Archive for AMIS

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…

nested sampling via SMC

Posted in Books, pictures, Statistics with tags , , , , , , , , , , , , on April 2, 2020 by xi'an

“We show that by implementing a special type of [sequential Monte Carlo] sampler that takes two im-portance sampling paths at each iteration, one obtains an analogous SMC method to [nested sampling] that resolves its main theoretical and practical issues.”

A paper by Queenslander Robert Salomone, Leah South, Chris Drovandi and Dirk Kroese that I had missed (and recovered by Grégoire after we discussed this possibility with our Master students). On using SMC in nested sampling. What are the difficulties mentioned in the above quote?

  1. Dependence between the simulated samples, since only the offending particle is moved by one or several MCMC steps. (And MultiNest is not a foolproof solution.)
  2. The error due to quadrature is hard to evaluate, with parallelised versions aggravating the error.
  3. There is a truncation error due to the stopping rule when the exact maximum of the likelihood function is unknown.

Not mentioning the Monte Carlo error, of course, which should remain at the √n level.

“Nested Sampling is a special type of adaptive SMC algorithm, where weights are assigned in a suboptimal way.”

The above remark is somewhat obvious for a fixed sequence of likelihood levels and a set of particles at each (ring) level. moved by a Markov kernel with the right stationary target. Constrained to move within the ring, which may prove delicate in complex settings. Such a non-adaptive version is however not realistic and hence both the level sets and the stopping rule need be selected from the existing simulation, respectively as a quantile of the observed likelihood and as a failure to modify the evidence approximation, an adaptation that is a Catch 22! as we already found in the AMIS paper.  (AMIS stands for adaptive mixture importance sampling.) To escape the quandary, the authors use both an auxiliary variable (to avoid atoms) and two importance sampling sequences (as in AMIS). And only a single particle with non-zero incremental weight for the (upper level) target. As the full details are a bit fuzzy to me, I hope I can experiment with my (quarantined) students on the full implementation of the method.

“Such cases asides, the question whether SMC is preferable using the TA or NS approach is really one of whether it is preferable to sample (relatively) easy distributions subject to a constraint or to sample potentially difficult distributions.”

A question (why not regular SMC?) I was indeed considering until coming to the conclusion section but did not find it treated in the paper. There is little discussion on the computing requirements either, as it seems the method is more time-consuming than a regular nested sample. (On the personal side,  I appreciated very much their “special thanks to Christian Robert, whose many blog posts on NS helped influence this work, and played a large partin inspiring it.”)

a new rule for adaptive importance sampling

Posted in Books, Statistics with tags , , , , , , , , , on March 5, 2019 by xi'an

Art Owen and Yi Zhou have arXived a short paper on the combination of importance sampling estimators. Which connects somehow with the talk about multiple estimators I gave at ESM last year in Helsinki. And our earlier AMIS combination. The paper however makes two important assumptions to reach optimal weighting, which is inversely proportional to the variance:

  1. the estimators are uncorrelated if dependent;
  2. the variance of the k-th estimator is of order a (negative) power of k.

The later is puzzling when considering a series of estimators, in that k appears to act as a sample size (as in AMIS), the power is usually unknown but also there is no reason for the power to be the same for all estimators. The authors propose to use ½ as the default, both because this is the standard Monte Carlo rate and because the loss in variance is then minimal, being 12% larger.

As an aside, Art Owen also wrote an invited discussion “the unreasonable effectiveness of Monte Carlo” of ” Probabilistic Integration: A Role in Statistical Computation?” by François-Xavier Briol, Chris  Oates, Mark Girolami (Warwick), Michael Osborne and Deni Sejdinovic, to appear in Statistical Science, discussion that contains a wealth of smart and enlightening remarks. Like the analogy between pseudo-random number generators [which work unreasonably well!] vs true random numbers and Bayesian numerical integration versus non-random functions. Or the role of advanced bootstrapping when assessing the variability of Monte Carlo estimates (citing a paper of his from 1992). Also pointing out at an intriguing MCMC paper by  Michael Lavine and Jim Hodges to appear in The American Statistician.

efficient adaptive importance sampling

Posted in Books, Statistics with tags , , , , , , , on June 22, 2018 by xi'an

Bernard Delyon and François Portier just recently arXived a paper on population or evolutionary importance sampling, pointed out to me by Víctor Elvira. Changing the proposal or importance sampler at each iteration. And averaging the estimates across iterations, but also mentioning AMIS. While drawing a distinction that I do not understand, since the simulation cost remains the same, while improving the variance of the resulting estimator. (But the paper points out later that their martingale technique of proof does not apply in this AMIS case.) Some interesting features of the paper are that

  • convergence occurs when the total number of simulations grows to infinity, which is the most reasonable scale for assessing the worth of the method;
  • some optimality in the oracle sense is established for the method;
  • an improvement is found by eliminating outliers and favouring update rate over simulation rate (at a constant cost). Unsurprisingly, the optimal weight of the t-th estimator is given by its inverse variance (with eqn (13) missing an inversion step). Although it relies on the normalised versions of the target and proposal densities, since it assumes the expectation of the ratio is equal to one.

When updating the proposal or importance distribution, the authors consider a parametric family with the update in the parameter being driven by moment or generalised moment matching, or Kullback reduction as in our population Monte Carlo paper. The interesting technical aspects of the paper include the use of martingale and empirical risk arguments. All in all, quite a pleasant surprise to see some follow-up to our work on that topic, more than 10 years later.

European statistics in Finland [EMS17]

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on August 2, 2017 by xi'an

While this European meeting of statisticians had a wide range of talks and topics, I found it to be more low key than the previous one I attended in Budapest, maybe because there was hardly any talk there in applied probability. (But there were some sessions in mathematical statistics and Mark Girolami gave a great entry to differential geometry and MCMC, in the spirit of his 2010 discussion paper. Using our recent trip to Montréal as an example of geodesic!) In the Bayesian software session [organised by Aki Vetahri], Javier Gonzáles gave a very neat introduction to Bayesian optimisation: he showed how optimisation can be turned into Bayesian inference or more specifically as a Bayesian decision problem using a loss function related to the problem of interest. The point in following a Bayesian path [or probabilist numerics] is to reduce uncertainty by the medium of prior measures on functions, although resorting [as usual] to Gaussian processes whose arbitrariness I somehow dislike within the infinity of priors (aka stochastic processes) on functions! One of his strong arguments was that the approach includes the possibility for design in picking the next observation point (as done in some ABC papers of Michael Guttman and co-authors, incl. the following talk at EMS 2017) but again the devil may be in the implementation when looking at minimising an objective function… The notion of the myopia of optimisation techniques was another good point: only looking one step ahead in the future diminishes the returns of the optimisation and an alternative presented at AISTATS 2016 [that I do not remember seeing in Càdiz] goes against this myopia.

Umberto Piccini also gave a talk on exploiting synthetic likelihoods in a Bayesian fashion (in connection with the talk he gave last year at MCqMC 2016). I wondered at the use of INLA for this Gaussian representation, as well as at the impact of the parameterisation of the summary statistics. And the session organised by Jean-Michel involved Jimmy Olson, Murray Pollock (Warwick) and myself, with great talks from both other speakers, on PaRIS and PaRISian algorithms by Jimmy, and on a wide range of exact simulation methods of continuous time processes by Murray, both managing to convey the intuition behind their results and avoiding the massive mathematics at work there. By comparison, I must have been quite unclear during my talk since someone interrupted me about how Owen & Zhou (2000) justified their deterministic mixture importance sampling representation. And then left when I could not make sense of his questions [or because it was lunchtime already].

multiple importance sampling

Posted in Books, Statistics, University life with tags , , , , , , , , on November 20, 2015 by xi'an

“Within this unified context, it is possible to interpret that all the MIS algorithms draw samples from a equal-weighted mixture distribution obtained from the set of available proposal pdfs.”

In a very special (important?!) week for importance sampling!, Elvira et al. arXived a paper about generalized multiple importance sampling. The setting is the same as in earlier papers by Veach and Gibas (1995) or Owen and Zhou (2000) [and in our AMIS paper], namely a collection of importance functions and of simulations from those functions. However, there is no adaptivity for the construction of the importance functions and no Markov (MCMC) dependence on the generation of the simulations.

multipl
“One of the goals of this paper is to provide the practitioner with solid theoretical results about the superiority of some specific MIS schemes.”

One first part deals with the fact that a random point taken from the conjunction of those samples is distributed from the equiweighted mixture. Which was a fact I had much appreciated when reading Owen and Zhou (2000). From there, the authors discuss the various choices of importance weighting. Meaning the different degrees of Rao-Blackwellisation that can be applied to the sample. As we discovered in our population Monte Carlo research [which is well-referred within this paper], conditioning too much leads to useless adaptivity. Again a sort of epiphany for me, in that a whole family of importance functions could be used for the same target expectation and the very same simulated value: it all depends on the degree of conditioning employed for the construction of the importance function. To get around the annoying fact that self-normalised estimators are never unbiased, the authors borrow Liu’s (2000) notion of proper importance sampling estimators, where the ratio of the expectations is returning the right quantity. (Which amounts to recover the correct normalising constant(s), I believe.) They then introduce five (5!) different possible importance weights that all produce proper estimators. However, those weights correspond to different sampling schemes, so do not apply to the same sample. In other words, they are not recycling weights as in AMIS. And do not cover the adaptive cases where the weights and parameters of the different proposals change along iterations. Unsurprisingly, the smallest variance estimator is the one based on sampling without replacement and an importance weight made of the entire mixture. But this result does not apply for the self-normalised version, whose variance remains intractable.

I find this survey of existing and non-existing multiple importance methods quite relevant and a must-read for my students (and beyond!). My reservations (for reservations there must be!) are that the study stops short of pushing further the optimisation. Indeed, the available importance functions are not equivalent in terms of the target and hence weighting them equally is sub-efficient. The adaptive part of the paper broaches upon this issue but does not conclude.

importance sampling with multiple MCMC sequences

Posted in Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , , , on October 2, 2015 by xi'an

Vivek Roy, Aixian Tan and James Flegal arXived a new paper, Estimating standard errors for importance sampling estimators with multiple Markov chains, where they obtain a central limit theorem and hence standard error estimates when using several MCMC chains to simulate from a mixture distribution as an importance sampling function. Just before I boarded my plane from Amsterdam to Calgary, which gave me the opportunity to read it completely (along with half a dozen other papers, since it is a long flight!) I first thought it was connecting to our AMIS algorithm (on which convergence Vivek spent a few frustrating weeks when he visited me at the end of his PhD), because of the mixture structure. This is actually altogether different, in that a mixture is made of unnormalised complex enough densities, to act as an importance sampler, and that, due to this complexity, the components can only be simulated via separate MCMC algorithms. Behind this characterisation lurks the challenging problem of estimating multiple normalising constants. The paper adopts the resolution by reverse logistic regression advocated in Charlie Geyer’s famous 1994 unpublished technical report. Beside the technical difficulties in establishing a CLT in this convoluted setup, the notion of mixing importance sampling and different Markov chains is quite appealing, especially in the domain of “tall” data and of splitting the likelihood in several or even many bits, since the mixture contains most of the information provided by the true posterior and can be corrected by an importance sampling step. In this very setting, I also think more adaptive schemes could be found to determine (estimate?!) the optimal weights of the mixture components.