arXiv subjects

Judith Rousseau

Publications and source records attributed to Judith Rousseau.

At least 19 recordsLinked to original sources

arXiv2026

The Bernstein-von Mises theorem and efficiency for semiparametric inference in multivariate Hawkes processes

In this paper, we study semiparametric inference for linear multivariate Hawkes processes, a class of point processes widely used to describe self and mutually exciting phenomena. We establish a convolution theorem giving the best limiting distribution for a regular estimator of smooth functional. Then, in the Bayesian setting, we prove a semiparametric Bernstein-von Mises (BvM) theorem for nonparametric random series priors. We apply this result to histogram and wavelet based priors. Taken together, the convolution and BvM theorems show that, from a frequentist point of view, semiparametric Bayesian procedures have asymptotically the optimal behavior. Deriving the BvM property for random series priors led us to prove L2 posterior contraction, complementing for these priors the results of Donnet, Rivoirard and Rousseau (2020).

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arXiv2026

Uncertainty Decomposition for Bayes-Filtered Transformers via Bayesian Predictive Inference

Bayes-filtered transformers are transformers meta-learned on sequences from a prior predictive distribution to approximate the corresponding posterior predictive distribution. They output total predictive uncertainty in a single forward pass but never explicitly represent a posterior distribution, making the standard route to separating aleatoric from epistemic uncertainty unavailable. We address this challenge through the lens of Bayesian predictive inference (BPI). Our main result is a predictive Central Limit Theorem (CLT) for supervised settings under conditions that are among the weakest known in the BPI literature. The CLT characterises the posterior of the limiting predictive distribution given an observed context as asymptotically Gaussian; the variance of this Gaussian quantifies epistemic uncertainty. We apply the framework to TabPFN, a Bayes-filtered transformer that is a state-of-the-art foundation model for tabular prediction. The resulting credible bands achieve near-nominal frequentist coverage as context length grows, and the decomposition largely matches standard desiderata: epistemic uncertainty shrinks with context length and is highest in sparsely observed regions within the span of the context data, while aleatoric uncertainty dominates near decision boundaries where classes overlap.

stat.ML↗

arXiv2026

Persuasive Privacy

We propose a novel framework for measuring privacy from a Bayesian game-theoretic perspective. This framework enables the creation of new, purpose-driven privacy definitions that are rigorously justified, while also allowing for the assessment of existing privacy guarantees through game theory. We show that pure and probabilistic differential privacy are special cases of our framework, and provide new interpretations of the post-processing inequality in this setting. Further, we demonstrate that privacy guarantees can be established for deterministic algorithms, which are overlooked by current privacy standards.

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arXiv2025

Estimation in linear high dimensional Hawkes processes: a Bayesian approach

In this paper we study the frequentist properties of Bayesian approaches in linear high dimensional Hawkes processes in a sparse regime where the number of interaction functions acting on each component of the Hawkes process is much smaller than the dimension. We consider two types of loss function: the empirical $L_1$ distance between the intensity functions of the process and the $L_1$ norm on the parameters (background rates and interaction functions). Our results are the first results to control the $L_1$ norm on the parameters under such a framework. They are also the first results to study Bayesian procedures in high dimensional Hawkes processes.

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arXiv2025

Adaptive Diffusion Guidance via Stochastic Optimal Control

Guidance is a cornerstone of modern diffusion models, playing a pivotal role in conditional generation and enhancing the quality of unconditional samples. However, current approaches to guidance scheduling--determining the appropriate guidance weight--are largely heuristic and lack a solid theoretical foundation. This work addresses these limitations on two fronts. First, we provide a theoretical formalization that precisely characterizes the relationship between guidance strength and classifier confidence. Second, building on this insight, we introduce a stochastic optimal control framework that casts guidance scheduling as an adaptive optimization problem. In this formulation, guidance strength is not fixed but dynamically selected based on time, the current sample, and the conditioning class, either independently or in combination. By solving the resulting control problem, we establish a principled foundation for more effective guidance in diffusion models.

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arXiv2024

Convergence of Diffusion Models Under the Manifold Hypothesis in High-Dimensions

Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio, and video generation as well as many more applications in science and beyond. The \textit{manifold hypothesis} states that high-dimensional data often lie on lower-dimensional manifolds within the ambient space, and is widely believed to hold in provided examples. While recent results have provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models, making this a very fruitful research direction. In this work, we study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of score learning. In terms of sampling complexity, we obtain rates independent of the ambient dimension w.r.t.\ the Wasserstein distance. We do this by developing a new framework connecting diffusion models to the well-studied theory of extrema of Gaussian Processes.

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arXiv2024

Bayesian spatiotemporal modelling of political violence and conflict events using discrete-time Hawkes processes

The monitoring of conflict risk in the humanitarian sector is largely based on simple historic averages. The overarching goal of this work is to assess the potential for using a more statistically rigorous approach to monitor the risk of political violence and conflict events in practice, and thereby improve our understanding of their temporal and spatial patterns, to inform preventative measures. In particular, a Bayesian, spatiotemporal variant of the Hawkes process is fitted to data gathered by the Armed Conflict Location and Event Data (ACLED) project to obtain sub-national estimates of conflict risk in South Asia over time and space. Our model can effectively estimate the risk level of these events within a statistically sound framework, with a more precise understanding of uncertainty than was previously possible. The model also provides insights into differences in behaviours between countries and conflict types. We also show how our model can be used to monitor short and long term trends, and that it is more stable and robust to outliers compared to current practices that rely on historical averages.

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arXiv2024

Nonparametric regression on random geometric graphs sampled from submanifolds

We consider the nonparametric regression problem when the covariates are located on an unknown smooth compact submanifold of a Euclidean space. Under defining a random geometric graph structure over the covariates we analyze the asymptotic frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion in the graph Laplacian eigenbasis. Under Holder smoothness assumption on the regression function and the density of the covariates over the submanifold, we prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index.

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arXiv2024

Empirical Bayes in Bayesian learning: understanding a common practice

In applications of Bayesian procedures, once a class of priors has been chosen, it may be tempting to fix the prior's hyperparameters from the data, in an empirical Bayes (EB) fashion, usually by their maximum marginal likelihood estimates (MMLE). This is a quite common but questionable practice, lacking a rigorous theoretical basis. We provide a theoretical framework where this form of EB is regarded as a computational strategy for approximating a genuine Bayesian posterior distribution and prove its general properties for parametric models. While computing the MMLE may still be demanding, we prove novel results that allow us to provide a simple proxy. These results establish the limit behavior of the MMLE in quite general settings, including both identifiable and non-identifiable models - specifically, overfitted mixture models - significantly filling a gap in the literature. Moreover, we study higher order merging, showing that, when not degenerate, the EB posterior approximates at a faster rate an oracle-Bayes posterior distribution based on the prior law that, within the given class of priors, expresses the most information on the true model's parameters. This is a faster approximation than classic Bernstein-von Mises results. Our work provides formal content to common beliefs on this popular practice.

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arXiv2023

Nonparametric Bayesian intensity estimation for covariate-driven inhomogeneous point processes

This work studies nonparametric Bayesian estimation of the intensity function of an inhomogeneous Poisson point process in the important case where the intensity depends on covariates, based on the observation of a single realisation of the point pattern over a large area. It is shown how the presence of covariates allows to borrow information from far away locations in the observation window, enabling consistent inference in the growing domain asymptotics. In particular, optimal posterior contraction rates under both global and point-wise loss functions are derived. The rates in global loss are obtained under conditions on the prior distribution resembling those in the well established theory of Bayesian nonparametrics, combined with concentration inequalities for functionals of stationary processes to control certain random covariate-dependent loss functions appearing in the analysis. The local rates are derived with an ad-hoc study that builds on recent advances in the theory of P\'olya tree priors, extended to the present multivariate setting with a novel construction that makes use of the random geometry induced by the covariates.

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arXiv2023

Asymptotics of approximate Bayesian computation when summary statistics converge at heterogeneous rates

We consider the asymptotic properties of Approximate Bayesian Computation (ABC) for the realistic case of summary statistics with heterogeneous rates of convergence. We allow some statistics to converge faster than the ABC tolerance, other statistics to converge slower, and cover the case where some statistics do not converge at all. We give conditions for the ABC posterior to converge, and provide an explicit representation of the shape of the ABC posterior distribution in our general setting; in particular, we show how the shape of the posterior depends on the number of slow statistics. We then quantify the gain brought by the local linear post-processing step.

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arXiv2023

Wasserstein convergence in Bayesian and frequentist deconvolution models

We study the multivariate deconvolution problem of recovering the distribution of a signal from independent and identically distributed observations additively contaminated with random errors (noise) from a known distribution. For errors with independent coordinates having ordinary smooth densities, we derive an inversion inequality relating the $L^1$-Wasserstein distance between two distributions of the signal to the $L^1$-distance between the corresponding mixture densities of the observations. This smoothing inequality outperforms existing inversion inequalities. As an application of the inversion inequality to the Bayesian framework, we consider $1$-Wasserstein deconvolution with Laplace noise in dimension one using a Dirichlet process mixture of normal densities as a prior measure on the mixing distribution (or distribution of the signal). We construct an adaptive approximation of the sampling density by convolving the Laplace density with a well-chosen mixture of normal densities and show that the posterior measure concentrates around the sampling density at a nearly minimax rate, up to a log-factor, in the $L^1$-distance. The same posterior law is also shown to automatically adapt to the unknown Sobolev regularity of the mixing density, thus leading to a new Bayesian adaptive estimation procedure for mixing distributions with regular densities under the $L^1$-Wasserstein metric. We illustrate utility of the inversion inequality also in a frequentist setting by showing that an appropriate isotone approximation of the classical kernel deconvolution estimator attains the minimax rate of convergence for $1$-Wasserstein deconvolution in any dimension $d\geq 1$, when only a tail condition is required on the latent mixing density and we derive sharp lower bounds for these problems

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arXiv2023

Posterior Contraction Rates for Mat\'ern Gaussian Processes on Riemannian Manifolds

Gaussian processes are used in many machine learning applications that rely on uncertainty quantification. Recently, computational tools for working with these models in geometric settings, such as when inputs lie on a Riemannian manifold, have been developed. This raises the question: can these intrinsic models be shown theoretically to lead to better performance, compared to simply embedding all relevant quantities into $\mathbb{R}^d$ and using the restriction of an ordinary Euclidean Gaussian process? To study this, we prove optimal contraction rates for intrinsic Mat\'ern Gaussian processes defined on compact Riemannian manifolds. We also prove analogous rates for extrinsic processes using trace and extension theorems between manifold and ambient Sobolev spaces: somewhat surprisingly, the rates obtained turn out to coincide with those of the intrinsic processes, provided that their smoothness parameters are matched appropriately. We illustrate these rates empirically on a number of examples, which, mirroring prior work, show that intrinsic processes can achieve better performance in practice. Therefore, our work shows that finer-grained analyses are needed to distinguish between different levels of data-efficiency of geometric Gaussian processes, particularly in settings which involve small data set sizes and non-asymptotic behavior.

stat.ML↗

arXiv2023

Semiparametric posterior corrections

We present a new approach to semiparametric inference using corrected posterior distributions. The method allows us to leverage the adaptivity, regularization and predictive power of nonparametric Bayesian procedures to estimate low-dimensional functionals of interest without being restricted by the holistic Bayesian formalism. Starting from a conventional nonparametric posterior, we target the functional of interest by transforming the entire distribution with a Bayesian bootstrap correction. We provide conditions for the resulting $\textit{one-step posterior}$ to possess calibrated frequentist properties and specialize the results for several canonical examples: the integrated squared density, the mean of a missing-at-random outcome, and the average causal treatment effect on the treated. The procedure is computationally attractive, requiring only a simple, efficient post-processing step that can be attached onto any arbitrary posterior sampling algorithm. Using the ACIC 2016 causal data analysis competition, we illustrate that our approach can outperform the existing state-of-the-art through the propagation of Bayesian uncertainty.

stat.ME↗

arXiv2022

Scalable and adaptive variational Bayes methods for Hawkes processes

Hawkes processes are often applied to model dependence and interaction phenomena in multivariate event data sets, such as neuronal spike trains, social interactions, and financial transactions. In the nonparametric setting, learning the temporal dependence structure of Hawkes processes is generally a computationally expensive task, all the more with Bayesian estimation methods. In particular, for generalised nonlinear Hawkes processes, Monte-Carlo Markov Chain methods applied to compute the doubly intractable posterior distribution are not scalable to high-dimensional processes in practice. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed. In this work, we first unify existing variational Bayes approaches under a general nonparametric inference framework, and analyse the asymptotic properties of these methods under easily verifiable conditions on the prior, the variational class, and the nonlinear model. Secondly, we propose a novel sparsity-inducing procedure, and derive an adaptive mean-field variational algorithm for the popular sigmoid Hawkes processes. Our algorithm is parallelisable and therefore computationally efficient in high-dimensional setting. Through an extensive set of numerical simulations, we also demonstrate that our procedure is able to adapt to the dimensionality of the parameter of the Hawkes process, and is partially robust to some type of model mis-specification.

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arXiv2022

A flexible, random histogram kernel for discrete-time Hawkes processes

Hawkes processes are a self-exciting stochastic process used to describe phenomena whereby past events increase the probability of the occurrence of future events. This work presents a flexible approach for modelling a variant of these, namely discrete-time Hawkes processes. Most standard models of Hawkes processes rely on a parametric form for the function describing the influence of past events, referred to as the triggering kernel. This is likely to be insufficient to capture the true excitation pattern, particularly for complex data. By utilising trans-dimensional Markov chain Monte Carlo inference techniques, our proposed model for the triggering kernel can take the form of any step function, affording significantly more flexibility than a parametric form. We first demonstrate the utility of the proposed model through a comprehensive simulation study. This includes univariate scenarios, and multivariate scenarios whereby there are multiple interacting Hawkes processes. We then apply the proposed model to several case studies: the interaction between two countries during the early to middle stages of the COVID-19 pandemic, taking Italy and France as an example, and the interaction of terrorist activity between two countries in close spatial proximity, Indonesia and the Philippines, and then within three regions of the Philippines.

stat.ME↗

arXiv2022

Estimating a density near an unknown manifold: a Bayesian nonparametric approach

We study the Bayesian density estimation of data living in the offset of an unknown submanifold of the Euclidean space. In this perspective, we introduce a new notion of anisotropic H\"older for the underlying density and obtain posterior rates that are minimax optimal and adaptive to the regularity of the density, to the intrinsic dimension of the manifold, and to the size of the offset, provided that the latter is not too small -- while still allowed to go to zero. Our Bayesian procedure, based on location-scale mixtures of Gaussians, appears to be convenient to implement and yields good practical results, even for quite singular data.

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arXiv2022

Evidence estimation in finite and infinite mixture models and applications

Estimating the model evidence - or mariginal likelihood of the data - is a notoriously difficult task for finite and infinite mixture models and we reexamine here different Monte Carlo techniques advocated in the recent literature, as well as novel approaches based on Geyer (1994) reverse logistic regression technique, Chib (1995) algorithm, and Sequential Monte Carlo (SMC). Applications are numerous. In particular, testing for the number of components in a finite mixture model or against the fit of a finite mixture model for a given dataset has long been and still is an issue of much interest, albeit yet missing a fully satisfactory resolution. Using a Bayes factor to find the right number of components K in a finite mixture model is known to provide a consistent procedure. We furthermore establish the consistence of the Bayes factor when comparing a parametric family of finite mixtures against the nonparametric 'strongly identifiable' Dirichlet Process Mixture (DPM) model.

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