**V**eronicka Rockova (from Chicago Booth) gave a talk on this theme at the Oxford Stats seminar this afternoon. Starting with a survey of ABC, synthetic likelihoods, and pseudo-marginals, to motivate her approach via GANs, learning an approximation of the likelihood from the GAN discriminator. Her explanation for the GAN type estimate was crystal clear and made me wonder at the connection with Geyer’s 1994 logistic estimator of the likelihood (a form of discriminator with a fixed generator). She also expressed the ABC approximation hence created as the actual posterior times an exponential tilt. Which she proved is of order 1/n. And that a random variant of the algorithm (where the shift is averaged) is unbiased. Most interestingly requiring no calibration and no tolerance. Except indirectly when building the discriminator. And no summary statistic. Noteworthy tension between correct shape and correct location.

## Archive for synthetic likelihood

## Metropolis-Hastings via classification

Posted in pictures, Statistics, Travel, University life with tags ABC, ABC consistency, Chicago, Chicago Booth School of Business, deep learning, discriminant analysis, GANs, logistic regression, seminar, summary statistics, synthetic likelihood, University of Oxford, webinar, winter running on February 23, 2021 by xi'an## sequential neural likelihood estimation as ABC substitute

Posted in Books, Kids, Statistics, University life with tags ABC, AISTATS 2019, AMIS, autoregressive flow, Bayesian inference, Gaussian copula, Gaussian processes, indirect inference, JMLR, Kullback-Leibler divergence, MCMC, neural density estimator, neural network, noise-contrastive estimation, normalizing flow, Scotland, synthetic likelihood, University of Edinburgh, variational Bayes methods 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…

## uncertainty in the ABC posterior

Posted in Statistics with tags ABC, Bayesian Analysis, Gaussian processes, misspecified model, Monte Carlo error, prior predictive, synthetic likelihood on July 24, 2019 by xi'an**I**n the most recent Bayesian Analysis, Marko Järvenpää et al. (including my coauthor Aki Vehtari) consider an ABC setting where the number of available simulations of pseudo-samples is limited. And where they want to quantify the amount of uncertainty resulting from the estimation of the ABC posterior density. Which is a version of the Monte Carlo error in practical ABC, in that this is the difference between the ABC posterior density for a given choice of summaries and a given choice of tolerance, and the actual approximation based on a finite number of simulations from the prior predictive. As in earlier works by Michael Gutmann and co-authors, the focus stands in designing a sequential strategy to decide where to sample the next parameter value towards minimising a certain expected loss. And in adopting a Gaussian process modelling for the discrepancy between observed data and simulated data, hence generalising the synthetic likelihood approach. This allows them to compute the expectation and the variance of the unnormalised ABC posterior, based on plugged-in estimators. From where the authors derive a loss as the expected variance of the acceptance probability (although it is not parameterisation invariant). I am unsure I see the point for this choice in that there is no clear reason for the resulting sequence of parameter choices to explore the support of the posterior distribution in a relatively exhaustive manner. The paper also mentions alternatives where the next parameter is chosen at the location where “the uncertainty of the unnormalised ABC posterior is highest”. Which sounds more pertinent to me. And further avoids integrating out the parameter. I also wonder if ABC mis-specification analysis could apply in this framework since the Gaussian process is most certainly a “wrong” model. (When concluding this post, I realised I had written a similar entry two years ago about the earlier version of the paper!)

## prepaid ABC

Posted in Books, pictures, Statistics, University life with tags ABC, Approximate Bayesian computation, KU Leuven, Leuven, likelihood-free methods, machine learning, neural network, reproducible research, support vector machines, synthetic likelihood on January 16, 2019 by xi'an**M**erijn Mestdagha, Stijn Verdoncka, Kristof Meersa, Tim Loossensa, and Francis Tuerlinckx from the KU Leuven, some of whom I met during a visit to its Wallon counterpart Louvain-La-Neuve, proposed and arXived a new likelihood-free approach based on saving simulations on a large scale for future users. Future users interested in the *same* model. The *very same* model. This makes the proposal quite puzzling as I have no idea as to when situations with exactly the same experimental conditions, up to the sample size, repeat over and over again. Or even just repeat once. (Some particular settings may accommodate for different sample sizes and the same prepaid database, but others as in genetics clearly do not.) I am sufficiently puzzled to suspect I have missed the message of the paper.

“In various fields, statistical models of interest are analytically intractable. As a result, statistical inference is greatly hampered by computational constraint s. However, given a model, different users with different data are likely to perform similar computations. Computations done by one user are potentially useful for other users with different data sets. We propose a pooling of resources across researchers to capitalize on this. More specifically, we preemptively chart out the entire space of possible model outcomes in a prepaid database. Using advanced interpolation techniques, any individual estimation problem can now be solved on the spot. The prepaid method can easily accommodate different priors as well as constraints on the parameters. We created prepaid databases for three challenging models and demonstrate how they can be distributed through an online parameter estimation service. Our method outperforms state-of-the-art estimation techniques in both speed (with a 23,000 to 100,000-fold speed up) and accuracy, and is able to handle previously quasi inestimable models.”

I foresee potential difficulties with this proposal, like compelling all future users to rely on the same summary statistics, on the same prior distributions (the “representative amount of parameter values”), and requiring a massive storage capacity. Plus furthermore relying at its early stage on the most rudimentary form of an ABC algorithm (although not acknowledged as such), namely the rejection one. When reading the description in the paper, the proposed method indeed selects the parameters (simulated from a prior or a grid) that are producing pseudo-observations that are closest to the actual observations (or their summaries s). The subsample thus constructed is used to derive a (local) non-parametric or machine-learning predictor s=f(θ). From which a point estimator is deduced by minimising in θ a deviance d(s⁰,f(θ)).

The paper does not expand much on the theoretical justifications of the approach (including the appendix that covers a formal situation where the prepaid grid conveniently covers the observed statistics). And thus does not explain on which basis confidence intervals should offer nominal coverage for the prepaid method. Instead, the paper runs comparisons with Simon Wood’s (2010) synthetic likelihood maximisation (Ricker model with three parameters), the rejection ABC algorithm (species dispersion trait model with four parameters), while the Leaky Competing Accumulator (with four parameters as well) seemingly enjoys no alternative. Which is strange since the first step of the prepaid algorithm is an ABC step, but I am unfamiliar with this model. Unsurprisingly, in all these cases, given that the simulation has been done prior to the computing time for the prepaid method and not for either synthetic likelihood or ABC, the former enjoys a massive advantage from the start.

“The prepaid method can be used for a very large number of observations, contrary to the synthetic likelihood or ABC methods. The use of very large simulated data sets allows investigation of large-sample properties of the estimator”

To return to the general proposal and my major reservation or misunderstanding, for different experiments, the (true or pseudo-true) value of the parameter will not be the same, I presume, and hence the region of interest [or grid] will differ. While, again, the computational gain is *de facto* obvious [since the costly production of the reference table is not repeated], and, to repeat myself, makes the comparison with methods that do require a massive number of simulations from scratch massively in favour of the prepaid option, I do not see a convenient way of recycling these prepaid simulations for another setting, that is, when some experimental factors, sample size or collection, or even just the priors, do differ. Again, I may be missing the point, especially in a specific context like repeated psychological experiments.

While this may have some applications in reproducibility (but maybe not, if the goal is in fact to detect cherry-picking), I see very little use in repeating the same statistical model on different datasets. Even repeating observations will require additional nuisance parameters and possibly perturb the likelihood and/or posterior to large extents.