**Q**i Liu, Anindya Bhadra, and William Cleveland from Purdue have arXived a paper entitled *Divide and Recombine for Large and Complex Data: Model Likelihood Functions using MCMC*. Which is a variation on the earlier divide & … papers attempting at handling large datasets. The beginning is quite similar to these earlier papers in that the likelihood is split into sub-likelihoods, approximated from MCMC samples and recombined into an approximate full likelihood. As in for instance Scott et al. one approximation use for the subsample is to replace the likelihood with a Normal approximation, or a skew Normal generalisation, which remains a limited choice for heavy tailed likelihoods. Producing a Normal and skew-Normal approximation for the whole [data] likelihood, respectively. If I understand correctly, these approximations are missing a normalising constant to bring them to scale with the true likelihood, which I do not completely understand as the likelihood only needs to be defined up to a [constant] constant for most purposes, including Bayesian ones. The method of estimation of this constant proposed therein is called the *contour probability algorithm* and it consists in using a highest density region to compare a likelihood and its approximation. (Nothing to do with our adaptation of Gelfand and Dey (1994) based on HPDs, with Darren Wright. Nor with nested sampling.) Returning a form of qq-plot. This is rather exploratory, while hardly addressing the issue of the precision of such approximations and the resolution of conflicting proposals. And the comparison with all these other recent proposals for splitting likelihoods into manageable bits (proposals that are mentioned in the final section, including our recentering scheme with my student Changye Wu).

## Archive for nested sampling

## divide & reconquer

Posted in Books, Statistics, University life with tags arXiv, contour, contour algorithm, divide-and-conquer strategy, harmonic mean estimator, HPD region, large data problems, nested sampling, Purdue University, skewed distribution, sub-likelihood on February 5, 2018 by xi'an## WBIC, practically

Posted in Statistics with tags Bayes factor, Bayesian model selection, evidence, harmonic mean estimator, MCMC, nested sampling, Pima Indians, power posterior, thermodynamic integration, WBIC on October 20, 2017 by xi'an

“Thus far, WBIC has received no more than a cursory mention by Gelman et al. (2013)”

**I** had missed this 2015 paper by Nial Friel and co-authors on a practical investigation of Watanabe’s WBIC. Where WBIC stands for widely applicable Bayesian information criterion. The thermodynamic integration approach explored by Nial and some co-authors for the approximation of the evidence, thermodynamic integration that produces the log-evidence as an integral between temperatures t=0 and t=1 of a powered evidence, is eminently suited for WBIC, as the widely applicable Bayesian information criterion is associated with the specific temperature t⁰ that makes the power posterior equidistant, Kullback-Leibler-wise, from the prior and posterior distributions. And the expectation of the log-likelihood under this very power posterior equal to the (genuine) evidence. In fact, WBIC is often associated with the sub-optimal temperature 1/log(n), where n is the (effective?) sample size. (By comparison, if my minimalist description is unclear!, thermodynamic integration requires a whole range of temperatures and associated MCMC runs.) In an ideal Gaussian setting, WBIC improves considerably over thermodynamic integration, the larger the sample the better. In more realistic settings, though, including a simple regression and a logistic [Pima Indians!] model comparison, thermodynamic integration may do better for a given computational cost although the paper is unclear about these costs. The paper also runs a comparison with harmonic mean and nested sampling approximations. Since the integral of interest involves a power of the likelihood, I wonder if a safe version of the harmonic mean resolution can be derived from simulations of the genuine posterior. Provided the exact temperature t⁰ is known…

## Astrostatistics school

Posted in Mountains, pictures, R, Statistics, Travel, University life with tags ABC, ABC model choice, abcrf, abctools package, Alps, astronomy, Autrans, Bayesian inference, Bayesian Methods in Cosmology, big wall, cosmology, Dickey-Savage ratio, Fall, mountains, nested sampling, R, random forests, rock climbing, RStudio, socks, trail running, Vercors on October 17, 2017 by xi'an**W**hat a wonderful week at the Astrostat [Indian] summer school in Autrans! The setting was superb, on the high Vercors plateau overlooking both Grenoble [north] and Valence [west], with the colours of the Fall at their brightest on the foliage of the forests rising on both sides of the valley and a perfect green on the fields at the centre, with sun all along, sharp mornings and warm afternoons worthy of a late Indian summer, too many running trails [turning into X country ski trails in the Winter] to contemplate for a single week [even with three hours of running over two days], many climbing sites on the numerous chalk cliffs all around [but a single afternoon for that, more later in another post!]. And of course a group of participants eager to learn about Bayesian methodology and computational algorithms, from diverse [astronomy, cosmology and more] backgrounds, trainings and countries. I was surprised at the dedication of the participants travelling all the way from Chile, Péru, and Hong Kong for the sole purpose of attending the school. David van Dyk gave the first part of the school on Bayesian concepts and MCMC methods, Roberto Trotta the second part on Bayesian model choice and hierarchical models, and myself a third part on, surprise, surprise!, approximate Bayesian computation. Plus practicals on R.

As it happens Roberto had to cancel his participation and I turned for a session into Christian Roberto, presenting his slides in the most objective possible fashion!, as a significant part covered nested sampling and Savage-Dickey ratios, not exactly my favourites for estimating constants. David joked that he was considering postponing his flight to see me talk about these, but I hope I refrained from engaging into controversy and criticisms… If anything because this was not of interest for the participants. Indeed when I started presenting ABC through what I thought was a pedestrian example, namely Rasmus Baath’s socks, I found that the main concern was not running an MCMC sampler or a substitute ABC algorithm but rather an healthy questioning of the construction of the informative prior in that artificial setting, which made me quite glad I had planned to cover this example rather than an advanced model [as, e.g., one of those covered in the packages abc, abctools, or abcrf]. Because it generated those questions about the prior [why a Negative Binomial? why these hyperparameters? &tc.] and showed how programming ABC turned into a difficult exercise even in this toy setting. And while I wanted to give my usual warning about ABC model choice and argue for random forests as a summary selection tool, I feel I should have focussed instead on another example, as this exercise brings out so clearly the conceptual difficulties with what is taught. Making me quite sorry I had to leave one day earlier. [As did missing an extra run!] Coming back by train through the sunny and grape-covered slopes of Burgundy hills was an extra reward [and no one in the train commented about the local cheese travelling in my bag!]

## and another one on nested sampling

Posted in Books, Statistics with tags adaptive Monte Carlo algorithm, efficient importance sampling, Monte Carlo Statistical Methods, nested sampling on May 2, 2017 by xi'an**T**he same authors as those of the paper discussed last week arXived a paper on dynamic nested sampling.

“We propose modifying the nested sampling algorithm by dynamically varying the number of “live points” in order to maximise the accuracy of a calculation for some number of posterior sample.”

Some of the material is actually quite similar to the previous paper (to the point I had to check they were not the *same* paper). The authors rightly point out that the main source of variation in the nested sampling approximation is due to the Monte Carlo variability in the estimated volume of the level sets.

The main notion in that paper is that it is acceptable to have a varying number of “live” points in nested sampling provided the weights are correctly accordingly. Adding more of those points as a new “thread” in a region where the likelihood changes rapidly. Addition may occur at any level of the likelihood, in fact, and is determined in the paper by an importance weight being in the upper tail of the importance weights… While the description is rather vague [for instance I do not get the notation in (9)] and the criteria for adding threads somewhat arbitrary, I find interesting that several passes at different precision levels can improve the efficiency of the nested approximation at a given simulation cost. Remains the issue of whether or not this is a sufficient perk for attracting users of other simulation techniques to the nested galaxy…

## marginal likelihoods from MCMC

Posted in Books, pictures, Statistics, University life with tags ABC, arXiv, Bayesian Methods in Cosmology, curse of dimensionality, evidence, INLA, k-nearest neighbour, marginal likelihood, nested sampling, Planck experiment, San Antonio, satellite on April 26, 2017 by xi'an**A** new arXiv entry on ways to approximate marginal likelihoods based on MCMC output, by astronomers (apparently). With an application to the 2015 Planck satellite analysis of cosmic microwave background radiation data, which reminded me of our joint work with the cosmologists of the Paris Institut d’Astrophysique ten years ago. In the literature review, the authors miss several surveys on the approximation of those marginals, including our San Antonio chapter, on Bayes factors approximations, but mention our ABC survey somewhat inappropriately since it is not advocating the use of ABC for such a purpose. (They mention as well variational Bayes approximations, INLA, powered likelihoods, if not nested sampling.)

The proposal of this paper is to identify the marginal *m* [actually denoted *a* there] as the normalising constant of an unnormalised posterior density. And to do so the authors estimate the posterior by a non-parametric approach, namely a k-nearest-neighbour estimate. With the additional twist of producing a sort of Bayesian posterior on the constant *m*. [And the unusual notion of number density, used for the unnormalised posterior.] The Bayesian estimation of m relies on a Poisson sampling assumption on the k-nearest neighbour distribution. (Sort of, since k is actually fixed, not random.)

If the above sounds confusing and imprecise it is because I am myself rather mystified by the whole approach and find it difficult to see the point in this alternative. The Bayesian numerics does not seem to have other purposes than producing a MAP estimate. And using a non-parametric density estimate opens a Pandora box of difficulties, the most obvious one being the curse of dimension(ality). This reminded me of the commented paper of Delyon and Portier where they achieve super-efficient convergence when using a kernel estimator, but with a considerable cost and a similar sensitivity to dimension.

## seeking the error in nested sampling

Posted in pictures, Statistics, Travel with tags Berlin, curse of dimensionality, error assessment, John Skilling, Monte Carlo error, nested sampling, Nicolas Chopin on April 13, 2017 by xi'an**A** newly arXived paper on the error in nested sampling, written by Higson and co-authors, and read in Berlin, looks at the difficult task of evaluating the sampling error of nested sampling. The conclusion is essentially negative in that the authors recommend multiple runs of the method to assess the magnitude of the variability of the output by bootstrap, i.e. to call for the most empirical approach…

The core of this difficulty lies in the half-plug-in, half-quadrature, half-Monte Carlo (!) feature of the nested sampling algorithm, in that (i) the truncation of the unit interval is based on a expectation of the mass of each shell (i.e., the zone between two consecutive isoclines of the likelihood, (ii) the evidence estimator is a quadrature formula, and (iii) the level of the likelihood at the truncation is replaced with a simulated value that is not even unbiased (and correlated with the previous value in the case of an MCMC implementation). As discussed in our paper with Nicolas, the error in the evidence approximation is of the same order as other Monte Carlo methods in that it gets down like the square root of the number of terms at each iteration. Contrary to earlier intuitions that focussed on the error due to the quadrature.

But the situation is much less understood when the resulting sample is used for estimation of quantities related with the posterior distribution. With no clear approach to assess and even less correct the resulting error, since it is not solely a Monte Carlo error. As noted by the authors, the quadrature approximation to the univariate integral replaces the unknown prior weight of a shell with its Beta order statistic expectation *and* the average of the likelihood over the shell with a single (uniform???) realisation. Or the mean value of a transform of the parameter with a single (biased) realisation. Since most posterior expectations can be represented as integrals over likelihood levels of the average value over an iso-likelihood contour. The approach advocated in the paper involved multiple threads of an “unwoven nested sampling run”, which means launching n nested sampling runs with one living term from the n currents living points in the current nested sample. (Those threads may then later be recombined into a single nested sample.) This is the starting point to a nested flavour of bootstrapping, where threads are sampled with replacement, from which confidence intervals and error estimates can be constructed. (The original notion appears in Skilling’s 2006 paper, but I missed it.)

The above graphic is an attempt within the paper at representing the (marginal) posterior of a transform f(θ). That I do not fully understand… The notations are rather horrendous as X is not the data but the prior probability for the likelihood to be above a given bound which is actually the corresponding quantile. (There is no symbol for data and £ is used for the likelihood function as well as realisations of the likelihood function…) A vertical slice on the central panel gives the posterior distribution of f(θ) given the event that the likelihood is in the corresponding upper tail. Or given the corresponding shell (?).

## nested sampling for philogenies

Posted in Statistics with tags estimating constants, evidence, harmonic mean estimator, importance sampling, nested sampling, normalising constant, philogenetic trees, philogenetics on March 3, 2017 by xi'an

“Methods to estimate the marginal likelihood should be sensitive to the prior choice. Non-informative priors should increase the contribution of low-likelihood regions of parameter space in the estimated marginal likelihood. Consequently, the prior choice should affect the estimated evidence.”

In a most recent arXival, Maturana, Brewer, and Klaere discuss of the appeal of nested sampling for conducting model choice in philogenetic models. In comparison with the “generalized steppingstone sampling” method, which represents the evidence as a product of ratios of evidences (Fan et al., 2011). And which I do not think I have previously met, with all references provided therein relating to Bayesian philogenetics, apparently. The stepping stone approach relies on a sequence of tempered targets, moving from a reference distribution to the real target as a temperature β goes from zero to one. (The paper also mentions thermodynamic integration as too costly.) Nested sampling—much discussed on this blog!—is presented in this paper as having the ability to deal with partly convex likelihoods, although I do not really get how or why. (As there is nothing new in the fairly pedagogical pretentation of nested sampling therein.) Nothing appears to be mentioned about the difficulty to handle multimodal as high likelihood isolated regions are unlikely to be sampled from poorly weighted priors (by which I mean that a region with significant likelihood mass is unlikely to get sampled if the prior distribution gives little prior weight to that region). The novelty in the paper is to compare nested sampling with generalized steppingstone sampling and path sampling on several phylogenetic examples. I did not spot computing time mentioned there. As usual with examples, my reservation is that the conclusions drawn for one particular analysis of one (three) particular example(s) does not convey a general method about the power and generality of a method. Even though I acknowledge that nested sampling is on principle operational in wide generality.