transport, diffusions, and sampling

At the Sampling, Transport, and Diffusions workshop at the Flatiron Institute, on Day #2, Marilou Gabrié (École Polytechnique) gave the second introductory lecture on merging sampling and normalising flows targeting the target distribution, when driven by a divergence criterion like KL, that only requires the shape of the target density. I first wondered about ergodicity guarantees in simultaneous MCMC and map training due to the adaptation of the flow but the update of the map only depends on the current particle cloud in (8). From an MCMC perspective, it sounds somewhat paradoxical to see the independent sampler making such an unexpected come-back when considering that no insider information is available about the (complex) posterior to drive the [what-you-get-is-what-you-see] construction of the transport map. However, the proposed approach superposed local (random-walk like) and global (transport) proposals in Algorithm 1.

Qiang Liu followed on learning transport maps, with the  Interesting notion of causalizing a graph by removing intersections (which are impossible for an ODE, as discussed by Eric Vanden-Eijden’s talk yesterday) through  coupling. Which underlies his notion of rectified flows. Possibly connecting with the next lightning talk by Jonathan Weare on spurious modes created by a variational Monte Carlo sampler and the use of stochastic gradient, corrected by (case-dependent?) regularisation.

Then came a whole series of MCMC talks!

Sam Livingstone spoke on Barker’s proposal (an incoming Biometrika paper!) as part of a general class of transforms g of the MH ratio, using jump processes based on a nasty normalising constant related with g (tractable for the original Barker algorithm). I then realised I had missed his StatSci paper on how to speak to statistical physics researchers!

Charles Margossian spoke about using a massive number of short parallel runs (many-short-chain regime) from a recent paper written with Aki,  Andrew, and Lionel Riou-Durand (Warwick) among others. Which brings us back to the challenge of producing convergence diagnostics and precisely the Gelman-Rubin R statistic or its recent nR avatar (with its linear limitations and dependence on parameterisation, as opposed to fuller distributional criteria). The core of the approach is in using blocks of GPUs to improve and speed-up the estimation of the between-chain variance. (D for R².) I still wonder at a waste of simulations / computing power resulting from stopping the runs almost immediately after warm-up is over, since reaching the stationary regime or an approximation thereof should be exploited more efficiently. (Starting from a minimal discrepancy sample would also improve efficiency.)

Lu Zhang also talked on the issue of cutting down warmup, presenting a paper co-authored with Bob, Andrew, and Aki, recommending Laplace / variational approximations for reaching faster high-posterior-density regions, using an algorithm called Pathfinder that relies on ELBO checks to counter poor performances of Laplace approximations. In the spirit of the workshop, it could be profitable to further transform / push-forward the outcome by a transport map.

Yuling Yao (of stacking and Pareto smoothing fame!) gave an original and challenging (in a positive sense) talk on the many ways of bridging densities [linked with the remark he shared with me the day before] and their statistical significance. Questioning our usual reliance on arithmetic or geometric mixtures. Ignoring computational issues, selecting a bridging pattern sounds not different from choosing a parameterised family of embedding distributions. This new typology of models can then be endowed with properties that are more or less appealing. (Occurences of the Hyvärinen score and our mixtestin perspective in the talk!)

Miranda Holmes-Cerfon talked about MCMC on stratification (illustrated by this beautiful picture of nanoparticle random walks). Which means sampling under varying constraints and dimensions with associated densities under the respective Hausdorff measures. This sounds like a perfect setting for reversible jump and in a sense it is, as mentioned in the talks. Except that the moves between manifolds are driven by the proximity to said manifold, helping with a higher acceptance rate, and making the proposals easier to construct since projections (or the reverses) have a physical meaning. (But I could not tell from the talk why the approach was seemingly escaping the symmetry constraint set by Peter Green’s RJMCMC on the reciprocal moves between two given manifolds).

data assimilation and reduced modelling for high-D problems [CIRM]

Next summer, from 19 July till 27 August, there will be a six week program at CIRM on the above theme, bringing together scientists from both the academic and industrial communities. The program includes a one-week summer school followed by 5 weeks of research sessions on projects proposed by academic and industrial partners.

Confirmed speakers of the summer school (Jul 19-23) are:

• Albert Cohen (Sorbonne University)
• Masoumeh Dashti (University of Sussex)
• Eric Moulines (Ecole Polytechnique)
• Anthony Nouy (Ecole Centrale de Nantes)
• Claudia Schillings (Mannheim University)

Junior participants may apply for fellowships to cover part or the whole stay. Registration and application to fellowships will be open soon.

state of the art in sampling & clustering [workshop]

Next month, I am taking part in a workshop on sampling & clustering at the Max-Planck-Institut für Physik in Garching, Germany (near München). By giving a three hour introduction to ABC, as I did three years ago in Autrans. Being there and talking with local researchers if the sanitary conditions allow. From my office otherwise. Other speakers include Michael Betancourt on HMC and Johannes Buchner on nested sampling. The remote participation to this MPI workshop is both open and free, but participants must register before 18 September, namely tomorrow.

a perfectly normally distributed sample

When I saw this title on R-bloggers, I was wondering how “more perfect” a Normal sample could be when compared with the outcome of rnorm(n). Hence went checking the original blog on bayestestR in search of more information. Which was stating nothing more than how to generate a sample is perfectly normal by using the rnorm_perfect function. Still unsure of the meaning, I contacted one of the contributors who replied very quickly

…that’s actually a good question. I would say an empirical sample having characteristics as close as possible to a cannonic gaussian distribution.

and again leaving me hungering for more details. I thus downloaded the package bayestestR and opened the rnorm_perfect function. Which is simply the sequence of n-quantiles

stats::qnorm(seq(1/n, 1 – 1/n, length.out = n), mean, sd)

which I would definitely not call a sample as it has nothing random. And perfect?! Not really, unless one associates randomness and imperfection.

approximate Bayesian inference under informative sampling

In the first issue of this year Biometrika, I spotted a paper with the above title, written by Wang, Kim, and Yang, and thought it was a particular case of ABC. However, when I read it on a rare metro ride to Dauphine, thanks to my hurting knee!, I got increasingly disappointed as the contents had nothing to do with ABC. The purpose of the paper was to derive a consistent and convergent posterior distribution based on a estimator of the parameter θ that is… consistent and convergent under informative sampling. Using for instance a Normal approximation to the sampling distribution of this estimator. Or to the sampling distribution of the pseudo-score function, S(θ) [which pseudo-normality reminded me of Ron Gallant’s approximations and of my comments on them]. The paper then considers a generalisation to the case of estimating equations, U(θ), which may again enjoy a Normal asymptotic distribution. Involving an object that does not make direct Bayesian sense, namely the posterior of the parameter θ given U(θ)…. (The algorithm proposed to generate from this posterior (8) is also a mystery.) Since the approach requires consistent estimators to start with and aims at reproducing frequentist coverage properties, I am thus at a loss as to why this pseudo-Bayesian framework is adopted.