**A**t BNP13, Brian Trippe presented the AISTAT 2022 paper he recently wrote with Tin D. Nguyen and Tamara Broderick. Which made me read their 2021 paper on the topic. There, they note that coupling may prove challenging, which they blame on label switching. Considering a naïve Gibbs sampler on the space of partitions, meaning allocating each data-point to one of the existing partitions or to a singleton, they construct an optimal transport coupling under Hamming distance. Which appears to be achievable in O(NK³log{K}), if K is the maximal number of partitions among both chains. The paper does not goes deeply into the implementation, which involves [to quote] (a) *computing* the* distances between each pair of partitions in the Cartesian product of supports of the Gibbs conditionals* and (b) *solving the optimal transport problem*. Except in the appendix where the book-keeping necessary to achieve O(K²) for pairwise distances and the remaining complexity follows from the standard Orlin’s algorithm. What remains unclear from the paper is that, while the chains couple faster (fastest?), the resulting estimators do not necessarily improve upon budget-equivalent alternatives. (The reason for the failure of the single chain in Figure 2 is hard to fathom.)

## Archive for maximal coupling

## coupling for the Gibbs sampler

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags AISTATS 2022, BNP13, Hamming distance, label switching, Lago Llanquihue, maximal coupling, optimal transport, Orlin's algorithm, Orsono volcano, partition, single chain on November 27, 2022 by xi'an## unbiased MCMC with couplings [4pm, 26 Feb., Paris]

Posted in Books, pictures, Statistics, University life with tags AgroParisTech, All about that Bayes, burns, Claude Bernard, Harvard University, maximal coupling, MCMC, Paris, PSL, seminar, unbiased MCMC, Université Paris Dauphine on February 24, 2020 by xi'an**O**n Wednesday, 26 February, Pierre Jacob (Havard U, currently visiting Paris-Dauphine) is giving a seminar on unbiased MCMC methods with couplings at AgroParisTech, bvd Claude Bernard, Paris 5ième, Room 32, at 4pm in the All about that Bayes seminar.

MCMC methods yield estimators that converge to integrals of interest in the limit of the number of iterations. This iterative asymptotic justification is not ideal; first, it stands at odds with current trends in computing hardware, with increasingly parallel architectures; secondly, the choice of “burn-in” or “warm-up” is arduous. This talk will describe recently proposed estimators that are unbiased for the expectations of interest while having a finite computing cost and a finite variance. They can thus be generated independently in parallel and averaged over. The method also provides practical upper bounds on the distance (e.g. total variation) between the marginal distribution of the chain at a finite step and its invariant distribution. The key idea is to generate “faithful” couplings of Markov chains, whereby pairs of chains coalesce after a random number of iterations. This talk will provide an overview of this line of research.

## 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**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.

## convergences of MCMC and unbiasedness

Posted in pictures, Statistics, University life with tags asynchronous algorithms, Hastings-Metropolis sampler, impatient user, maximal coupling, MCMC convergence, optimal transport, parallelisation, Paris Dauphine, perfect sampling, unbiased MCMC on January 16, 2018 by xi'an**D**uring his talk on unbiased MCMC in Dauphine today, Pierre Jacob provided a nice illustration of the convergence modes of MCMC algorithms. With the stationary target achieved after 100 Metropolis iterations, while the mean of the target taking much more iterations to be approximated by the empirical average. Plus a nice connection between coupling time and convergence. Convergence to the target.During Pierre’s talk, some simple questions came to mind, from developing an “impatient user version”, as in perfect sampling, in order to stop chains that run “forever”, to optimising parallelisation in order to avoid problems of asynchronicity. While the complexity of coupling increases with dimension and the coupling probability goes down, the average coupling time varies but an unexpected figure is that the expected cost per iteration is of 2 simulations, irrespective of the chosen kernels. Pierre also made a connection with optimal transport coupling and stressed that the maximal coupling was for the proposal and not for the target.