**I**n connection with the talk I gave last July in Rennes for MCqMC 2018, I posted yesterday a preprint on arXiv of the work that my [soon to defend!] Dauphine PhD student Changye Wu and I did on an alternative PDMP. In this novel avatar of the zig-zag sampler, a non-reversible, continuous-time MCMC sampler, that we called the Coordinate Sampler, based on a piecewise deterministic Markov process. In addition to establishing the theoretical validity of this new sampling algorithm, we show in the same line as Deligiannidis et al. (2018) that the Markov chain it induces exhibits geometrical ergodicity for distributions which tails decay at least as fast as an exponential distribution and at most as fast as a Gaussian distribution. A few numerical examples (a 2D banana shaped distribution à la Haario et al., 1999, strongly correlated high-dimensional normals, a log-Gaussian Cox process) highlight that our coordinate sampler is more efficient than the zig-zag sampler, in terms of effective sample size.Actually, we had sent this paper before the summer as a NIPS [2018] submission, but it did not make it through [the 4900 submissions this year and] the final review process, being eventually rated above the acceptance bar but not that above!

## Archive for Rennes

## coordinate sampler as a non-reversible Gibbs-like MCMC sampler

Posted in Books, Kids, Statistics, University life with tags arXiv, Cox process, MCqMC 2018, NIPS 2018, PDMP, PhD students, Rennes, Université Paris Dauphine, Zig-Zag on September 12, 2018 by xi'an## IMS workshop [day 3]

Posted in pictures, R, Statistics, Travel, University life with tags Bayesian computation, Birch, delayed simulation, high dimensions, hypocoercivity, IMS, Institute for Mathematical Sciences, Lapland, MCqMC 2018, National University Singapore, non-reversible diffusion, NUS, ODE, partly deterministic processes, probabilistic programming, Rao-Blackwellisation, Rennes, Singapore, Wang-Landau algorithm, workshop on August 30, 2018 by xi'an**I** made the “capital” mistake of walking across the entire NUS campus this morning, which is quite green and pretty, but which almost enjoys an additional dimension brought by such an intense humidity that one feels having to get around this humidity!, a feature I have managed to completely erase from my memory of my previous visit there. Anyway, nothing of any relevance. oNE talk in the morning was by Markus Eisenbach on tools used by physicists to speed up Monte Carlo methods, like the Wang-Landau flat histogram, towards computing the partition function, or the distribution of the energy levels, definitely addressing issues close to my interest, but somewhat beyond my reach for using a different language and stress, as often in physics. (I mean, as often in physics talks I attend.) An idea that came out clear to me was to bypass a (flat) histogram target and aim directly at a constant slope cdf for the energy levels. (But got scared away by the Fourier transforms!)

Lawrence Murray then discussed some features of the Birch probabilistic programming language he is currently developing, especially a fairly fascinating concept of delayed sampling, which connects with locally-optimal proposals and Rao Blackwellisation. Which I plan to get back to later [and hopefully sooner than later!].

In the afternoon, Maria de Iorio gave a talk about the construction of nonparametric priors that create dependence between a sequence of functions, a notion I had not thought of before, with an array of possibilities when using the stick breaking construction of Dirichlet processes.

And Christophe Andrieu gave a very smooth and helpful entry to partly deterministic Markov processes (PDMP) in preparation for talks he is giving next week for the continuation of the workshop at IMS. Starting with the guided random walk of Gustafson (1998), which extended a bit later into the non-reversible paper of Diaconis, Holmes, and Neal (2000). Although I had a vague idea of the contents of these papers, the role of the velocity **ν** became much clearer. And premonitory of the advances made by the more recent PDMP proposals. There is obviously a continuation with the equally pedagogical talk Christophe gave at MCqMC in Rennes two months [and half the globe] ago, but the focus being somewhat different, it really felt like a new talk [my short term memory may also play some role in this feeling!, as I now remember the discussion of Hilderbrand (2002) for non-reversible processes]. An introduction to the topic I would recommend to anyone interested in this new branch of Monte Carlo simulation! To be followed by the most recently arXived hypocoercivity paper by Christophe and co-authors.

## indecent exposure

Posted in Statistics with tags ABC, Bayesian optimisation, Bretagne, Brittany, exponential families, image analysis, image processing, inference, Lugano, maximum likelihood estimation, MCqMC 2018, pre-processing, Rennes on July 27, 2018 by xi'an**W**hile attending my last session at MCqMC 2018, in Rennes, before taking a train back to Paris, I was confronted by this radical opinion upon our previous work with Matt Moores (Warwick) and other coauthors from QUT, where the speaker, Maksym Byshkin from Lugano, defended a new approach for maximum likelihood estimation using novel MCMC methods. Based on the point fixe equation characterising maximum likelihood estimators for exponential families, when theoretical and empirical moments of the natural statistic are equal. Using a Markov chain with stationary distribution the said exponential family, the fixed point equation can be turned into a zero divergence equation, requiring simulation of pseudo-data from the model, which depends on the unknown parameter. Breaking this circular argument, the authors note that simulating pseudo-data that reproduce the observed value of the sufficient statistic is enough. Which is related with Geyer and Thomson (1992) famous paper about Monte Carlo maximum likelihood estimation. From there I was and remain lost as I cannot see why a derivative of the expected divergence with respect to the parameter θ can be computed when this divergence is found by Monte Carlo rather than exhaustive enumeration. And later used in a stochastic gradient move on the parameter θ… Especially when the null divergence is imposed on the parameter. In any case, the final slide shows an application to a large image and an Ising model, solving the problem (?) in 140 seconds and suggesting indecency, when our much slower approach is intended to produce a complete posterior simulation in this context.

## Rennes street art [#3]

Posted in pictures, Running, Travel with tags Brittany, graffitis, MCqMC 2018, Rennes, street art on July 24, 2018 by xi'an## unusual clouds [jatp]

Posted in pictures, Travel, Wines with tags ÌPA, beer, Brittany, Cité d'Ys, clouds, jatp, MCqMC 2018, Rennes, summer light, sunset, thunderstorm on July 19, 2018 by xi'an## subset sampling

Posted in Statistics with tags MCMC algorithms, MCqMC 2018, nested sampling, Probabilistic Engineering Mechanics, random walk, Rennes, subset sampling, tail events, tail probabilities on July 13, 2018 by xi'an**A** paper by Au and Beck (2001) was mentioned during a talk at MCqMC 2018 in Rennes and I checked Probabilistic Engineering Mechanics for details. There is no clear indication that the subset simulation advocated therein is particularly effective. The core idea is to obtain the probability to belong to a small set A by a cascading formula, namely the product of the probability to belong to A¹, then the conditional probability to belong to A² given A¹, &tc. When the subsets A¹, A², …, A constitute a decreasing embedded sequence. The simulation conditional on being in one of the subsets is operated by a random-walk Metropolis-within-Gibbs scheme, with an additional rejection when the value is not in the said subset. (Surprisingly, the authors re-establish the validity of this scheme.) Hence the proposal faces similar issues as nested sampling, except that the nested subsets here are defined quite differently as they are essentially free, provided they can be easily evaluated. Each of the random walks need be scaled, the harder a task because this depends on the corresponding subset volume. The subsets themselves are rarely defined in a natural manner, except when being tail events. And need to be calibrated so that the conditional probability of falling into each remains large enough, the cost of free choice. The Markov chain on the previous subset can prove useful to build the next subset , but there is no general principle behind this remark. (If any, this is connected with X entropy.) But else, the past chains are very much wasted, compared with, say, an SMC treatment of the problem. The paper also notices that starting a Markov chain in the set means there is no burnin time and hence that the probability estimators are thus unbiased. (This creates a correlation between successive Markov chains, but I think it could be ignored if the starting point was chosen at random or after a random number of extra steps.) The authors further point out that the chain may fail to be ergodic, if the proposal distribution lacks energy to link connected regions of the current subset . They suggest using multiple chains with multiple starting points, which alleviates the issue only to some extent, as it ultimately depends on the spread of the starting points. As acknowledged in the paper.

## a thread to bin them all [puzzle]

Posted in Books, Kids, R, Travel with tags Brittany, clouds, FiveThirtyEight, mathematical puzzle, meerkats, R, Rennes, sunset, The Riddler on July 9, 2018 by xi'an

**T**he most recent riddle on the Riddler consists in finding the shorter sequence of digits (in 0,1,..,9) such that all 10⁴ numbers between 0 (or 0000) and 9,999 can be found as a group of consecutive four digits. This sequence is obviously longer than 10⁴+3, but how long? On my trip to Brittany last weekend, I wrote an R code first constructing the sequence at random by picking with high preference the next digit among those producing a new four-digit number

tenz=10^(0:3) wn2dg=function(dz) 1+sum(dz*tenz) seqz=rep(0,10^4) snak=wndz=sample(0:9,4,rep=TRUE) seqz[wn2dg(wndz)]=1 while (min(seqz)==0){ wndz[1:3]=wndz[-1];wndz[4]=0 wndz[4]=sample(0:9,1,prob=.01+.99*(seqz[wn2dg(wndz)+0:9]==0)) snak=c(snak,wndz[4]) sek=wn2dg(wndz) seqz[sek]=seqz[sek]+1}

which usually returns a value above 75,000. I then looked through the sequence to eliminate useless replicas

for (i in sample(4:(length(snak)-5))){ if ((seqz[wn2dg(snak[(i-3):i])]>1) &(seqz[wn2dg(snak[(i-2):(i+1)])]>1) &(seqz[wn2dg(snak[(i-1):(i+2)])]>1) &(seqz[wn2dg(snak[i:(i+3)])]>1)){ seqz[wn2dg(snak[(i-3):i])]=seqz[wn2dg(snak[(i-3):i])]-1 seqz[wn2dg(snak[(i-2):(i+1)])]=seqz[wn2dg(snak[(i-2):(i+1)])]-1 seqz[wn2dg(snak[(i-1):(i+2)])]=seqz[wn2dg(snak[(i-1):(i+2)])]-1 seqz[wn2dg(snak[i:(i+3)])]=seqz[wn2dg(snak[i:(i+3)])]-1 snak=snak[-i] seqz[wn2dg(snak[(i-3):i])]=seqz[wn2dg(snak[(i-3):i])]+1 seqz[wn2dg(snak[(i-2):(i+1)])]=seqz[wn2dg(snak[(i-2):(i+1)])]+1 seqz[wn2dg(snak[(i-1):(i+2)])]=seqz[wn2dg(snak[(i-1):(i+2)])]+1}}

until none is found. A first attempt produced 12,911 terms in the sequence. A second one 12,913. A third one 12,871. Rather consistent figures but not concentrated enough to believe in achieving a true minimum. An overnight run produced 12,779 as the lowest value. Checking the answer the week after, it appears that 10⁴+3 *is* the correct answer!