Metropolis-adjusted interacting particle sampling

In recent years, various interacting particle samplers have been developed to sample from complex target distributions, such as those found in Bayesian inverse problems. These samplers are motivated by the mean-field limit perspective and implemented as ensembles of particles that move in the product state space according to coupled stochastic differential equations. The ensemble approximation and numerical time stepping used to simulate these systems can introduce bias and affect the invariance of the particle system with respect to the target distribution. To correct for this, we investigate the use of a Metropolization step, similar to the Metropolis-adjusted Langevin algorithm. We examine Metropolization of either the whole ensemble or smaller subsets of the ensemble, and prove basic convergence of the resulting ensemble Markov chain to the target distribution. Our numerical results demonstrate the benefits of this correction in numerical examples for popular interacting particle samplers such as ALDI, CBS, and stochastic SVGD.

Multimodal Laplace-based sampling

Building on the approximation result in Hellinger distance, we study the use of a Laplace-
based approximation as a proposal distribution for both importance sampling and the in-
dependent Metropolis–Hastings algorithm, and analyze the efficiency of these methods. For
importance sampling, we quantify efficiency via the effective sample size and establish con-
vergence rates in the regime of increasingly concentrated posterior distributions. For the
independent Metropolis–Hastings algorithm, we derive convergence rates for the average ac-
ceptance probability, the lag-one autocorrelation, and the spectral gap. Finally, we validate
our theoretical findings through numerical experiments.

From Filtering to Inversion: Kalman-Based Methods for Nonlinear Parameter Estimation in Geotechnics

The Kalman filter is a well-established method for data assimilation. Its extensions - the Ensemble Kalman Filter (EnKF) and the Unscented Kalman Filter (UKF) - are widely used for filtering in nonlinear dynamical systems while avoiding explicit linearization. In recent years, the underlying principles of the EnKF and the UKF have been extended to Kalman inversion algorithms for inverse problems, such as Ensemble Kalman Inversion (EKI) and Unscented Kalman Inversion (UKI). These methods have proven effective for solving inverse problems and performing uncertainty quantification in complex models while remaining computationally scalable. This perspective is illustrated for both EKI and UKI through examples from geotechnics.

Slice Sampling using Approximations

Slice sampling is a well-established Markov chain Monte Carlo method for drawing (approximate) samples from a posterior distribution that is often known only up to a normalizing constant. The method is based on choosing a new state on a slice, i.e., a superlevel set of the (unnormalized) target density with respect to a reference measure. However, slice sampling algorithms typically require multiple evaluations of the target density per iteration and can therefore become computationally expensive, particularly in high-dimensional settings. To mitigate these costs, in this talk we examine how deterministic approximations of the target density can be incorporated into slice sampling schemes. We demonstrate the effectiveness of our methods with several numerical experiments in the context of Bayesian inference.

Stochastic Discontinuous Galerkin Methods for PDE-Based Models with Random Coefficient

Many mathematical models arising in science and engineering involve uncertain input data, such as material properties, boundary conditions, or source terms. These uncertainties are naturally described by partial differential equations with random coefficients, making uncertainty quantification an essential component in the reliable prediction of physical processes. 

This talk is concerned with stochastic discontinuous Galerkin methods for convection–diffusion equations with random coefficients. Random fields are represented by means of the Karhunen–Loève expansion, while the stochastic Galerkin framework is employed to transform the underlying problem into a coupled deterministic system. For the spatial discretization, discontinuous Galerkin methods are used due to their robustness and favorable approximation properties for convection-dominated problems together with efficient adaptive strategies for resolving boundary and interior layers. Since stochastic Galerkin discretizations typically lead to large coupled algebraic systems, low-rank iterative techniques are considered to reduce the computational complexity. Several numerical examples demonstrate the performance of the proposed methods and highlight their potential for the efficient treatment of uncertainty in PDE-based models.
 

From Brier to 'h% correct': communicating predictive accuracy as hit rates

Communicating the accuracy of probabilistic forecasts and estimates to non-quantitative audiences remains a persistent obstacle in applied work. Strictly proper scoring rules are the principled tools for measuring error, but their numerical values rarely carry intuitive meaning: neither lay decision-makers nor domain experts typically possess a frame of reference for what constitutes a "low" or "high" score. We propose a general reparametrization. For any strictly proper scoring rule with finite expected loss, the score of a probabilistic forecast admits a certainty-equivalent hit rate h:
the accuracy that a binary classifier would have to achieve to deliver the same expected utility to the population of decision-makers implicitly weighted by the scoring rule. The construction rests on the Schervish-type representation of proper scoring rules as mixtures of cost-loss decision problems. Instead of reporting the expected loss, we suggest reporting the certainty-equivalent hit rate — "this forecast is h% accurate". Most importantly, this framing retains propriety — h is maximized by truthful reporting — and preserves a defensible decision-theoretic meaning. For symmetric rules such as the Brier score, h is simply 1 minus the (normalized) error; for asymmetric rules, h also depends on the specification of a type-I/type-II error split. Non-finite rules such as the logarithmic score admit no such finite equivalent. The approach is legitimate but not harmless: because the certainty-equivalent hit rate depends on the chosen scoring rule, h can be inflated or deflated by the analyst's rule choice. We discuss the resulting scope for manipulation and argue that a transparent disclosure of the underlying scoring rule is necessary. Whether certainty-equivalent accuracy scores improve uncertainty communication in practice is an open empirical question.