Inference of brain functional connectivity networks from resting-state fMRI data is a key focus in neuroimaging. This paper introduces new Bayesian approaches for inferring a functional connectivity graph from multivariate resting-state fMRI time series of a single subject. Our methods rely on novel Bayesian priors on correlation matrices and a dedicated prior elicitation framework, which translates prior beliefs about the expected level and variability of correlations into interpretable hyperparameter choices, enabling the construction of expert-informed priors. When combined with a Gaussian likelihood, these priors also exhibit computational advantages. Compared to most existing methods for this problem that estimate constant weights, our model provides distributional weights defined by the posterior distributions for the connectivity graph, yielding more robust point estimates through the regularizing effect of expert-informed priors, evaluating uncertainty, and enabling a range of post-inference analyses. In particular, we derive a procedure for identifying significant connectivities based on posterior distributions of weights and credible sets. To the best of our knowledge, only one existing Bayesian functional connectivity model is applicable to single-subject resting-state fMRI data, making our approach a valuable addition to the field and demonstrating superior performance in our experiments.

This paper establishes the optimal sub-Gaussian variance proxy for truncated Gaussian and truncated exponential random variables. The proofs rely on first characterizing the optimal variance proxy as the unique solution to a set of two equations and then observing that for these two truncated distributions, one may find explicit solutions to this set of equations. Moreover, we establish the conditions under which the optimal variance proxy coincides with the variance, thereby characterizing the strict sub-Gaussianity of the truncated random variables. Specifically, we demonstrate that truncated Gaussian variables exhibit strict sub-Gaussian behavior if and only if they are symmetric, meaning their truncation is symmetric with respect to the mean. Conversely, truncated exponential variables are shown to never exhibit strict sub-Gaussian properties. These findings contribute to the understanding of these prevalent probability distributions in statistics and machine learning, providing a valuable foundation for improved and optimal modeling and decision-making processes.

Compositional score-based approaches to simulation-based inference (SBI) approximate the posterior over a shared parameter given n independent observations by aggregating individually learned posterior scores: currently, there are two main propositions of such methods (Geffner et al. (2023), Linhart et al. (2026)). As the resulting composite score does not correspond to the score of any distribution along the forward diffusion path of the true multi-observation posterior, sampling from it via a reverse SDE leads to an irreducible bias. Annealed Langevin dynamics provides a principled alternative: it treats the composite score as the genuine score of a sequence of tractable bridging densities and samples from them in succession. When properly tuned, it could lead to a controllable bias. However, its hyperparameters, namely step sizes, the number of steps per level, and the number of annealing levels, have so far been chosen empirically. We derive Wasserstein bounds for annealed Langevin with approximate scores and translate them into explicit decision rules for these hyperparameters that guarantee a prescribed sampling accuracy, while highlighting different theoretical aspects of each composite score formulation. In the Gaussian setting, we obtain closed-form expressions for all relevant quantities and prove that the bridging densities of Linhart et al. (2026) consistently admit larger step sizes and require fewer total Langevin steps than those of Geffner et al. (2023). Furthermore, we show empirically that the tuning obtained in the Gaussian setting generalizes to more complex problems, thus providing a well-understood and theoretically grounded starting point for practitioners using compositional score-based approaches.

We introduce a general framework that extends Bayesian inference by allowing the researcher to explicitly encode confidence in each source of uncertainty within the model. This mechanism provides a new handle for model design and regularisation control. Building on this framework, we develop a general approach for inducing sparsity in statistical models and illustrate its use in linear and logistic regression, as well as in Bayesian neural networks.

We propose a Bayesian framework for uncertainty quantification and comparison in brain connectivity graph analysis. Standard graph-based approaches typically rely on point estimates of correlation matrices, overlooking the uncertainty induced by high-dimensional estimation from limited data. Our methodology constructs and compares credible hyperrectangles derived from posterior distributions, providing interpretable tools for subject-level inference and longitudinal monitoring. We develop scalable algorithms for estimating these regions in high dimensions and establish theoretical guarantees in the inverse-Wishart model for resting-state fMRI data, including a Bernstein–von Mises theorem for correlation matrices and control of a Bayesian family-wise error rate. The proposed framework enables principled detection of significant connectivity differences both globally and locally while preserving joint dependency structures. While demonstrating competitive performance against multiple-testing procedures on synthetic datasets, our approach also facilitates the direct comparison of two distinct scans from a single patient, a capability currently absent from the literature. We leverage this novelty on real datasets to improve interpretability. Beyond fMRI data, the approach provides a general framework for comparison problems in high-dimensional dependent settings.

We present a novel approach to ecological risk assessment by recasting the species sensitivity distribution (SSD) method within a Bayesian nonparametric (BNP) framework. Widely mandated by environmental regulatory bodies globally, SSD has faced criticism due to its historical reliance on parametric assumptions when modelling species variability. By adopting nonparametric mixture models, we address this limitation, establishing a statistically robust foundation for SSD. Our BNP approach offers several advantages, including its efficacy in handling small datasets or censored data, which are common in ecological risk assessment, and its ability to provide principled uncertainty quantification alongside simultaneous density estimation and clustering. We utilize a specific nonparametric prior as the mixing measure, chosen for its robust clustering properties, a crucial consideration given the lack of strong prior beliefs about the number of components. Through simulation studies and analysis of real datasets, we demonstrate the superiority of our BNP-SSD over classical SSD methods. We also provide a BNP-SSD Shiny application, making our methodology available to the Ecotoxicology community. Moreover, we exploit the inherent clustering structure of the mixture model to explore patterns in species sensitivity. Our findings underscore the effectiveness of the proposed approach in improving ecological risk assessment methodologies.

Inference from interaction maps, such as centromere identification from genome-wide chromosome conformation capture techniques – notably Hi-C – can be formulated as a generic inverse problem: infer a set of parameters given a map summarizing pairwise interactions between entities through blocks of variable numbers and sizes. In this work, we introduce a data-driven approach that leverages shared structure between these maps, such as global alignment between localized patterns, while handling the variability in number and size of entities arising in real-world data. Our approach relies on a transformer architecture capable of handling such variability and a custom simulator to generate abundant, yet computationally cheap synthetic data for training. Applied to the problem of centromere localization, the method accurately recovers their genomic positions across a wide range of species of various genome sizes.

Variational methods are widely used to approximate posterior distributions in Bayesian inference when exact computation is infeasible. We study when such approximations inherit posterior consistency. Our first result shows that, on a general metric space, a uniform bound on the Kullback–Leibler divergence from the approximating measures to a tight sequence of target measures forces the approximating sequence to be tight. It follows that if the target posteriors converge weakly to a Dirac mass at the true parameter, then any variational sequence with bounded Kullback–Leibler divergence to the targets is also consistent. We also give simple logarithmic-moment conditions that verify this boundedness condition, and illustrate them for smooth generalised posterior distributions.

We approach multivariate mode estimation through Gibbs distributions and introduce GERVE (Gibbs-measure Entropy-Regularised Variational Estimation), a likelihood-free framework that approximates Gibbs measures directly from samples by maximizing an entropy-regularised variational objective with natural-gradient updates. GERVE brings together kernel density estimation, mean-shift, variational inference, and annealing in a single platform for mode estimation. It fits Gaussian mixtures that concentrate on high-density regions and yields cluster assignments from responsibilities, with reduced sensitivity to the chosen number of components. We provide theory in two regimes: as the Gibbs temperature approaches zero, mixture components converge to population modes; at fixed temperature, maximisers of the empirical objective exist, are consistent, and are asymptotically normal. We also propose a bootstrap procedure for per-mode confidence ellipses and stability scores. Simulation and real-data studies show accurate mode recovery and emergent clustering, robust to mixture overspecification. GERVE is a practical likelihood-free approach when the number of modes or groups is unknown and full density estimation is impractical.

We address the online unconstrained submodular maximization problem (Online USM), in a setting with stochastic bandit feedback. In this framework, a decision-maker receives noisy rewards from a non monotone submodular function taking values in a known bounded interval. This paper proposes Double-Greedy - Explore-then-Commit (DG-ETC), adapting the Double-Greedy approach from the offline and online full-information settings. DG-ETC satisfies a O(d log(dT)) problem-dependent upper bound for the 1/2-approximate pseudo-regret, as well as a O(dT(2/3) log(dT)(1/3)) problem-free one at the same time, outperforming existing approaches. In particular, we introduce a problem-dependent notion of hardness characterizing the transition between logarithmic and polynomial regime for the upper bounds.

Diagnosing convergence of Markov chain Monte Carlo is crucial and remains an essentially unsolved problem. Among the most popular methods, the potential scale reduction factor, commonly named $\hat{R}$, is an indicator that monitors the convergence of output chains to a target distribution, based on a comparison of the between- and within-variances. Several improvements have been suggested since its introduction in the 90s. Here, we aim at better understanding the $\hat{R}$ behavior by proposing a localized version that focuses on quantiles of the target distribution. This new version relies on key theoretical properties of the associated population value. It naturally leads to proposing a new indicator $\hat{R}_\infty$, which is shown to allow both for localizing the Markov chain Monte Carlo convergence in different quantiles of the target distribution, and at the same time for handling some convergence issues not detected by other $\hat{R}$ versions.

We investigate the problem of characterizing the optimal variance proxy for sub-Gaussian random variables,whose moment-generating function exhibits bounded growth at infinity. We apply a general characterization method to discrete random variables with equally spaced atoms. We thoroughly study 3-mass distributions, thereby generalizing the well-studied Bernoulli case. We also prove that the discrete uniform distribution over $N$ points is strictly sub-Gaussian. Finally, we provide an open-source Python package that combines analytical and numerical approaches to compute optimal sub-Gaussian variance proxies across a wide range of distributions.

Simulation-based inference (SBI) has become a widely used framework in applied sciences for estimating the parameters of stochastic models that best explain experimental observations. A central question in this setting is how to effectively combine multiple observations in order to improve parameter inference and obtain sharper posterior distributions. Recent advances in score-based diffusion methods address this problem by constructing a compositional score, obtained by aggregating individual posterior scores within the diffusion process. While it is natural to suspect that the accumulation of individual errors may significantly degrade sampling quality as the number of observations grows, this important theoretical issue has so far remained unexplored. In this paper, we study the compositional score produced by the GAUSS algorithm of Linhart et al. (2024) and establish an upper bound on its mean squared error in terms of both the individual score errors and the number of observations. We illustrate our theoretical findings on a Gaussian example, where all analytical expressions can be derived in a closed form.

We introduce a new multimodal optimization approach called Natural Variational Annealing (NVA) that combines the strengths of three foundational concepts to simultaneously search for multiple global and local modes of black-box nonconvex objectives. First, it implements a simultaneous search by using variational posteriors, such as, mixtures of Gaussians. Second, it applies annealing to gradually trade off exploration for exploitation. Finally, it learns the variational search distribution using natural-gradient learning where updates resemble well-known and easy-to-implement algorithms. The three concepts come together in NVA giving rise to new algorithms and also allowing us to incorporate "fitness shaping", a core concept from evolutionary algorithms. We assess the quality of search on simulations and compare them to methods using gradient descent and evolution strategies. We also provide an application to a real-world inverse problem in planetary science.

We present a method to improve the calibration of deep ensembles in the small data regime in the presence of unlabeled data. Our approach, which we name U-ensembles, is extremely easy to implement: given an unlabeled set, for each unlabeled data point, we simply fit a different randomly selected label with each ensemble member. We provide a theoretical analysis based on a PAC-Bayes bound which guarantees that for such a labeling we obtain low negative log-likelihood and high ensemble diversity on testing samples. Empirically, through detailed experiments, we find that for low to moderately-sized training sets, U-ensembles are more diverse and provide better calibration than standard ensembles.

We address the problem of stochastic combinatorial semi-bandits, where a player selects among $P$ actions from the power set of a set containing $d$ base items. Adaptivity to the problem's structure is essential in order to obtain optimal regret upper bounds. As estimating the coefficients of a covariance matrix can be manageable in practice, leveraging them should improve the regret. We design ``optimistic'' covariance-adaptive algorithms relying on online estimations of the covariance structure, called OLS-UCB-C and COS-V (only the variances for the latter). They both yields improved gap-free regret. Although COS-V can be slightly suboptimal, it improves on computational complexity by taking inspiration from Thompson Sampling approaches. It is the first sampling-based algorithm satisfying a $\sqrt{T}$ gap-free regret (up to poly-logs). We also show that in some cases, our approach efficiently leverages the semi-bandit feedback and outperforms bandit feedback approaches, not only in exponential regimes where $P\gg d$ but also when $P\leq d$, which is not covered by existing analyses.

We present a method to improve the calibration of deep ensembles in the small data regime in the presence of unlabeled data. Our approach, which we name $\nu$-ensembles, is extremely easy to implement: given an unlabeled set, for each unlabeled data point, we simply fit a different randomly selected label with each ensemble member. We provide a theoretical analysis based on a PAC-Bayes bound which guarantees that for such a labeling we obtain low negative log-likelihood and high ensemble diversity on testing samples. Empirically, through detailed experiments, we find that for low to moderately-sized training sets, $\nu$-ensembles are more diverse and provide better calibration than standard ensembles, sometimes significantly.

Constraint-based methods and noise-based methods are two distinct families of methods proposed for uncovering causal graphs from observational data. However, both operate under strong assumptions that may be challenging to validate or could be violated in real-world scenarios. In response to these challenges, there is a growing interest in hybrid methods that amalgamate principles from both methods, showing robustness to assumption violations. This paper introduces a novel comprehensive framework for hybridizing constraint-based and noise-based methods designed to uncover causal graphs from observational time series. The framework is structured into two classes. The first class employs a noise-based strategy to identify a super graph, containing the true graph, followed by a constraint-based strategy to eliminate unnecessary edges. In the second class, a constraint-based strategy is applied to identify a skeleton, which is then oriented using a noise-based strategy. The paper provides theoretical guarantees for each class under the condition that all assumptions are satisfied, and it outlines some properties when assumptions are violated. To validate the efficacy of the framework, two algorithms from each class are experimentally tested on simulated data, realistic ecological data, and real datasets sourced from diverse applications. Notably, two novel datasets related to Information Technology monitoring are introduced within the set of considered real datasets. The experimental results underscore the robustness and effectiveness of the hybrid approaches across a broad spectrum of datasets.

Large classes of problems can be formulated as inverse problems, where the goal is to find parameter values that best explain some observed measures. The relationship between parameters and observations is typically highly non-linear, with relatively high dimensional observations and correlated multidimensional parameters. To deal with these constraints via inverse regression strategies, we consider the Gaussian Local Linear Mapping (GLLiM) model, a special instance of mixture of expert models. We propose a general scheme to design a Bayesian nonparametric GLLiM model to avoid any commitment to an arbitrary number of experts. A tractable estimation algorithm is designed using variational Bayesian expectation-maximisation. We establish posterior consistency for the number of mixture components after the merge-truncate-merge algorithm post-processing. Illustrations on simulated data show good results in terms of recovering the true number of experts and the regression function.