**I**n the June issue of Biometrika, which had been sitting for a few weeks on my desk under my teapot!, Jeremy Heng and Pierre Jacob published a paper on unbiased estimators for Hamiltonian Monte Carlo using couplings. (Disclaimer: I was not involved with the review or editing of this paper.) Which extends to HMC environments the earlier paper of Pierre Jacob, John O’Leary and Yves Atchadé, to be discussed soon at the Royal Statistical Society. The fundamentals are the same, namely that an unbiased estimator can be produced from a converging sequence of estimators and that it can be *de facto* computed if two Markov chains with the same marginal can be coupled. The issue with Hamiltonians is to figure out how to couple their dynamics. In the Gaussian case, it is relatively easy to see that two chains with the same initial momentum meet periodically. In general, there is contraction within a compact set (Lemma 1). The coupling extends to a time discretisation of the Hamiltonian flow by a leap-frog integrator, still using the same momentum. Which roughly amounts in using the same random numbers in both chains. When defining a relaxed meeting (!) where both chains are within δ of one another, the authors rely on a drift condition (8) that reminds me of the early days of MCMC convergence and seem to imply the existence of a small set “where the target distribution [density] is strongly log-concave”. And which makes me wonder if this small set could be used instead to create renewal events that would in turn ensure both stationarity and unbiasedness without the recourse to a second coupled chain. When compared on a Gaussian example with couplings on Metropolis-Hastings and MALA (Fig. 1), the coupled HMC sees hardly any impact of the dimension of the target (in the average coupling time), with a much lower value. However, I wonder at the relevance of the meeting time as an assessment of efficiency. In the sense that the coupling time is not a convergence time but reflects as well on the initial conditions. I acknowledge that this allows for an averaging over parallel implementations but I remain puzzled by the statement that this leads to “estimators that are consistent in the limit of the number of replicates, rather than in the usual limit of the number of Markov chain iterations”, since a particularly poor initial distribution could on principle lead to a mode of the target being never explored or on the coupling time being ever so rarely too large for the computing abilities at hand.

## Archive for discussion paper

## unbiased Hamiltonian Monte Carlo with couplings

Posted in Books, Kids, Statistics, University life with tags Biometrika, discussion paper, Hamiltonian Monte Carlo, leapfrog integrator, maximal coupling, Royal Statistical Society, unbiased MCMC on October 25, 2019 by xi'an## latent nested nonparametric priors

Posted in Books, Statistics with tags Bayesian Analysis, Bayesian non-parametrics, clustering, completely random measures, Dirichlet mixture priors, discussion paper, hierarchical Bayesian modelling, iris data on September 23, 2019 by xi'an**A** paper on an extended type of non-parametric priors by Camerlenghi et al. [all good friends!] is about to appear in Bayesian Analysis, with a discussion open for contributions (**until October 15**). While a fairly theoretical piece of work, it validates a Bayesian approach for non-parametric clustering of separate populations with, broadly speaking, common clusters. More formally, it constructs a new family of models that allows for a partial or complete equality between two probability measures, but does not force full identity when the associated samples do share some common observations. Indeed, the more traditional structures prohibit one or the other, from the Dirichlet process (DP) prohibiting two probability measure realisations from being equal or partly equal to some hierarchical DP (HDP) already allowing for common atoms across measure realisations, but prohibiting complete identity between two realised distributions, to nested DP offering one extra level of randomness, but with an infinity of DP realisations that prohibits common atomic support besides completely identical support (and hence distribution).

The current paper imagines *two* realisations of random measures written as a sum of a common random measure and of one of two separate almost independent random measures: (14) is the core formula of the paper that allows for partial or total equality. An extension to a setting larger than facing *two* samples seems complicated if only because of the number of common measures one has to introduce, from the totally common measure to measures that are only shared by a subset of the samples. Except in the simplified framework when a single and universally common measure is adopted (with enough justification). The randomness of the model is handled via different completely random measures that involved something like four degrees of hierarchy in the Bayesian model.

Since the example is somewhat central to the paper, the case of one or rather two two-component Normal mixtures with a common component (but with different mixture weights) is handled by the approach, although it seems that it was already covered by HDP. Having exactly the same term (i.e., with the very same weight) is not, but this may be less interesting in real life applications. Note that alternative & easily constructed & parametric constructs are already available in this specific case, involving a limited prior input and a lighter computational burden, although the Gibbs sampler behind the model proves extremely simple on the paper. (One may wonder at the robustness of the sampler once the case of identical distributions is visited.)

Due to the combinatoric explosion associated with a higher number of observed samples, despite obvious practical situations, one may wonder at any feasible (and possibly sequential) extension, that would further keep a coherence under marginalisation (in the number of samples). And also whether or not multiple testing could be coherently envisioned in this setting, for instance when handling all hospitals in the UK. Another consistency question covers the Bayes factor used to assess whether the two distributions behind the samples are or not identical. (One may wonder at the importance of the question, hopefully applied to more relevant dataset than the Iris data!)

## Bayesian conjugate gradients [open for discussion]

Posted in Books, pictures, Statistics, University life with tags Bayesian Analysis, Bayesian methods for hackers, discussion paper, probabilistic numerics, probabilistic programming, University of Warwick on June 25, 2019 by xi'an**W**hen fishing for an illustration for this post on Google, I came upon this Bayesian methods for hackers cover, a book about which I have no clue whatsoever (!) but that mentions probabilistic programming. Which serves as a perfect (?!) introduction to the call for discussion in Bayesian Analysis of the incoming Bayesian conjugate gradient method by Jon Cockayne, Chris Oates (formerly Warwick), Ilse Ipsen and Mark Girolami (still partially Warwick!). Since indeed the paper is about probabilistic numerics à la Mark and co-authors. Surprisingly dealing with solving the deterministic equation Ax=b by Bayesian methods. The method produces a posterior distribution on the solution x⁰, given a fixed computing effort, which makes it pertain to the anytime algorithms. It also relates to an earlier 2015 paper by Christian Hennig where the posterior is on A⁻¹ rather than x⁰ (which is quite a surprising if valid approach to the problem!) The computing effort is translated here in computations of projections of random projections of Ax, which can be made compatible with conjugate gradient steps. Interestingly, the choice of the prior on x is quite important, including setting a low or high convergence rate… **Deadline is August 04!**

## the end of the Series B’log…

Posted in Books, Statistics, University life with tags blogging, discussion paper, Journal of the Royal Statistical Society, Series B, Series B'log on September 22, 2017 by xi'an**T**oday is the last and final day of Series B’log as David Dunson, Piotr Fryzlewicz and myself have decided to stop the experiment, *faute de combattants*. (As we say in French.) The authors nicely contributed long abstracts of their papers, for which I am grateful, but with a single exception, no one came out with comments or criticisms, and the idea to turn some Series B papers into discussion papers does not seem to appeal, at least in this format. Maybe the concept will be rekindled in another form in the near future, but for now we let it lay down. So be it!

## beyond objectivity, subjectivity, and other ‘bjectivities

Posted in Statistics with tags Andrew Gelman, Christian Hennig, discussion paper, Errol Street, frequentist inference, London, objectivism, Read paper, Royal Statistical Society, RSS, Series A, statistical modelling, subjective versus objective Bayes, subjectivity on April 12, 2017 by xi'an**H**ere is my discussion of Gelman and Hennig at the Royal Statistical Society, which I am about to deliver!