**A** follow-up episode to the SlideShare m’a tuer [sic] saga: After the 20 November closure of my xianblog account and my request for an explanation, I was told by Linkedin that a complaint has been made about one of my talks for violation of copyright. Most surprisingly, at least at first, it was about the slides for the graduate lectures I gave ten years ago at CREST on (re)reading Jaynes’ Probability Theory. While the slides contain a lot of short quotes from the Logic of Science, somewhat necessarily since I discuss the said book, there are also many quotes from Jeffreys’ Theory of Probability and “t’is but a scratch” on the contents of this lengthy book… Plus, the pdf file appears to be accessible on several sites, including one with an INRIA domain. Since I had to fill a “Counter-Notice of Copyright Infringement” to unlock the rest of the depository, I just hope no legal action is going to be taken about this lecture. But I remain puzzled at the reasoning behind the complaint, unwilling to blame radical Jaynesians for it! As an aside, here are the registered 736 views of the slides for the past year:

## Archive for Og

## unbiased consistent nested sampling via sequential Monte Carlo [a reply]

Posted in pictures, Statistics, Travel with tags auxiliary variable, Brisbane, evidence, marginal likelihood, nested sampling, Og, particle filter, QUT, unbiasedness on June 13, 2018 by xi'an*Rob Salomone sent me the following reply on my comments of yesterday about their recently arXived paper.*

“Which never occurred as the number one difficulty there, as the simplest implementation runs a Markov chain from the last removed entry, independently from the remaining entries. Even stationarity is not an issue sinceI believe that the first occurrence within the level set is distributed from the constrained prior.”

“And then, in a twist that is not clearly explained in the paper, the focus moves to an improved nested sampler that moves one likelihood value at a time, with a particle step replacing a singleparticle. (Things get complicated when several particles may take the very same likelihood value, but randomisation helps.) At this stage the algorithm is quite similar to the original nested sampler. Except for the unbiased estimation of the constants, thefinal constant, and the replacement of exponential weights exp(-t/N) by powers of (N-1/N)”

**is**a special case of SMC (with the weights replaced with a suboptimal choice).

## unbiased consistent nested sampling via sequential Monte Carlo

Posted in pictures, Statistics, Travel with tags auxiliary variable, Brisbane, evidence, marginal likelihood, nested sampling, Og, particle filter, QUT, unbiasedness on June 12, 2018 by xi'an

“Moreover, estimates of the marginal likelihood are unbiased.” (p.2)

Rob Salomone, Leah South, Chris Drovandi and Dirk Kroese (from QUT and UQ, Brisbane) recently arXived a paper that frames the nested sampling in such a way that marginal likelihoods can be unbiasedly (and consistently) estimated.

“Why isn’t nested sampling more popular with statisticians?” (p.7)

A most interesting question, especially given its popularity in cosmology and other branches of physics. A first drawback pointed out in the c is the requirement of independence between the elements of the sample produced at each iteration. Which never occurred as the number one difficulty there, as the simplest implementation runs a Markov chain from the last removed entry, independently from the remaining entries. Even stationarity is not an issue since I believe that the first occurrence within the level set is distributed from the constrained prior.

A second difficulty is the use of quadrature which turns integrand into step functions at random slices. Indeed, mixing Monte Carlo with numerical integration makes life much harder, as shown by the early avatars of nested sampling that only accounted for the numerical errors. (And which caused Nicolas and I to write our critical paper in Biometrika.) There are few studies of that kind in the literature, the only one I can think of being [my former PhD student] Anne Philippe‘s thesis twenty years ago.

The third issue stands with the difficulty in parallelising the method. Except by jumping k points at once, rather than going one level at a time. While I agree this makes life more complicated, I am also unsure about the severity of that issue as k nested sampling algorithms can be run in parallel and aggregated in the end, from simple averaging to something more elaborate.

The final blemish is that the nested sampling estimator has a stopping mechanism that induces a truncation error, again maybe a lesser problem given the overall difficulty in assessing the total error.

The paper takes advantage of the ability of SMC to produce unbiased estimates of a sequence of normalising constants (or of the normalising constants of a sequence of targets). For nested sampling, the sequence is made of the prior distribution restricted to an embedded sequence of level sets. With another sequence restricted to bands (likelihood between two likelihood boundaries). If all restricted posteriors of the second kind and their normalising constant are known, the full posterior is known. Apparently up to the main normalising constant, i.e. the marginal likelihood., *ℨ*, except that it is also the sum of all normalising constants. Handling this sequence by SMC addresses the four concerns of the four authors, apart from the truncation issue, since the largest likelihood bound need be set for running the algorithm.

When the sequence of likelihood bounds is chosen based on the observed likelihoods so far, the method becomes adaptive. Requiring again the choice of a stopping rule that may induce bias if stopping occurs too early. And then, in a twist that is not clearly explained in the paper, the focus moves to an improved nested sampler that moves one likelihood value at a time, with a particle step replacing a single particle. (Things get complicated when several particles may take the very same likelihood value, but randomisation helps.) At this stage the algorithm is quite similar to the original nested sampler. Except for the unbiased estimation of the constants, the final constant, and the replacement of exponential weights exp(-t/N) by powers of (N-1/N).

The remainder of this long paper (61 pages!) is dedicated to practical implementation, calibration and running a series of comparisons. A nice final touch is the thanks to the ‘Og for its series of posts on nested sampling, which “helped influence this work, and played a large part in inspiring it.”

In conclusion, this paper is certainly a worthy exploration of the nested sampler, providing further arguments towards a consistent version, with first and foremost an (almost?) unbiased resolution. The comparison with a wide range of alternatives remains open, in particular time-wise, if evidence is the sole target of the simulation. For instance, the choice of this sequence of targets in an SMC may be improved by another sequence, since changing one particle at a time does not sound efficient. The complexity of the implementation and in particular of the simulation from the prior under more and more stringent constraints need to be addressed.