Archive for Wasserstein distance

right place, wrong version

Posted in Statistics with tags , , , , , , , , , on August 12, 2020 by xi'an

frontier of simulation-based inference

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on June 11, 2020 by xi'an

“This paper results from the Arthur M. Sackler Colloquium of the National Academy of Sciences, `The Science of Deep Learning,’ held March 13–14, 2019, at the National Academy of Sciences in Washington, DC.”

A paper by Kyle Cranmer, Johann Brehmer, and Gilles Louppe just appeared in PNAS on the frontier of simulation-based inference. Sounding more like a tribune than a research paper producing new input. Or at least like a review. Providing a quick introduction to simulators, inference, ABC. Stating the shortcomings of simulation-based inference as three-folded:

  1. costly, since required a large number of simulated samples
  2. loosing information through the use of insufficient summary statistics or poor non-parametric approximations of the sampling density.
  3. wasteful as requiring new computational efforts for new datasets, primarily for ABC as learning the likelihood function (as a function of both the parameter θ and the data x) is only done once.

And the difficulties increase with the dimension of the data. While the points made above are correct, I want to note that ideally ABC (and Bayesian inference as a whole) only depends on a single dimension observation, which is the likelihood value. Or more practically that it only depends on the distance from the observed data to the simulated data. (Possibly the Wasserstein distance between the cdfs.) And that, somewhat unrealistically, that ABC could store the reference table once for all. Point 3 can also be debated in that the effort of learning an approximation can only be amortized when exactly the same model is re-employed with new data, which is likely in industrial applications but less in scientific investigations, I would think. About point 2, the paper misses part of the ABC literature on selecting summary statistics, e.g., the culling afforded by random forests ABC, or the earlier use of the score function in Martin et al. (2019).

The paper then makes a case for using machine-, active-, and deep-learning advances to overcome those blocks. Recouping other recent publications and talks (like Dennis on One World ABC’minar!). Once again presenting machine-learning techniques such as normalizing flows as more efficient than traditional non-parametric estimators. Of which I remain unconvinced without deeper arguments [than the repeated mention of powerful machine-learning techniques] on the convergence rates of these estimators (rather than extolling the super-powers of neural nets).

“A classifier is trained using supervised learning to discriminate two sets of data, although in this case both sets come from the simulator and are generated for different parameter points θ⁰ and θ¹. The classifier output function can be converted into an approximation of the likelihood ratio between θ⁰ and θ¹ (…) learning the likelihood or posterior is an unsupervised learning problem, whereas estimating the likelihood ratio through a classifier is an example of supervised learning and often a simpler task.”

The above comment is highly connected to the approach set by Geyer in 1994 and expanded in Gutmann and Hyvärinen in 2012. Interestingly, at least from my narrow statistician viewpoint!, the discussion about using these different types of approximation to the likelihood and hence to the resulting Bayesian inference never engages into a quantification of the approximation or even broaches upon the potential for inconsistent inference unlocked by using fake likelihoods. While insisting on the information loss brought by using summary statistics.

“Can the outcome be trusted in the presence of imperfections such as limited sample size, insufficient network capacity, or inefficient optimization?”

Interestingly [the more because the paper is classified as statistics] the above shows that the statistical question is set instead in terms of numerical error(s). With proposals to address it ranging from (unrealistic) parametric bootstrap to some forms of GANs.

the buzz about nuzz

Posted in Books, Mountains, pictures, Statistics with tags , , , , , , , , , , , , , on April 6, 2020 by xi'an

“…expensive in these terms, as for each root, Λ(x(s),v) (at the cost of one epoch) has to be evaluated for each root finding iteration, for each node of the numerical integral

When using the ZigZag sampler, the main (?) difficulty is in producing velocity switch as the switches are produced as interarrival times of an inhomogeneous Poisson process. When the rate of this process cannot be integrated out in an analytical manner, the only generic approach I know is in using Poisson thinning, obtained by finding an integrable upper bound on this rate, generating from this new process and subsampling. Finding the bound is however far from straightforward and may anyway result in an inefficient sampler. This new paper by Simon Cotter, Thomas House and Filippo Pagani makes several proposals to simplify this simulation, Nuzz standing for numerical ZigZag. Even better (!), their approach is based on what they call the Sellke construction, with Tom Sellke being a probabilist and statistician at Purdue University (trivia: whom I met when spending a postdoctoral year there in 1987-1988) who also wrote a fundamental paper on the opposition between Bayes factors and p-values with Jim Berger.

“We chose as a measure of algorithm performance the largest Kolmogorov-Smirnov (KS) distance between the MCMC sample and true distribution amongst all the marginal distributions.”

The practical trick is rather straightforward in that it sums up as the exponentiation of the inverse cdf method, completed with a numerical resolution of the inversion. Based on the QAGS (Quadrature Adaptive Gauss-Kronrod Singularities) integration routine. In order to save time Kingman’s superposition trick only requires one inversion rather than d, the dimension of the variable of interest. This nuzzled version of ZIgZag can furthermore be interpreted as a PDMP per se. Except that it retains a numerical error, whose impact on convergence is analysed in the paper. In terms of Wasserstein distance between the invariant measures. The paper concludes with a numerical comparison between Nuzz and random walk Metropolis-Hastings, HMC, and manifold MALA, using the number of evaluations of the likelihood as a measure of time requirement. Tuning for Nuzz is described, but not for the competition. Rather dramatically the Nuzz algorithm performs worse than this competition when counting one epoch for each likelihood computation and better when counting one epoch for each integral inversion. Which amounts to perfect inversion, unsurprisingly. As a final remark, all models are more or less Normal, with very smooth level sets, maybe not an ideal range



Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , on January 10, 2020 by xi'an

First, I really have to congratulate my friend Jim Hobert for a great organisation of the meeting adopting my favourite minimalist principles (no name tag, no “goodies” apart from the conference schedule, no official talks). Without any pretense at objectivity, I also appreciated very much the range of topics and the sweet frustration of having to choose between two or three sessions each time. Here are some notes taken during some talks (with no implicit implication for the talks no mentioned, re. above frustration! as well as very short nights making sudden lapse in concentration highly likely).

On Day 1, Paul Fearnhead’s inaugural plenary talk was on continuous time Monte Carlo methods, mostly bouncy particle and zig-zag samplers, with a detailed explanation on the simulation of the switching times which likely brought the audience up to speed even if they had never heard of them. And an opening on PDMPs used as equivalents to reversible jump MCMC, reminding me of the continuous time (point process) solutions of Matthew Stephens for mixture inference (and of Preston, Ripley, Møller).

The same morn I heard of highly efficient techniques to handle very large matrices and p>n variables selections by Akihiko Nishimura and Ruth Baker on a delayed acceptance ABC, using a cheap proxy model. Somewhat different from indirect inference. I found the reliance on ESS somewhat puzzling given the intractability of the likelihood (and the low reliability of the frequency estimate) and the lack of connection with the “real” posterior. At the same ABC session, Umberto Picchini spoke on a joint work with Richard Everitt (Warwick) on linking ABC and pseudo-marginal MCMC by bootstrap. Actually, the notion of ABC likelihood was already proposed as pseudo-marginal ABC by Anthony Lee, Christophe Andrieu and Arnaud Doucet in the discussion of Fearnhead and Prangle (2012) but I wonder at the focus of being unbiased when the quantity is not the truth, i.e. the “real” likelihood. It would seem more appropriate to attempt better kernel estimates on the distribution of the summary itself. The same session also involved David Frazier who linked our work on ABC for misspecified models and an on-going investigation of synthetic likelihood.

Later, there was a surprise occurrence of the Bernoulli factory in a talk by Radu Herbei on Gaussian process priors with accept-reject algorithms, leading to exact MCMC, although the computing implementation remains uncertain. And several discussions during the poster session, incl. one on the planning of a 2021 workshop in Oaxaca centred on objective Bayes advances as we received acceptance of our proposal by BIRS today!

On Day 2, David Blei gave a plenary introduction to variational Bayes inference and latent Dirichlet allocations, somewhat too introductory for my taste although other participants enjoyed this exposition. He also mentioned a recent JASA paper on the frequentist consistency of variational Bayes that I should check. Speaking later with PhD students, they really enjoyed this opening on an area they did not know that well.

A talk by Kengo Kamatani (whom I visited last summer) on improved ergodicity rates for heavy tailed targets and Crank-NIcholson modifications to the random walk proposal (which uses an AR(1) representation instead of the random walk). With the clever idea of adding the scale of the proposal as an extra parameter with a prior of its own. Gaining one order of magnitude in the convergence speed (i.e. from d to 1 and from d² to d, where d is the dimension), which is quite impressive (and just published in JAP).Veronica Rockova linked Bayesian variable selection and machine learning via ABC, with conditions on the prior for model consistency. And a novel approach using part of the data to learn an ABC partial posterior, which reminded me of the partial  Bayes factors of the 1990’s although it is presumably unrelated. And a replacement of the original rejection ABC via multi-armed bandits, where each variable is represented by an arm, called ABC Bayesian forests. Recalling the simulation trick behind Thompson’s approach, reproduced for the inclusion or exclusion of variates and producing a fixed estimate for the (marginal) inclusion probabilities, which makes it sound like a prior-feeback form of empirical Bayes. Followed by a talk of Gregor Kastner on MCMC handling of large time series with specific priors and a massive number of parameters.

The afternoon also had a wealth of exciting talks and missed opportunities (in the other sessions!). Which ended up with a strong if unintended French bias since I listened to Christophe Andrieu, Gabriel Stolz, Umut Simsekli, and Manon Michel on different continuous time processes, with Umut linking GANs, multidimensional optimal transport, sliced-Wasserstein, generative models, and new stochastic differential equations. Manon Michel gave a highly intuitive talk on creating non-reversibility, getting rid of refreshment rates in PDMPs to kill any form of reversibility.

BayesComp 2020 at a glance

Posted in Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , on December 18, 2019 by xi'an