Archive for Euler-Maruyama discretisation

mostly Monte Carlo in June

Posted in Statistics, University life with tags , , , , , , , , , , , , , , , , , , on May 30, 2026 by xi'an

The last episode of the academic year for our mostly Monte Carlo seminar, next week:

On Friday 05/06/26, from 3-5pm at PariSanté Campus

15h00: Sam Livingstoke (University College London)

Skew-symmetric numerical schemes for stochastic differential equations: strong convergence and multi-level extension
I will discuss recent work fusing together two strands of the applied mathematics and statistics literature, one concerned with developing flexible probability distributions for data that rely on a small number of parameters, and another concerned with developing numerical integration schemes to simulate stochastic processes.  The specific case that I will focus on uses the skew-symmetric family of probability distributions introduced by Adelchi Azzalini and co-authors to approximate the transition kernels of diffusion processes over small time steps, producing alternative numerical schemes to the classical Euler-Maruyama approach.  Applying the scheme to the overdamped Langevin diffusion leads to an unadjusted version of the Barker proposal Metropolis-Hastings algorithm.  In earlier work weak accuracy was established over finite and infinite time scales, crucially without needing a globally Lipschitz assumption on the drift of the stochastic differential equation.  I will review this and then discuss more recent work establishing strong convergence in the mean-squared sense using a novel coupling between the numerical and exact processes.  This also enables the development of a multi-level Monte Carlo scheme, which I will discuss the merits of with particular focus on the superlinear drift case, as compared to Euler and Tamed Euler alternatives.
This is joint work with Yuga Iguchi, Giorgos Vasdekis & Rui-Yang Zhang.
16h00: Dana Naderi (Université Paris Dauphine PSL)
Approximating evidence via bounded harmonic means

Efficient Bayesian model selection relies on the model evidence or marginal likelihood, whose computation often requires evaluating an intractable integral. The harmonic mean estimator (HME) has long been a standard method of approximating the evidence. While computationally simple, the version introduced by Newton and Raftery (1994) potentially suffers from infinite variance. To overcome this issue, Gelfand and Dey (1994) defined a standardized representation of the estimator based on an instrumental function and Robert and Wraith (2009) later proposed to use higher posterior density (HPD) indicators as instrumental functions. Following this approach, a practical method is proposed, based on an elliptical covering of the HPD region with non-overlapping ellipsoids. The resulting estimator, called the Elliptical Covering Marginal Likelihood Estimator (ECMLE), not only eliminates the infinite-variance issue of the original HME and allows exact volume computations, but is also able to be used in multimodal settings. Through several examples, we illustrate that ECMLE outperforms other recent methods such as THAMES and its improved version (Metodiev et al. 2025). Moreover, ECMLE demonstrates lower variance a key challenge that subsequent HME variants have sought to address-and provides more stable evidence approximations, even in challenging settings.

This is joint work with Kaniav Kamari, Dareen Wraith & myself (X).

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ABC in Lapland

Posted in Mountains, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , on March 15, 2023 by xi'an

Greetings from Levi, Lapland! Sonia Petrone beautifully started the ABC workshop with a (the!) plenary Sunday night talk on quasi-Bayes in the spirit of both Fortini & Petrone (2020) and the more recent Fong, Holmes, and Walker (2023). The talk got me puzzled by wondering the nature of convergence, in that it happens no matter what the underlying distribution (or lack thereof) of the data is, in that, even without any exchangeability structure, the predictive is converging. The quasi stems from a connection with the historical Smith and Markov (1978) sequential update approximation for the posterior attached with mixtures of distributions. Which itself relates to both Dirichlet posterior updates and Bayesian bootstrap à la Newton & Raftery. Appropriate link when the convergence seems to stem from the sequence of predictives instead of the underlying distribution, if any, pulling Bayes by its own bootstrap…! Chris Holmes also talked the next day about this approach, esp. about a Bayesian approach to causality that does not require counterfactuals, in connection with a recent arXival of his (on my reading list).

Carlo Alberto presented both his 2014 SABC (simulated annealing) algorithm with a neat idea of reducing waste in the tempering schedule and a recent summary selection approach based on an auto-encoder function of both y and noise to reduce to sufficient statistic. A similar idea was found in Yannik Schälte’s talk (slide above). Who was returning to Richard Wiilkinson’s exact ABC¹³ with adaptive sequential generator, also linking to simulated annealing and ABC-SMC¹² to the rescue. Notion of amortized inference. Seemingly approximating data y with NN and then learn parameter by a normalising flow.

David Frazier talked on Q-posterior²³ approach, based on Fisher’s identity, for approximating score function, which first seemed to require some exponential family structure on a completed model (but does not, after discussing with David!), Jack Jewson on beta divergence priors²³ for uncertainty on likelihoods, better than KLD divergence on e-contamination situations, any impact on ABC? Masahiro Fujisawa back to outliers impact on ABC, again with e-contaminations (with me wondering at the impact of outliers on NN estimation).

In the afternoon session (due to two last minute cancellations, we skipped (or [MCMC] skied) one afternoon session, which coincided with a bright and crispy day, how convenient! ), Massi Tamborino (U of Warwick) FitzHugh-Nagumo process, with impossibilities to solve the inference problem differently, for instance Euler-Maruyama does not always work, numerical schemes are inducing a bias. Back to ABC with the hunt for a summary that get rid of the noise, as in Carlo Alberto’s work. Yuexi Wang talked about her works on adversarial ABC inspired from GANs. Another instance where noise is used as input. True data not used in training? Imke Botha discussed an improvement to ensemble Kalman inversion which, while biased, gains over both regular SMC timewise and ensemble Kalman inversion in precision, and Chaya Weerasinghe focussed on Bayesian forecasting in state space models under model misspecification, via approximate Bayesian computation, using an auxiliary model to produce summary statistics as in indirect inference.

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