Archive for Poisson equation

BayesComp 2025.2

Posted in Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on June 19, 2025 by xi'an


The main BayesComp²⁵ conference started with Pierre Jacob’s plenary talk on his recent advances on coupling for unbiased MCMC—currently ERC grantee on that topic—.  Raising lazy questions like using a different target or transition kernel for the second chain in the coupling, connecting the Poisson equation and control variates, handling the signed issue with the unbiased approximations. Interestingly, they obtain an unbiased estimator of the asymptotic variance of the unbiased estimator. And a correction for self-normalised importance sampling, which has some connections with our 1996 (?) pinball sampler. Also an evaluation of the median of means, rather than the average of means, which is a thing I had been (lazily) contemplating for a while  (On the greedy side, as I was writing my recovery exam for my Monte Carlo course, I realised the results Pierre presented could be somewhat recycled into exam problems!)

My first parallel session was on gradient-based methods with a talk by Francesca Crucinio on proximal particle Langevin algorithms (similar to the one she gave in PariSanté last year), a talk by Zhihao Wang on stereographic multiple try Metropolis(-Rosenbluth-Teller) that unsurprisingly recovers ergodicity thanks to the compactness of the ball. For which I wonder why a Normal proposal makes complete sense since one could consider a mover after the projection instead and why iid rather than repelling multiple proposals are used… The last speaker was just out from the plane from California, Siddharth Vishwanath who spoke about repelling-attracting HMC. With very nice animations of HMC, if reaching the main point of using both negative and positive frictions a few minutes before the session finished. The method preserves volume and potential, if not energy.

Speaking of which (energy), I find myself struggling with my less than 6 hours of sleep since arrival during the first afternoon session, despite a fiery hot spot lunch, which means in plainer terms that I alas dozed in and out of the talks. The second session saw Jack Jewson exposing in deeper details the exact PDMP algorithm for Gibbs measures  Jeremias Knoblauch mentioned yesterday. And Jonathan Huggins as well, using Gaussian processes as proxies for expected likelihoods, with lower guarantees than pseudo-marginal versions. In a mildly connected way, Robin Ryder went through the resolution of the ecological inference challenge they produce with Nicolas Chopin and Théo Valdoire (all authors with whom I am connected, Théo being a brillant student of our MASH Master last year and now in Harvard, hopefully till the end of his PhD!)

On the extra-academic curriculum, I had a yummy dinner in the Maxwell Hawker (street) food centre, incl. Xiao Long Bao that cooled down fast enough to avoid the usual scalding effect, plus rojak a mixed fruit and vegetable fried in a peanut sauce that I had never tasted before, popiah (ditto), chili noodles, and an appam with durian deepfried balls as a fabulous and unexpected dessert.

control variates [seminar]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on November 5, 2021 by xi'an

Today, Petros Dellaportas (whom I have know since the early days of MCMC, when we met in CIRM) gave a seminar at the Warwick algorithm seminar on control variates for MCMC, reminding me of his 2012 JRSS paper. Based on the Poisson equation and using a second control variate to stabilise the Monte Carlo approximation do the first control variate. The difference with usual control variates is finding a first approximate G(x)-q(y|x)G(Y) to F-πF. And the first Poisson equation is using α(x,y)q(y|x) rather than π. Then the second expands log α(x,y)q(y|x) to achieve a manageable term.

Abstract: We provide a general methodology to construct control variates for any discrete time random walk Metropolis and Metropolis-adjusted Langevin algorithm Markov chains that can achieve, in a post-processing manner and with a negligible additional computational cost, impressive variance reduction when compared to the standard MCMC ergodic averages. Our proposed estimators are based on an approximate solution of the Poisson equation for a multivariate Gaussian target densities of any dimension.

I wonder if there were a neural network version that would first build G from scratch and later optimise it towards solving the Poisson equation. As in this recent arXival I haven’t read (yet).

MCMC with control variates

Posted in Books, Statistics, University life with tags , , , , , , , , , , on February 17, 2012 by xi'an

In the latest issue of JRSS Series B (74(1), Jan, 2012), I just noticed that no paper is “from my time” as co-editor, i.e. that all of them have been submitted after I completed my term in Jan. 2010. Given the two year delay, this is not that surprising, but it also means I can make comments on some papers w/o reservation! A paper I had seen earlier (as a reader, not as an editor nor as a referee!) is Petros Dellaportas’ and Ioannis Kontoyiannis’ Control variates for estimation based on  reversible Markov chain Monte Carlo samplers. The idea is one of post-processing MCMC output, by stabilising the empirical average via control variates. There are two difficulties, one in finding control variates, i.e. functions $\Psi(\cdot)$ with zero expectation under the target distribution, and another one in estimating the optimal coefficient in a consistent way. The paper solves the first difficulty by using the Poisson equation, namely that G(x)-KG(x) has zero expectation under the stationary distribution associated with the Markov kernel K. Therefore, if KG can be computed in closed form, this is a generic control variate taking advantage of the MCMC algorithm. Of course, the above if is a big if: it seems difficult to find closed form solutions when using a Metropolis-Hastings algorithm for instance and the paper only contains illustrations within the conjugate prior/Gibbs sampling framework. The second difficulty is also met by Dellaportas and Kontoyiannis, who show that the asymptotic variance of the resulting central limit can be equal to zero in some cases.