## coupled filters

Posted in Kids, Statistics, University life with tags , , , , , , , , , on July 11, 2016 by xi'an

Pierre Jacob, Fredrik Lindsten, and Thomas Schön recently arXived a paper on coupled particle filters. A coupling problem that proves to be much more complicated than expected, due to the discrete nature of particle filters. The starting point of the paper is the use of common (e.g., uniform) random numbers for the generation of each entry in the particle system at each time t, which maximal correlation gets damaged by the resampling steps (even when using the same uniforms). One suggestion for improving the correlation between entries at each time made in the paper is to resort to optimal transport, using the distance between particles as the criterion. A cheaper alternative is inspired from multi-level Monte Carlo. It builds a joint multinomial distribution by optimising the coupling probability. [Is there any way to iterate this construct instead of considering only the extreme cases of identical values versus independent values?] The authors also recall a “sorted sampling” method proposed by Mike Pitt in 2002, which is to rely on the empirical cdfs derived from the particle systems and on the inverse cdf technique, which is the approach I would have first considered. Possibly with a smooth transform of both ecdf’s in order to optimise the inverse cdf move.  Actually, I have trouble with the notion that the ancestors of a pair of particles should matter. Unless one envisions a correlation of the entire path, but I am ensure how one can make paths correlated (besides coupling). And how this impacts likelihood estimation. As shown in the above excerpt, the coupled approximations produce regular versions and, despite the negative bias, fairly accurate evaluations of likelihood ratios, which is all that matters in an MCMC implementation. The paper also proposes a smoothing algorithm based on Rhee and Glynn (2012) debiasing technique, which operates on expectations against the smoothing distribution (conditional on a value of the parameter θ). Which may connect with the notion of simulating correlated paths. The interesting part is that, due to the coupling, the Rhee and Glynn unbiased estimator has a finite (if random) stopping time.

## twisted filters

Posted in Statistics with tags , , , , , , , , , , on October 6, 2012 by xi'an

Nick Witheley (Bristol) and Anthony Lee (Warwick) just posted an interesting paper called ‘Twisted particle filters‘ on arXiv. (Presumably unintentionally, the title sounds like Twisted Sister, pictured above, even though I never listened to this [particularly] heavy kind of hard rock! Twisting is customarily used in the large deviation literature.)

The twisted particle paper studies the impact of the choice of something similar to, if subtly different from, an importance function on the approximation of the marginal density (or evidence) for HMMS. (In essence, the standard particle filter is modified for only one particle in the population.) The core of the paper is to compare those importance functions in a fixed N-large n setting. As in simpler importance sampling cases, there exists an optimal if impractical choice of importance function, leading to a zero variance estimator of the evidence. Nick and Anthony produce an upper bound on the general variance as well. One of the most appealing features of the paper is that the authors manage a convergence result in n rather than N. (Although the algorithms are obviously validated in the more standard large N sense.)

The paper is quite theoretical and dense (I was about to write heavy in connection with the above!), with half its length dedicated to proofs. It relies on operator theory, with eigen-functions behind the optimal filter, while not unrelated with Pierre Del Moral’s works. (It took me a while to realise that the notation ω was not the elemental draw from the σ-algebra but rather the realisation of the observed sequence! And I had to keep recalling that θ was the lag operator and not a model parameter [there is no model parameter].)