**A**n ICLR 2019 paper by Neklyudov, Egorov and Vetrov on an optimal choice of the proposal in an independent Metropolis algorithm I discovered via an X validated question. Namely whether or not the expected Metropolis-Hastings acceptance ratio is always one (which it is not when the support of the proposal is restricted). The paper mentions the domination of the Accept-Reject algorithm by the associated independent Metropolis-Hastings algorithm, which has actually been stated in our Monte Carlo Statistical Methods (1999, Lemma 6.3.2) and may prove even older. The authors also note that the expected acceptance probability is equal to one minus the total variation distance between the joint defined as target x Metropolis-Hastings proposal distribution and its time-reversed version. Which seems to suffer from the same difficulty as the one mentioned in the X validated question. Namely that it only holds when the support of the Metropolis-Hastings proposal is at least the support of the target (or else when the support of the joint defined as target x Metropolis-Hastings proposal distribution is somewhat symmetric. Replacing total variation with Kullback-Leibler then leads to a manageable optimisation target if the proposal is a parameterised independent distribution. With a GAN version when the proposal is not explicitly available. I find it rather strange that one still seeks independent proposals for running Metropolis-Hastings algorithms as the result will depend on the family of proposals considered and as performances will deteriorate with dimension (the authors mention a 10% acceptance rate, which sounds quite low). [As an aside, ICLR 2020 will take part in Addis Abeba next April.]

## Archive for adaptive Monte Carlo algorithm

## an independent sampler that maximizes the acceptance rate of the MH algorithm

Posted in Books, Kids, Statistics, University life with tags accept-reject algorithm, adaptive Monte Carlo algorithm, Addis Abeba, Bayesian GANs, Ethiopia, ICLR 2019, importance sampling, Kullback-Leibler divergence, Monte Carlo Statistical Methods, optimal acceptance rate, optimisation, reversibility, simulation, total variation on September 3, 2019 by xi'an## probably ABC [and provably robust]

Posted in Books, pictures, Statistics, Travel with tags ABC, ABC-SMC, adaptive Monte Carlo algorithm, Bayesian asymptotics, CREST, Gaussian processes, likelihood-free methods, misspecified model, oracle inequalities on August 8, 2017 by xi'an**T**wo weeks ago, James Ridgway (formerly CREST) arXived a paper on misspecification and ABC, a topic on which David Frazier, Judith Rousseau and I have been working for a while now [and soon to be arXived as well]. Paper that I re-read on a flight to Amsterdam [hence the above picture], written as a continuation of our earlier paper with David, Gael, and Judith. One specificity of the paper is to use an exponential distribution on the distance between the observed and simulated sample within the ABC distribution. Which reminds me of the resolution by Bissiri, Holmes, and Walker (2016) of the intractability of the likelihood function. James’ paper contains oracle inequalities between the ABC approximation and the genuine distribution of the summary statistics, like a bound on the distance between the expectations of the summary statistics under both models. Which writes down as a sum of a model bias, of two divergences between empirical and theoretical averages, on smoothness penalties, and on a prior impact term. And a similar bound on the distance between the expected distance to the oracle estimator of θ under the ABC distribution [and a Lipschitz type assumption also found in our paper]. Which first sounded weird [to me] as I would have expected the true posterior, until it dawned on me that the ABC distribution is the one used for the estimation [a passing strike of over-Bayesianism!]. While the oracle bound could have been used directly to discuss the rate of convergence of the exponential rate λ to zero [with the sample size n], James goes into the interesting alternative direction of setting a prior on λ, an idea that dates back to Olivier Catoni and Peter Grünwald. Or rather a pseudo-posterior on λ, a common occurrence in the PAC-Bayesian literature. In one of his results, James obtains a dependence of λ on the dimension m of the summary [as well as the root dependence on the sample size n], which seems to contradict our earlier independence result, until one realises this scale parameter is associated with a distance variable, itself scaled in m.

The paper also contains a non-parametric part, where the parameter θ is the unknown distribution of the data and the summary the data itself. Which is quite surprising as I did not deem it possible to handle non-parametrics with ABC. Especially in a misspecified setting (although I have trouble perceiving what this really means).

“We can use most of the Monte Carlo toolbox available in this context.”

The theoretical parts are a bit heavy on notations and hard to read [as a vacation morning read at least!]. They are followed by a Monte Carlo implementation using SMC-ABC. And pseudo-marginals [at least formally as I do not see how the specific features of pseudo-marginals are more that an augmented representation here]. And adaptive multiple pseudo-samples that reminded me of the Biometrika paper of Anthony Lee and Krys Latuszynski (Warwick). Therefore using indeed most of the toolbox!

## and another one on nested sampling

Posted in Books, Statistics with tags adaptive Monte Carlo algorithm, efficient importance sampling, Monte Carlo Statistical Methods, nested sampling on May 2, 2017 by xi'an**T**he same authors as those of the paper discussed last week arXived a paper on dynamic nested sampling.

“We propose modifying the nested sampling algorithm by dynamically varying the number of “live points” in order to maximise the accuracy of a calculation for some number of posterior sample.”

Some of the material is actually quite similar to the previous paper (to the point I had to check they were not the *same* paper). The authors rightly point out that the main source of variation in the nested sampling approximation is due to the Monte Carlo variability in the estimated volume of the level sets.

The main notion in that paper is that it is acceptable to have a varying number of “live” points in nested sampling provided the weights are correctly accordingly. Adding more of those points as a new “thread” in a region where the likelihood changes rapidly. Addition may occur at any level of the likelihood, in fact, and is determined in the paper by an importance weight being in the upper tail of the importance weights… While the description is rather vague [for instance I do not get the notation in (9)] and the criteria for adding threads somewhat arbitrary, I find interesting that several passes at different precision levels can improve the efficiency of the nested approximation at a given simulation cost. Remains the issue of whether or not this is a sufficient perk for attracting users of other simulation techniques to the nested galaxy…

## inference with Wasserstein distance

Posted in Books, Statistics, University life with tags 17w5025, adaptive Monte Carlo algorithm, Banff, BIRS, Canada, empirical distribution, Harvard University, numerical transport, optimal transport, statistical inference, synthetic data, Wasserstein distance on January 23, 2017 by xi'an**T**oday, Pierre Jacob posted on arXiv a paper of ours on the use of the Wasserstein distance in statistical inference, which main focus is exploiting this distance to create an automated measure of discrepancy for ABC. Which is why the full title is Inference in generative models using the Wasserstein distance. Generative obviously standing for the case when a model can be generated from but cannot be associated with a closed-form likelihood. We had all together discussed this notion when I visited Harvard and Pierre last March, with much excitement. (While I have not contributed much more than that round of discussions and ideas to the paper, the authors kindly included me!) The paper contains theoretical results for the consistency of statistical inference based on those distances, as well as computational on how the computation of these distances is practically feasible and on how the Hilbert space-filling curve used in sequential quasi-Monte Carlo can help. The notion further extends to dependent data via delay reconstruction and residual reconstruction techniques (as we did for some models in our empirical likelihood BCel paper). I am quite enthusiastic about this approach and look forward discussing it at the 17w5015 BIRS ABC workshop, next month!