Archive for embarrassingly parallel MCMC

coupling-based approach to f-divergences diagnostics for MCMC

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , on October 27, 2025 by xi'an

Adrien Corenflos (University of Warwick) and Hai-Dang Dau (NUS) just arXived their paper on MCMC diagnostics that Adrien told me about last month, while in Warwick.

“This [f-divergence] bound is clearly suboptimal since it does not vary in t and does not take into account the mixing of the Markov chain. We present a scheme where the weights are ‘harmonized’ as the Markov chain progresses, reflecting its mixing through the notion of coupling.”

They start by opposing the classical ergodic average and embarrassingly parallel estimates obtained by N parallel chains culled of their B initial values, to couplings used in standard diagnoses. Opting for the parallel perspective, maybe rekindling the diagnostic war of the early 1990s! The evaluation tool in the paper is based on f-divergences, like the χ² divergence which naturally relates to the effective sample size when considering weighted atomic measures. When consistent, these weighted approximations produce upper bounds on the f-divergence, with exact convergence in case of independence.

In my opinion the most exciting part of the paper stands with the ability to modify these weights along MCMC iterations, since the naïve sequential importance sampling argument I also use in class keeps them constant! The trick is to (be able to) couple randomly chosen parallel chains, with the weights being averaged at each coupling event. The resulting algorithm preserves expectation (in the importance sampling sense) and consistency (in the particle sense). Furthermore, the f-divergence bound based on the weights can only decrease between iterations, which reminds me of interleaving. And exponential convergence of the weights to uniform ones (under the strong assumption of a uniformly lower bounded probability of coupling). The paper concludes with interesting remarks on perfect sampling, Rao-Blackwellisation, control variates, and backward sampling.

A long-standing gap exists between the theoretical analysis of Markov chain Monte Carlo convergence, which is often based on statistical divergences, and the diagnostics used in practice. We introduce the first general convergence diagnostics for Markov chain Monte Carlo based on any f χ² -divergence, allowing users to directly monitor, among others, the Kullback–Leibler and the divergences as well as the Hellinger and the total variation distances. Our first key contribution is a coupling-based ‘weight harmonization’ scheme that produces a direct, computable, and consistent weighting of interacting Markov chains with respect to their target distribution. The second key contribution is to show how such consistent weightings of empirical measures can be used to provide upper bounds to f -divergences in general. We prove that these bounds are guaranteed to tighten over time and converge to zero as the chains approach stationarity, providing a concrete diagnostic.

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All about that [Bayes] seminar [24 Jan]

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , on January 13, 2025 by xi'an

The next All about that (Bayes) seminar will take place on Friday 24 Jan at SCAI, on the Jussieu campus, with the following talks. (Appearances to the contrary, I was not in the least involved in the program!)

13h30 – 14h30 Joshua Bon (OCEAN, Université Paris Dauphine) – Bayesian score calibration for approximate models

 Scientists continue to develop increasingly complex mechanistic models to reflect their knowledge more realistically. Statistical inference using these models can be challenging since the corresponding likelihood function is often intractable and model simulation may be computationally burdensome. Fortunately, in many of these situations, it is possible to adopt a surrogate model or approximate likelihood function. It may be convenient to conduct Bayesian inference directly with the surrogate, but this can result in bias and poor uncertainty quantification. In this paper (https://arxiv.org/abs/2211.05357) we propose a new method for adjusting approximate posterior samples to reduce bias and produce more accurate uncertainty quantification. We do this by optimizing a transform of the approximate posterior that maximizes a scoring rule. Our approach requires only a (fixed) small number of complex model simulations and is numerically stable. We demonstrate beneficial corrections to several approximate posteriors using our method on several examples of increasing complexity.

14h30 – 15h30 Giacomo Zanella (Bocconi University) – Entropy contraction of the Gibbs sampler under log-concavity

In this talk I will present recent work (https://arxiv.org/abs/2410.00858) on the non-asymptotic analysis of the Gibbs sampler, a classical and popular MCMC algorithm for sampling. In particular, under the assumption that the probability measure π of interest is strongly log-concave, we show that the random scan Gibbs sampler contracts in relative entropy, and provide a sharp characterization of the associated contraction rate. The result implies that, under appropriate conditions, the number of full evaluations of π required for the Gibbs sampler to converge is independent of the dimension. If time permits, I will also discuss connections and applications of the above results to the problem of zero-order parallel sampling, as well as extensions to Hit-and-Run and Metropolis-within-Gibbs.

Based on joint work with Filippo Ascolani and Hugo Lavenant.

16h00 – 17h00 Paul Bastide (Université Paris Cité) – Goodness of Fit for Bayesian Generative Models with Applications in Population Genetics

In population genetics, inference about intractable likelihood models is common, and simulation methods, including Approximate Bayesian Computation (ABC) and Simulation-Based Inference (SBI), are essential. ABC/SBI methods work by simulating instrumental data sets of the models under study and comparing them with the observed data set y⁰. Advanced machine learning tools are used for tasks such as model selection and parameter inference. The present work focuses on model criticism. This type of analysis, called goodness of fit (GoF), is important for model validation. It can also be used for model pruning when the number of candidates to be considered is excessive, especially in the context where data simulation is expensive. We introduce two new GoF tests based on the local outlier factor (LOF), an indicator that was initially defined for outlier and novelty detection. We test whether y⁰ is distributed from the prior predictive distribution (pre-inference GoF) and whether there is a parameter value such that y⁰ is distributed from the likelihood with that value (post-inference GoF).  We evaluate the performance of our two GoF tests on simulated datasets from three different model settings of varying complexity, and on a dataset of single nucleotide polymorphism (SNP) markers for the evaluation of complex evolutionary scenarios of modern human populations.

Joint work with Guillaume Le Mailloux, Jean-Michel Marin and Arnaud Estoup.

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deep and embarrassingly parallel MCMC

Posted in Books, pictures, Statistics with tags , , , , , , , on April 9, 2019 by xi'an

Diego Mesquita, Paul Blomstedt, and Samuel Kaski (from Helsinki, like the above picture) just arXived a paper on embarrassingly parallel MCMC. Following a series of papers discussed on this ‘og in the past. They use a deep learning approach of Dinh et al. (2017) to the computation of the probability density of a convoluted and non-volume-preserving transform of a given random variable to turn multiple samples from sub-posteriors [corresponding to the k k-th roots of the true posterior] into a sample from the true posterior. If I understand correctly the argument [on page 4], the deep neural network provides a density estimate that apparently does better than traditional non-parametric density estimates. Maybe by being more efficient than a Parzen-Rosenblat estimator which is of order the number of simulations… For any value of θ, the estimate of the true target is the product of these estimates and for a value of θ simulated from one of the subposteriors an importance weight naturally ensues. However, for a one-dimensional transform of θ, h(θ), I would prefer estimating first the density of h(θ) for each sample and then construct an importance weight. If only to avoid the curse of dimension.

On various benchmarks, like the banana-shaped 2D target above, the proposed method (NAP) does better. Even in relatively high dimensions. Given that the overall computing times are not produced, with only the calibration that the same number of subsamples were produced for all methods, it would be interesting to test the same performances on even higher dimensions and larger population sizes.

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