Archive for unbiased MCMC

what a party!

Posted in pictures, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , , , on September 13, 2021 by xi'an

We ended up having a terrific b’day party last Thursday after noon, with about 30 friends listening in Institut Henri Poincaré to Florence, Pierre, and Sylvia giving lectures on my favourite themes, namely ABC, MCMC, and mixture inference. Incl. subtle allusions to my many idiosyncrasies in three different flavours!  And a limited number of anecdotes, incl. the unavoidable Cancún glasses disaster! We later headed to a small Ethiopian restaurant located on the other side of the Panthéon, rue de l’Ecole Polytechnique (rather than on the nearby rue Laplace!), which was going to be too tiny for us, especially in these COVID times, until the sky cleared up and the restaurant set enough tables in the small street to enjoy their injeras and wots till almost midnight. The most exciting episode of the evening came when someone tried to steal some of our bags that had been stored in a back room and when Tony spotted the outlier and chased him till the thief dropped the bags..! Thanks to Tony for saving the evening and our computers!!! To Éric, Jean-Michel and Judith for organising this 9/9 event (after twisting my arm just a wee bit). And to all my friends who joined the party, some from far away…

congrats, Pierre!!!

Posted in Statistics with tags , , , , , , , , , on March 3, 2021 by xi'an

simulating hazard

Posted in Books, Kids, pictures, Statistics, Travel with tags , , , , , , , , , , , , on May 26, 2020 by xi'an

A rather straightforward X validated question that however leads to an interesting simulation question: when given the hazard function h(·), rather than the probability density f(·), how does one simulate this distribution? Mathematically h(·) identifies the probability distribution as much as f(·),

1-F(x)=\exp\left\{ \int_{-\infty}^x h(t)\,\text{d}t \right\}=\exp\{H(x)\}

which means cdf inversion could be implemented in principle. But in practice, assuming the integral is intractable, what would an exact solution look like? Including MCMC versions exploiting one fixed point representation or the other.. Since

f(x)=h(x)\,\exp\left\{ \int_{-\infty}^x h(t)\,\text{d}t \right\}

using an unbiased estimator of the exponential term in a pseudo-marginal algorithm would work. And getting an unbiased estimator of the exponential term can be done by Glynn & Rhee debiasing. But this is rather costly… Having Devroye’s book under my nose [at my home desk] should however have driven me earlier to the obvious solution to… simply open it!!! A whole section (VI.2) is indeed dedicated to simulations when the distribution is given by the hazard rate. (Which made me realise this problem is related with PDMPs in that thinning and composition tricks are common to both.) Besides the inversion method, ie X=H⁻¹(U), Devroye suggests thinning a Poisson process when h(·) is bounded by a manageable g(·). Or a generic dynamic thinning approach that converges when h(·) is non-increasing.

unbiased MCMC with couplings [4pm, 26 Feb., Paris]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , on February 24, 2020 by xi'an

On 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 , , , , , , on October 25, 2019 by xi'an

In 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.

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