## black box MCMC

Posted in Books, Statistics with tags , , , , , , , , on July 17, 2021 by xi'an

“…back-box methods, despite using no information of the proposal distribution, can actually give better estimation accuracy than the typical importance sampling [methods]…”

Earlier this week I was pointed out to Liu & Lee’s black box importance sampling, published in AISTATS 2017. (which I did not attend). Already found in Briol et al. (2015) and Oates, Girolami, and Chopin (2017), the method starts from Charles Stein‘s “unbiased estimator of the loss” (that was a fundamental tool in my own PhD thesis!), a variation on integration by part:

$\mathbb E_p[\nabla\log p(X) f(X)+\nabla f(X)]=0$

for differentiable functions f and p cancelling at the boundaries. It also holds for the kernelised extension

$\mathbb E_p[k_p(X,x')]=0$

for all x’, where the integrand is a 1-d function of an arbitrary kernel k(x,x’) and of the score function ∇log p. This null expectation happens to be a minimum since

$\mathbb E_{X,X'\sim q}[k_p(X,X')]\ge 0$

and hence importance weights can be obtained by minimising

$\sum_{ij} w_i w_j k_p(x_i,x_j)$

in w (from the unit simplex), for a sample of iid realisations from a possibly unknown distribution with density q. Liu & Lee show that this approximation converges faster than the standard Monte Carlo speed √n, when using Hilbertian properties of the kernel through control variates. Actually, the same thing happens when using a (leave-one-out) non-parametric kernel estimate of q rather than q. At least in theory.

“…simulating n parallel MCMC chains for m steps, where the length m of the chains can be smaller than what is typically used in MCMC, because it just needs to be large enough to bring the distribution `roughly’ close to the target distribution”

A practical application of the concept is suggested in the above quote. As a corrected weight for interrupted MCMC. Or when using an unadjusted Langevin algorithm. Provided the minimisation of the objective quadratic form is fast enough, the method can thus be used as a benchmark for regular MCMC implementation.

## Russian roulette still rolling

Posted in Statistics with tags , , , , , , , , , , , , on March 22, 2017 by xi'an

Colin Wei and Iain Murray arXived a new version of their paper on doubly-intractable distributions, which is to be presented at AISTATS. It builds upon the Russian roulette estimator of Lyne et al. (2015), which itself exploits the debiasing technique of McLeish et al. (2011) [found earlier in the physics literature as in Carter and Cashwell, 1975, according to the current paper]. Such an unbiased estimator of the inverse of the normalising constant can be used for pseudo-marginal MCMC, except that the estimator is sometimes negative and has to be so as proved by Pierre Jacob and co-authors. As I discussed in my post on the Russian roulette estimator, replacing the negative estimate with its absolute value does not seem right because a negative value indicates that the quantity is close to zero, hence replacing it with zero would sound more appropriate. Wei and Murray start from the property that, while the expectation of the importance weight is equal to the normalising constant, the expectation of the inverse of the importance weight converges to the inverse of the weight for an MCMC chain. This however sounds like an harmonic mean estimate because the property would also stand for any substitute to the importance density, as it only requires the density to integrate to one… As noted in the paper, the variance of the resulting Roulette estimator “will be high” or even infinite. Following Glynn et al. (2014), the authors build a coupled version of that solution, which key feature is to cut the higher order terms in the debiasing estimator. This does not guarantee finite variance or positivity of the estimate, though. In order to decrease the variance (assuming it is finite), backward coupling is introduced, with a Rao-Blackwellisation step using our 1996 Biometrika derivation. Which happens to be of lower cost than the standard Rao-Blackwellisation in that special case, O(N) versus O(N²), N being the stopping rule used in the debiasing estimator. Under the assumption that the inverse importance weight has finite expectation [wrt the importance density], the resulting backward-coupling Russian roulette estimator can be proven to be unbiased, as it enjoys a finite expectation. (As in the generalised harmonic mean case, the constraint imposes thinner tails on the importance function, which then hampers the convergence of the MCMC chain.) No mention is made of achieving finite variance for those estimators, which again is a serious concern due to the similarity with harmonic means…

## Florid’AISTATS

Posted in pictures, R, Statistics, Travel, University life with tags , , , , , , , , , on August 31, 2016 by xi'an

The next AISTATS conference is taking place in Florida, Fort Lauderdale, on April 20-22. (The website keeps the same address one conference after another, which means all my links to the AISTATS 2016 conference in Cadiz are no longer valid. And that the above sunset from Florida is named… cadiz.jpg!) The deadline for paper submission is October 13 and there are two novel features:

1. Fast-track for Electronic Journal of Statistics: Authors of a small number of accepted papers will be invited to submit an extended version for fast-track publication in a special issue of the Electronic Journal of Statistics (EJS) after the AISTATS decisions are out. Details on how to prepare such extended journal paper submission will be announced after the AISTATS decisions.
2. Review-sharing with NIPS: Papers previously submitted to NIPS 2016 are required to declare their previous NIPS paper ID, and optionally supply a one-page letter of revision (similar to a revision letter to journal editors; anonymized) in supplemental materials. AISTATS reviewers will have access to the previous anonymous NIPS reviews. Other than this, all submissions will be treated equally.

I find both initiatives worth applauding and replicating in other machine-learning conferences. Particularly in regard with the recent debate we had at Annals of Statistics.