. (Well, as far as I can tell!) The mystery itself is a wee bit stretched and the final resolution slightly disappointing, implying some unlikely behaviour from the major characters. In particular, I do not buy the explanation motivating the arson itself. Terrible cover too. And not a great title in English (Watch me or Look at me would have been better) given the many books, movies and songs with the same title. Nonetheless, I liked very much the overall atmosphere of the book, enough to recommend it.

Filed under: Books, Travel Tagged: autism, Þóra Guðmundsdóttir, financial crisis, Horfðu á mig, Iceland, Reykjavik, Yrsa Sigurðardóttir
Categories: Bayesian Bloggers

an ISBA tee-shirt?!

Xian's Og - Fri, 2014-09-19 07:18

 Sonia Petrone announced today at BAYSM’14 that a competition was open for the design of an official ISBA tee-shirt! The deadline is October 15 and the designs are to be sent to Clara Grazian, currently at CEREMADE, Université Dauphine [that should be enough to guess her email!]. I will most certainly submit my mug design. And maybe find enough free time to design a fake eleven Paris with moustache tee-shirt. With Bayes’ [presumed] portrait of course…

Filed under: Kids, pictures, University life Tagged: BAYSM 2014, ISBA, Markov chain, mug, tee-shirt, werewolf
Categories: Bayesian Bloggers

Am Belvedere

Xian's Og - Thu, 2014-09-18 17:14
Categories: Bayesian Bloggers

up [and down] Pöstlingberg

Xian's Og - Thu, 2014-09-18 07:18

Early morning today, following my Linz guests’ advice, I went running towards the top of Pöstlingberg, a hill 250m over Linz and the Danube river. A perfect beacon thus avoiding wrong turns and extra-mileage, but still a wee climb on a steep path for the last part. The reward of the view from the top was definitely worth the [mild] effort and I even had enough time to enjoy a good Austrian breakfast before my ABC talk

Filed under: Mountains, pictures, Running, Travel, University life Tagged: Austria, breakfast, Danube, Donau, IFAS, Linz, montain running, Pöstlingberg, seminar, sunrise
Categories: Bayesian Bloggers

my life as a mixture [slides]

Xian's Og - Wed, 2014-09-17 17:14

Here are the slides of my talk today at the BAYSM’14 conference in Vienna. Mostly an overview of some of my papers on mixtures, with the most recent stuff…

Filed under: pictures, Statistics, Travel, University life Tagged: Austria, BAYSM 2014, church, mixtures, Neue Jesuitenkirche, slides, Universitätkirche, Vienna, Wien
Categories: Bayesian Bloggers

position at Warwick

Xian's Og - Wed, 2014-09-17 07:18

A new position for the of Professor Of Statistics and Data Science / Director of the [newly created] Warwick Data Science Institute has been posted. To quote from the job description, “the position arises from the Department of Statistics’ commitment, in collaboration with the Warwick Mathematics Institute and the Department of Computer Science, to a coherent methodological approach to the fundamentals of Data Science and the challenges of complex data sets (for example big data).”  The interview date is November 27, 2014. All details available here.

Filed under: Statistics, University life Tagged: academic position, big data, data science, England, job opening, professor of statistics, University of Warwick, Warwick Data Science Institute
Categories: Bayesian Bloggers

arriving in Linz

Xian's Og - Tue, 2014-09-16 17:14
Categories: Bayesian Bloggers

Le Monde [short] guide to Vienna

Xian's Og - Tue, 2014-09-16 07:18

An interesting (?) coincidence: Le Monde weekend edition has its tourist page dedicated to Vienna! As usual, it is a list of places recommended by a local, Le Vienne de Robert Stadler, which includes

Maybe a wee bit limited a scope (albeit the house designed by Wittgenstein sounds definitely worth the trip!). For a wider range of Vienna highlights for BAYSM 2014 participants, The New York Times offers 36 hours in Vienna. With apparently no intersection with the above list. (But the same imbalance towards restaurants and bars!)

Filed under: Books, Travel Tagged: 36 hours in, Austria, Le Monde, Ludwig Wittgenstein, The New York Times, Vienna
Categories: Bayesian Bloggers

Statistics first slides

Xian's Og - Mon, 2014-09-15 17:14

Today I started my new course of Statistics for our third year undergraduates. In English! A point that came as a surprise for the students but I got no complaint (so far) and they started asking questions in English during the class. The slides are “under construction” and this first chapter borrows a fair chunk from Andrew’s blog entries. Including the last slide on the six Kaiser Fung quotes, which was posted yesterday night. The next chapter is going to be more standard, with statistical models, limit theorems, and exponential families.

Filed under: Books, Kids, Statistics, University life
Categories: Bayesian Bloggers

Series B reaches 5.721 impact factor!

Xian's Og - Sun, 2014-09-14 17:14

I received this email from Wiley with the great figure that JRSS Series B has now reached a 5.721 impact factor. Which makes it the first journal in Statistics from this perspective. Congrats to editors Gareth Roberts, Piotr Fryzlewicz and Ingrid Van Keilegom for this achievement! An amazing jump from the 2009 figure of 2.84…!

Filed under: Books, Statistics, University life Tagged: impact factor, John Wiley, JRSSB, Series B
Categories: Bayesian Bloggers

xkcd [interview & book]

Xian's Og - Sat, 2014-09-13 17:14

Of interest for xkcd fans: What If?: Serious Scientific Answers to Absurd Hypothetical Questions is out! Actually, it is currently the #1 bestseller on amazon! (A physics book makes it to the top of the bestseller list, a few weeks after a theoretical economics book got there. Nice! Actually, a statistics book also made it to the top: Nate Silver’s The SIgnal and the Noise….) I did not read the book, but it is made of some of the questions answered by Randall Munroe (the father of xkcd) on his what if blog. In connection with this publication, Randall Munroe is interviewed on FiveThirtyEight (Nate Silver’s website), as kindly pointed out to me by Bill Jefferys. The main message is trying to give people a feeling about numbers, a rough sense of numeracy. Which was also the purpose of the guesstimation books.

Filed under: Books, Kids, Statistics Tagged: Amazon, bestseller, book review, FiveThirtyEight, Guesstimation, Nate Silver, what if?, xkcd
Categories: Bayesian Bloggers

available dark [book review]

Xian's Og - Fri, 2014-09-12 17:14

“Paved roads had long ago surrended to gravel tracks that disappeared into a desert of snow covered lava. Black spires like a forest of charred trees blotted out the stars near the horizon.”

This is the last book I read from my Amazon package: Available Dark by Elizabeth Hand. I cannot remember how I came to order it… Maybe a confusion with another fantasy author like Elizabeth Moon? Or simply because the story was taking place between MaineFinland and Iceland?! Anyway, I read the book within two days during a short hiking trip to the volcano region of Central France. The plot has indeed a mesmerizing quality that made me keep reading further and further at ungodly hours. (With the help of an US jetlag.) It is original and intense enough to overcome the major difficulty that the central character, Cas, is far from sympathetic, from specialising in corpse photography to being almost constantly on drugs. But the construction of the plot and the introduction of the characters, always seen from Cas’ viewpoint, are well-done, even though the ending is both precipitated and unrealistic. Too many coincidences. The original setup of this novel is the Finnish black metal scene, with its undercurrents of satanism, ritual murders, and church burnings. Rather accurate judging from the wikipedia page on the topic! What I appreciated most was the description of the first impression of Iceland on Cas, when she landed from Helsinki. “The trip to Reykjavik [from the airport] was like a bus tour through Mordor. Black lava fields, an endless waste broken here and there by ruined machinery or a building of stained corrugated metal.” So I may consider reading another novel in the series in a near future…

Filed under: Books, Mountains, Travel Tagged: Available Dark, black metal, Elizabeth Hand, Finland, Helsinki, Iceland, Reykjavik
Categories: Bayesian Bloggers

3,000 posts and 1,000,000 views so far…

Xian's Og - Fri, 2014-09-12 07:18

As the ‘Og went over its [first] million views and 3,000 posts since its first post in October 2008, the most popular entries (lots of book reviews, too many obituaries, and several guest posts):

In{s}a(ne)!! 9,330 “simply start over and build something better” 8,514 George Casella 6,712 About 4,853 Bayesian p-values 4,468 Sudoku via simulated annealing 4,150 Julien on R shortcomings 3,673 Solution manual to Bayesian Core on-line 3,040 Solution manual for Introducing Monte Carlo Methods with R 2,954 #2 blog for the statistics geek?! 2,706 Of black swans and bleak prospects 2,596 Gelman’s course in Paris, next term! 2,451 the Art of R Programming [guest post] 2,242 Parallel processing of independent Metropolis-Hastings algorithms 2,208 Bayes’ Theorem 1,925 Bayes on the Beach 2010 [2] 1,778 Do we need an integrated Bayesian/likelihood inference? 1,742 Théorème vivant 1,617 Dennis Lindley (1923-2013) 1,613 Coincidence in lotteries 1,543 The mistborn trilogy 1,532 Julian Besag 1945-2010 1,529 Frequency vs. probability 1,448 Bayes’ Theorem in the 21st Century, really?! 1,401 the cartoon introduction to statistics 1,398 understanding computational Bayesian statistics 1,369 The Search for Certainty 1,274 Bayesian modeling using WinBUGS 1,273 Particle MCMC discussion 1,256 Reference prior for logistic regression 1,215 Tornado in Central Park 1,142 Harmonic mean estimators 1,138 A ridiculous email 1,134 Andrew gone NUTS! 1,132 Top 15 all-timers? 1,130 Millenium 1 [movie] 1,121 Monte Carlo Statistical Methods third edition 1,102 Introducing Monte Carlo Methods with R: a first course 1,090

and the most frequent search terms (excluding those connected with my name), with again two beach towns at the top!

benidorm 1,804 surfers paradise 1,050 george casella 785 mont blanc 705 introducing monte carlo methods with r 587 marie curie 500 mistborn 480 millenium 413 i love r 411 andrew wyeth 398 abele blanc 385 bayesian p value 375 bayesian p-value 374 walter bonatti 351 nested sampling 333 particle mcmc 332 dumplings 298


Filed under: Books, Kids, Statistics Tagged: book reviews, guest post
Categories: Bayesian Bloggers

my life as a mixture [BAYSM 2014, Wien]

Xian's Og - Thu, 2014-09-11 17:14

Next week I am giving a talk at BAYSM in Vienna. BAYSM is the Bayesian Young Statisticians meeting so one may wonder why, but with Chris Holmes and Mike West, we got invited as more… erm… senior speakers! So I decided to give a definitely senior talk on a thread pursued throughout my career so far, namely mixtures. Plus it also relates to works of the other senior speakers. Here is the abstract for the talk:

Mixtures of distributions are fascinating objects for statisticians in that they both constitute a straightforward extension of standard distributions and offer a complex benchmark for evaluating statistical procedures, with a likelihood both computable in a linear time and enjoying an exponential number of local models (and sometimes infinite modes). This fruitful playground appeals in particular to Bayesians as it constitutes an easily understood challenge to the use of improper priors and of objective Bayes solutions. This talk will review some ancient and some more recent works of mine on mixtures of distributions, from the 1990 Gibbs sampler to the 2000 label switching and to later studies of Bayes factor approximations, nested sampling performances, improper priors, improved importance samplers, ABC, and a inverse perspective on the Bayesian approach to testing of hypotheses.

I am very grateful to the scientific committee for this invitation, as it will give me the opportunity to meet the new generation, learn from them and in addition discover Vienna where I have never been, despite several visits to Austria. Including its top, the Großglockner. I will also give a seminar in Linz the day before. In the Institut für Angewandte Statistik.

Filed under: Books, Kids, Mountains, pictures, Statistics, Travel, University life Tagged: Austria, Bayes factor, Bayesian tests of hypotheses, BAYSM, Gibbs sampling, importance sampling, label switching, Linz, mixtures, Vienna, WU Wirtschaftsuniversität Wien
Categories: Bayesian Bloggers

O’Bayes 2015: back in València

Xian's Og - Thu, 2014-09-11 07:18

The next O’Bayes meeting (more precisely the International Workshop on Objective Bayes Methodology, O-Bayes15), will take place in València, Spain, on June 1-4, 2015. This is the second time an O’Bayes conference takes place in València, after the one José Miguel Bernardo organised in 1998 there.  The principal objectives of O-Bayes15 will be to facilitate the exchange of recent research developments in objective Bayes theory, methodology and applications, and related topics (like limited information Bayesian statistics), to provide opportunities for new researchers, and to establish new collaborations and partnerships. Most importantly, O-Bayes15 will be dedicated to our friend Susie Bayarri, to celebrate her life and contributions to Bayesian Statistics. Check the webpage of O-Bayes15 for the program (under construction) and the practical details. Looking forward to the meeting and hopeful for a broadening of the basis of the O’Bayes community and of its scope!

Filed under: pictures, Statistics, Travel, University life Tagged: José Miguel Bernardo, O-Bayes 2015, objective Bayes, Spain, Susie Bayarri, Valencia conferences
Categories: Bayesian Bloggers

Scottish polls…

Xian's Og - Wed, 2014-09-10 17:14

As much as I love Scotland, or because of it, I would not dream of suggesting to Scots that one side of the referendum sounds better than the other. However, I am rather annoyed at the yoyo-like reactions to the successive polls about the result, because, just like during the US elections, each poll is analysed separately rather than being pooled with the earlier ones in a reasonable meta-analysis… Where is Nate Silver when we need him?!

Filed under: pictures, Statistics, Travel Tagged: elections, Glasgow, Hillhead, independence, Nate Silver, poll, Scotland, Scottish independence referendum, United Kingdom
Categories: Bayesian Bloggers

random generators… unfit for ESP testing?!

Xian's Og - Tue, 2014-09-09 17:14

“The term psi denotes anomalous processes of information or energy transfer that are currently unexplained in terms of known physical or biological mechanisms.”

When re-reading [in the taxi to Birmingham airport] Bem’s piece on “significant” ESP tests, I came upon the following hilarious part that I could not let pass:

“For most psychological experiments, a random number table or the random function built into most programming languages provides an adequate tool for randomly assigning participants to conditions or sequencing stimulus presentations. For both methodological and conceptual reasons, however, psi researchers have paid much closer attention to issues of randomization.

At the methodological level, the problem is that the random functions included in most computer languages are not very good in that they fail one or more of the mathematical tests used to assess the randomness of a sequence of numbers (L’Ecuyer, 2001), such as Marsaglia’s rigorous Diehard Battery of Tests of Randomness (1995). Such random functions are sometimes called pseudo random number generators (PRNGs) because they [are] not random in the sense of being indeterminate because once the initial starting number (the seed) is set, all future numbers in the sequence are fully determined.”

Well, pseudo-random generators included in all modern computer languages that I know have passed tests like diehard. It would be immensely useful to learn of counterexamples as those using the corresponding language should be warned!!!

“In contrast, a hardware-based or “true” RNG is based on a physical process, such as radioactive decay or diode noise, and the sequence of numbers is indeterminate in the quantum mechanical sense. This does not in itself guarantee that the resulting sequence of numbers can pass all the mathematical tests of randomness (…) Both Marsaglia’s own PRNG algorithm and the “true” hardware-based Araneus Alea I RNG used in our experiments pass all his diehard tests (…) At the conceptual level, the choice of a PRNG or a hardware-based RNG bears on the interpretation of positive findings. In the present context, it bears on my claim that the experiments reported in this article provide evidence for precognition or retroactive influence.”

There is no [probabilistic] validity in the claim that hardware random generators are more random than pseudo-random ones. Hardware generators may be unpredictable even by the hardware conceptor, but the only way to check they produce generations from a uniform distribution follows exactly the same pattern as for PRNG. And the lack of reproducibility of the outcome makes it impossible to check the reproducibility of the study. But here comes the best part of the story!

“If an algorithm-based PRNG is used to determine the successive left-right positions of the target pictures, then the computer already “knows” the upcoming random number before the participant makes his or her response; in fact, once the initial seed number is generated, the computer implicitly knows the entire sequence of left/right positions. As a result, this information is potentially available to the participant through real-time clairvoyance, permitting us to reject the more extraordinary claim that the direction of the causal arrow has actually been reversed.”

Extraordinary indeed… But not more extraordinary than conceiving that a [psychic] participant in the experiment may “see” the whole sequence of random numbers!

“In contrast, if a true hardware-based RNG is used to determine the left/right positions, the next number in the sequence is indeterminate until it is actually generated by the quantum physical process embedded in the RNG, thereby ruling out the clairvoyance alternative. This argues for using a true RNG to demonstrate precognition or retroactive influence. But alas, the use of a true RNG opens the door to the psychokinesis interpretation: The participant might be influencing the placement of the upcoming target rather than perceiving it, a possibility supported by a body of empirical evidence testing psychokinesis with true RNGs (Radin, 2006, pp.154–160).”

Good! I was just about to make the very same objection! If someone can predict the whole sequence of [extremely long integer] values of a PRNG, it gets hardly any more irrational to imagine that he or she can mentally impact a quantum mechanics event. (And hopefully save Schröninger’s cat in the process.) Obviously, it begs the question as to how a subject could forecast a location of the picture that depends on the random generation but not forecast the result of the random generation.

“Like the clairvoyance interpretation, the psychokinesis interpretation also permits us to reject the claim that the direction of the causal arrow has been reversed. Ironically, the psychokinesis alternative can be ruled out by using a PRNG, which is immune to psychokinesis because the sequence of numbers is fully determined and can even be checked after the fact to confirm that its algorithm has not been perturbed. Over the course of our research program—and within the experiment just reported—we have obtained positive results using both PRNGs and a true RNG, arguably leaving precognition/reversed causality the only nonartifactual interpretation that can account for all the positive results.”

This is getting rather confusing. Avoid using a PRNG for fear the subject infers about the sequence and avoid using a RNG for fear of the subject tempering with the physical generator. An omniscient psychic would be able to hand both types of generators, wouldn’t he or she!?!

“This still leaves open the artifactual alternative that the output from the RNG is producing inadequately randomized sequences containing patterns that fortuitously match participants’ response biases.”

This objection shows how little confidence the author has in the randomness tests he previously mentioned: a proper random generator is not inadequately randomized. And if chance only rather than psychic powers is involved, there is no explanation for the match with the participants’ response. Unless those participants are so clever as to detect the flaws in the generator…

“In the present experiment, this possibility is ruled out by the twin findings that erotic targets were detected significantly more frequently than randomly interspersed nonerotic targets and that the nonerotic targets themselves were not detected significantly more frequently than chance. Nevertheless, for some of the other experiments reported in this article, it would be useful to have more general assurance that there are not patterns in the left/right placements of the targets that might correlate with response biases of participants. For this purpose, Lise Wallach, Professor of Psychology at Duke University, suggested that I run a virtual control experiment using random inputs in place of human participants.”

Absolutely brilliant! This test replacing the participants with random generators has shown that the subjects’ answers do not correspond to an iid sequence from a uniform distribution. It would indeed require great psychic powers to reproduce a perfectly iid U(0,1) sequence! And the participants were warned about the experiment so naturally expected to see patterns in the sequence of placements.

Filed under: Books, Statistics Tagged: Birmingham, DieHard, ESP, hardware random generator, PRNG, pseudo-random generator, random simulation, randomness
Categories: Bayesian Bloggers

ABC@NIPS: call for papers

Xian's Og - Tue, 2014-09-09 07:18

In connection with the previous announcement of ABC in Montréal, a call for papers that came out today:

NIPS 2014 Workshop: ABC in Montreal

December 12, 2014
Montréal, Québec, Canada

Approximate Bayesian computation (ABC) or likelihood-free (LF) methods have developed mostly beyond the radar of the machine learning community, but are important tools for a large segment of the scientific community. This is particularly true for systems and population biology, computational psychology, computational chemistry, etc. Recent work has both applied machine learning models and algorithms to general ABC inference (NN, forests, GPs) and ABC inference to machine learning (e.g. using computer graphics to solve computer vision using ABC). In general, however, there is significant room for collaboration between the two communities.

The workshop will consist of invited and contributed talks, poster spotlights, and a poster session. Rather than a panel discussion we will encourage open discussion between the speakers and the audience!

Examples of topics of interest in the workshop include (but are not limited to):

* Applications of ABC to machine learning, e.g., computer vision, inverse problems
* ABC in Systems Biology, Computational Science, etc
* ABC Reinforcement Learning
* Machine learning simulator models, e.g., NN models of simulation responses, GPs etc.
* Selection of sufficient statistics
* Online and post-hoc error
* ABC with very expensive simulations and acceleration methods (surrogate modeling, choice of design/simulation points)
* ABC with probabilistic programming
* Posterior evaluation of scientific problems/interaction with scientists
* Post-computational error assessment
* Impact on resulting ABC inference
* ABC for model selection

We invite submissions in NIPS 2014 format with a maximum of 4 pages, excluding references. Anonymity is not required. Relevant works that have been recently published or presented elsewhere are allowed, provided that previous publications are explicitly acknowledged. Please submit papers in PDF format to .
This workshop has been endorsed by ISBA. As part of their sponsorship, ISBA will be awarding a limited number of travel awards to PhD students and young researchers. The organizing committee may nominate particularly strong submissions for this award.

In addition to the general ISBA endorsement, ABC in Montréal has been endorsed by the BayesComp section of ISBA.

Important Dates:
Submission Deadline: October 9, 2014
Author Notification: October 26, 2014
Workshop: December 12 or 13, 2014

Invited Speakers:
Michael Blum, Laboratoire TIMC-IMAG, Grenoble
Juliane Liepe, Imperial College London
Vikash Mansinghka, MIT
Frank Wood, Oxford

Neil Lawrence, University of Sheffield
Ted Meeds, University of Amsterdam
Christian Robert, Université Paris-Dauphine
Max Welling, University of Amsterdam
Richard Wilkinson, University of Nottingham

The organizers can be contacted at

Filed under: Statistics, Travel, University life Tagged: ABC, BayesComp, Canada, ISBA@NIPS, likelihood-free methods, machine learning, Montréal, NIPS 2014, Québec, simulation
Categories: Bayesian Bloggers

single variable transformation approach to MCMC

Xian's Og - Mon, 2014-09-08 17:14

I read the newly arXived paper “On Single Variable Transformation Approach to Markov Chain Monte Carlo” by Dey and Bhattacharya on the pleasant train ride between Bristol and Coventry last weekend. The paper actually follows several earlier papers by the authors that I have not read in detail. The notion of single variable transform is to add plus or minus the same random noise to all components of the current value of the Markov chain, instead of the standard d-dimensional random walk proposal of the reference Metropolis-Hastings algorithm, namely all proposals are of the form

meaning the chain proceeds [after acceptance] along one and only one of the d diagonals. The authors’ arguments are that (a) the proposal is cheaper and (b) the acceptance rate is higher. What I find questionable in this argument is that this does not directly matter in the evaluation of the performances of the algorithm. For instance, higher acceptance in a Metropolis-Hasting algorithm does not imply faster convergence and smaller asymptotic variance. (This goes without mentioning the fact that the comparative Figure 1 is so variable with the dimension as to be of limited worth. Figure 1 and 2 are also found in an earlier arXived paper of the authors.) For instance, restricting the moves along the diagonals of the Euclidean space implies that there is a positive probability to make two successive proposals along the same diagonal, which is a waste of time. When considering the two-dimensional case, joining two arbitrary points using an everywhere positive density g upon ε means generating two successive values from g, which is equivalent cost-wise to generating a single noise from a two-dimensional proposal. Without the intermediate step of checking the one-dimensional move along one diagonal. So much for a gain. In fine, the proposal found in this paper sums up as being a one-at-a-time version of a standard random walk Metropolis-Hastings algorithm.

Filed under: Books, Statistics, Travel Tagged: arXiv, asymptotic variance, Metropolis-Hastings, mixing speed, random walk
Categories: Bayesian Bloggers