## Archive for particle filter

## no filter [no jatp]

Posted in Kids, Mountains, pictures with tags California, climate change, Concord, forest fires, getty images, global warming, jatp, Limeridge, particle filter, red sun, The New Abnormal, USA on September 14, 2020 by xi'an## IMS workshop [day 4]

Posted in pictures, Statistics, Travel, University life with tags Feynman-Kac formalism, National University Singapore, NUS, optimal transport, particle filter, particle filters, particle MCMC, pseudo-marginal MCMC, sunrise, transportation model on August 31, 2018 by xi'an **W**hile I did not repeat the mistake of yesterday morning, just as well because the sun was unbearably strong!, I managed this time to board a bus headed in the wrong direction and as a result went through several remote NUS campi! Missing the first talk of the day as a result. By Youssef Marzouk, with a connection between sequential Monte Carlo and optimal transport. Transport for sampling, that is. The following talk by Tiangang Cui was however related, with Marzouk a co-author, as it aimed at finding linear transforms towards creating Normal approximations to the target to be used as proposals in Metropolis algorithms. Which may sound like something already tried a zillion times in the MCMC literature, except that the setting was rather specific to some inverse problems, imposing a generalised Normal structure on the transform, then optimised by transport arguments. It is unclear to me [from just attending the talk] how complex this derivation is and how dimension steps in, but the produced illustrations were quite robust to an increase in dimension.

The remaining talks for the day were mostly particular, from Anthony Lee introducing a new and almost costless way of producing variance estimates in particle filters, exploiting only the ancestry of particles, to Mike Pitt discussing the correlated pseudo-marginal algorithm developed with George Deligiannidis and Arnaud Doucet. Which somewhat paradoxically managed to fight the degeneracy [i.e., the need for a number of terms increasing like the time index T] found in independent pseudo-marginal resolutions, moving down to almost log(T)… With an interesting connection to the quasi SMC approach of Mathieu and Nicolas. And Sebastian Reich also stressed the links with optimal transport in a talk about data assimilation that was way beyond my reach. The day concluded with fireworks, through a magistral lecture by Professeur Del Moral on a continuous time version of PMCMC using the Feynman-Kac terminology. Pierre did a superb job during his lecture towards leading the whole room to the conclusion.

## 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.

## MCMC with multiple tries

Posted in Books, pictures, Statistics, University life with tags All Blacks, delayed acceptance, ensemble Monte Carlo, MCMC, Monte Carlo Statistical Methods, multiple-try Metropolis algorithm, particle filter, population Monte Carlo, rugby, survey on April 5, 2018 by xi'an**E**arlier this year, Luca Martino wrote and arXived a review on multiple try MCMC. As its name suggests, the starting point of this algorithm is to propose N potential moves simultaneously instead of one, possibly according to N different proposal (conditional) densities, and to select one by a normalised importance sampling weight. The move is accepted by a Metropolis-Hastings step based on the ratio of the normalisation constants [at the current and at the one-before-current stages]. Besides the cost of computing the summation and generating the different variates, this method also faces the drawback of requiring N-1 supplementary simulations that are only used for achieving detailed balance and computing a backward summation of importance weights. (A first section of the review is dedicated to independent Metropolis-Hastings proposals, q(θ), which make life simpler, but are less realistic in my opinion since some prior knowledge or experimentation is necessary to build a relevant distribution q(θ).) An alternative covered in the survey is ensemble Monte Carlo (Neal, 2011), which produces a whole sample at each iteration, with target the product of the initial targets. This reminded me of our pinball sampler, which aimed at producing a spread-out sample while keeping the marginal correct. Although the motivation sounds closer to a particle sampler. Especially with this associated notion of an empirical approximation of the target. The next part of the review is about delayed rejection, which is a natural alternative approach to speeding up MCMC by considering several possibilities, if sequentially. Started in Antonietta Mira‘s 1999 PhD thesis. The difficulty with this approach is that the acceptance probability gets increasingly complex as the number of delays grows, which may annihilate its appeal relative to simultaneous multiple tries.