Archive for variational Bayes methods

sequential neural likelihood estimation as ABC substitute

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , , , , , , on May 14, 2020 by xi'an

A JMLR paper by Papamakarios, Sterratt, and Murray (Edinburgh), first presented at the AISTATS 2019 meeting, on a new form of likelihood-free inference, away from non-zero tolerance and from the distance-based versions of ABC, following earlier papers by Iain Murray and co-authors in the same spirit. Which I got pointed to during the ABC workshop in Vancouver. At the time I had no idea as to autoregressive flows meant. We were supposed to hold a reading group in Paris-Dauphine on this paper last week, unfortunately cancelled as a coronaviral precaution… Here are some notes I had prepared for the meeting that did not take place.

A simulator model is a computer program, which takes a vector of parameters θ, makes internal calls to a random number generator, and outputs a data vector x.”

Just the usual generative model then.

“A conditional neural density estimator is a parametric model q(.|φ) (such as a neural network) controlled by a set of parameters φ, which takes a pair of datapoints (u,v) and outputs a conditional probability density q(u|v,φ).”

Less usual, in that the outcome is guaranteed to be a probability density.

“For its neural density estimator, SNPE uses a Mixture Density Network, which is a feed-forward neural network that takes x as input and outputs the parameters of a Gaussian mixture over θ.”

In which theoretical sense would it improve upon classical or Bayesian density estimators? Where are the error evaluation, the optimal rates, the sensitivity to the dimension of the data? of the parameter?

“Our new method, Sequential Neural Likelihood (SNL), avoids the bias introduced by the proposal, by opting to learn a model of the likelihood instead of the posterior.”

I do not get the argument in that the final outcome (of using the approximation within an MCMC scheme) remains biased since the likelihood is not the exact likelihood. Where is the error evaluation? Note that in the associated Algorithm 1, the learning set is enlarged on each round, as in AMIS, rather than set back to the empty set ∅ on each round.

…given enough simulations, a sufficiently flexible conditional neural density estimator will eventually approximate the likelihood in the support of the proposal, regardless of the shape of the proposal. In other words, as long as we do not exclude parts of the parameter space, the way we propose parameters does not bias learning the likelihood asymptotically. Unlike when learning the posterior, no adjustment is necessary to account for our proposing strategy.”

This is a rather vague statement, with the only support being that the Monte Carlo approximation to the Kullback-Leibler divergence does converge to its actual value, i.e. a direct application of the Law of Large Numbers! But an interesting point I informally made a (long) while ago that all that matters is the estimate of the density at x⁰. Or at the value of the statistic at x⁰. The masked auto-encoder density estimator is based on a sequence of bijections with a lower-triangular Jacobian matrix, meaning the conditional density estimate is available in closed form. Which makes it sounds like a form of neurotic variational Bayes solution.

The paper also links with ABC (too costly?), other parametric approximations to the posterior (like Gaussian copulas and variational likelihood-free inference), synthetic likelihood, Gaussian processes, noise contrastive estimation… With experiments involving some of the above. But the experiments involve rather smooth models with relatively few parameters.

“A general question is whether it is preferable to learn the posterior or the likelihood (…) Learning the likelihood can often be easier than learning the posterior, and it does not depend on the choice of proposal, which makes learning easier and more robust (…) On the other hand, methods such as SNPE return a parametric model of the posterior directly, whereas a further inference step (e.g. variational inference or MCMC) is needed on top of SNL to obtain a posterior estimate”

A fair point in the conclusion. Which also mentions the curse of dimensionality (both for parameters and observations) and the possibility to work directly with summaries.

Getting back to the earlier and connected Masked autoregressive flow for density estimation paper, by Papamakarios, Pavlakou and Murray:

“Viewing an autoregressive model as a normalizing flow opens the possibility of increasing its flexibility by stacking multiple models of the same type, by having each model provide the source of randomness for the next model in the stack. The resulting stack of models is a normalizing flow that is more flexible than the original model, and that remains tractable.”

Which makes it sound like a sort of a neural network in the density space. Optimised by Kullback-Leibler minimisation to get asymptotically close to the likelihood. But a form of Bayesian indirect inference in the end, namely an MLE on a pseudo-model, using the estimated model as a proxy in Bayesian inference…

ISBA2020 program

Posted in Kids, Statistics, Travel, University life with tags , , , , , , , , , , , , on January 29, 2020 by xi'an

The scheduled program for ISBA 2020 is now on-line. And full of exciting sessions, many with computational focus. With dear hopes that the nCo-2019 epidemics will have abated by then (and not solely for the sake of the conference, most obviously!). While early registration ends by 15 April, the deadline for junior travel support ends up this month. And so does the deadline for contributions.

MCMC, with common misunderstandings

Posted in Books, pictures, R, Statistics, University life with tags , , , , , , , , , , , , on January 27, 2020 by xi'an

As I was asked to write a chapter on MCMC methods for an incoming Handbook of Computational Statistics and Data Science, published by Wiley, rather than cautiously declining!, I decided to recycle the answers I wrote on X validated to what I considered to be the most characteristic misunderstandings about MCMC and other computing methods, using as background the introduction produced by Wu Changye in his PhD thesis. Waiting for the opinion of the editors of the Handbook on this Q&A style. The outcome is certainly lighter than other recent surveys like the one we wrote with Peter Green, Krys Latuszinski, and Marcelo Pereyra, for Statistics and Computing, or the one with Victor Elvira, Nick Tawn, and Changye Wu.

BayesComp’20

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.

AABI9 tidbits [& misbits]

Posted in Books, Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on December 10, 2019 by xi'an

Today’s Advances in Approximate Bayesian Inference symposium, organised by Thang Bui, Adji Bousso Dieng, Dawen Liang, Francisco Ruiz, and Cheng Zhang, took place in front of Vancouver Harbour (and the tentalising ski slope at the back) and saw more than 400 participants, drifting away from the earlier versions which had a stronger dose of ABC and much fewer participants. There were students’ talks in a fair proportion, as well (and a massive number of posters). As of below, I took some notes during some of the talks with no pretense at exhaustivity, objectivity or accuracy. (This is a blog post, remember?!) Overall I found the day exciting (to the point I did not suffer at all from the usal naps consecutive to very short nights!) and engaging, with a lot of notions and methods I had never heard about. (Which shows how much I know nothing!)

The first talk was by Michalis Titsias, Gradient-based Adaptive Markov Chain Monte Carlo (jointly with Petros Dellaportas) involving as its objective function the multiplication of the variance of the move and of the acceptance probability, with a proposed adaptive version merging gradients, variational Bayes, neurons, and two levels of calibration parameters. The method advocates using this construction in a burnin phase rather than continuously, hence does not require advanced Markov tools for convergence assessment. (I found myself less excited by adaptation than earlier, maybe because it seems like switching one convergence problem for another, with additional design choices to be made.)The second talk was by Jakub Swiatkowsk, The k-tied Normal Distribution: A Compact Parameterization of Gaussian Mean Field Posteriors in Bayesian Neural Networks, involving mean field approximation in variational inference (loads of VI at this symposium!), meaning de facto searching for a MAP estimator, and reminding me of older factor analysis and other analyse de données projection methods, except it also involved neural networks (what else at NeurIPS?!)The third talk was by Michael Gutmann, Robust Optimisation Monte Carlo, (OMC) for implicit data generated models (Diggle & Graton, 1982), an ABC talk at last!, using a formalisation through the functional representation of the generative process and involving derivatives of the summary statistic against parameter, in that sense, with the (Bayesian) random nature of the parameter sample only induced by the (frequentist) randomness in the generative transform since a new parameter “realisation” is obtained there as the one providing minimal distance between data and pseudo-data, with no uncertainty or impact of the prior. The Jacobian of this summary transform (and once again a neural network is used to construct the summary) appears in the importance weight, leading to OMC being unstable, beyond failing to reproduce the variability expressed by the regular posterior or even the ABC posterior. It took me a while to wonder `where is Wally?!’ (the prior) as it only appears in the importance weight.

The fourth talk was by Sergey Levine, Reinforcement Learning, Optimal , Control, and Probabilistic Inference, back to Kullback-Leibler as the objective function, with linkage to optimal control (with distributions as actions?), plus again variational inference, producing an approximation in sequential settings. This sounded like a type of return of the MaxEnt prior, but the talk pace was so intense that I could not follow where the innovations stood.

The fifth talk was by Iuliia Molchanova, on Structured Semi-Implicit Variational Inference, from BAyesgroup.ru (I did not know of a Bayesian group in Russia!, as I was under the impression that Bayesian statistics were under-represented there, but apparently the situation is quite different in machine learning.) The talk brought an interesting concept of semi-implicit variational inference, exploiting some form of latent variables as far as I can understand, using mixtures of Gaussians.

The sixth talk was by Rianne van den Berg, Normalizing Flows for Discrete Data, and amounted to covering three papers also discussed in NeurIPS 2019 proper, which I found somewhat of a suboptimal approach to an invited talk, as it turned into a teaser for following talks or posters. But the teasers it contained were quite interesting as they covered normalising flows as integer valued controlled changes of variables using neural networks about which I had just became aware during the poster session, in connection with papers of Papamakarios et al., which I need to soon read.

The seventh talk was by Matthew Hoffman: Langevin Dynamics as Nonparametric Variational Inference, and sounded most interesting, both from title and later reports, as it was bridging Langevin with VI, but I alas missed it for being “stuck” in a tea-house ceremony that lasted much longer than expected. (More later on that side issue!)

After the second poster session (with a highly original proposal by Radford Neal towards creating  non-reversibility at the level of the uniform generator rather than later on), I thus only attended Emily Fox’s Stochastic Gradient MCMC for Sequential Data Sources, which superbly reviewed (in connection with a sequence of papers, including a recent one by Aicher et al.) error rate and convergence properties of stochastic gradient estimator methods there. Another paper I need to soon read!

The one before last speaker, Roman Novak, exposed a Python library about infinite neural networks, for which I had no direct connection (and talks I have always difficulties about libraries, even without a four hour sleep night) and the symposium concluded with a mild round-table. Mild because Frank Wood’s best efforts (and healthy skepticism about round tables!) to initiate controversies, we could not see much to bite from each other’s viewpoint.