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 multivariate nonlinear Hawkes processes, Monte-Carlo Markov Chain (MCMC) methods used to sample from the posterior distribution do not scale well to the dimension of the process. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed, however, these methods do not allow to perform model selection on the graph of interactions of the Hawkes model. In this work, we propose a novel adaptive Bayesian variational method that performs model selection and can estimate a sparse graphical parameter. For the popular sigmoid Hawkes processes, we design a parallel algorithm which is scalable to high-dimensional point processes and large sequences of events. Furthermore, we 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. Finally, through an extensive set of numerical simulations, we demonstrate that our method is able to adapt to the dimensionality of the parameter of the Hawkes process, and is partially robust to certain types of model misspecification.

We focus on the estimation of the intensity of a Poisson process in the presence of a uniform noise. We propose a kernel-based procedure fully calibrated in theory and practice. We show that our adaptive estimator is optimal from the oracle and minimax points of view, and provide new lower bounds when the intensity belongs to a Sobolev ball. By developing the Goldenshluger-Lepski methodology in the case of deconvolution for Poisson processes, we propose an optimal data-driven selection of the kernel bandwidth. Our method is illustrated on the spatial distribution of replication origins and sequence motifs along the human genome.

This paper studies the estimation of the conditional density $f(x,\cdot)$ of $Y_i$ given $X_i=x$, from the observation of an i.i.d. sample $(X_i,Y_i)\in \mathbb R^d$, $i\in \{1,\dots,n\}.$ We assume that $f$ depends only on $r$ unknown components with typically $r\ll d$.We provide an adaptive fully-nonparametric strategy based on kernel rules to estimate $f$. To select the bandwidth of our kernel rule, we propose a new fast iterative algorithm inspired by the Rodeo algorithm (Wasserman and Lafferty, 2006) to detect the sparsity structure of $f$. More precisely, in the minimax setting, our pointwise estimator, which is adaptive to both the regularity and the sparsity, achieves the quasi-optimal rate of convergence. Our results also hold for (unconditional) density estimation. The computational complexity of our method is only $O(dn \log n)$. A deep numerical study shows nice performances of our approach.

Sparse linear inverse problems appear in a variety of settings, but often the noise contaminating observations cannot accurately be described as bounded by or arising from a Gaussian distribution. Poisson observations in particular are a characteristic feature of several real-world applications. Previous work on sparse Poisson inverse problems encountered several limiting technical hurdles. This paper describes a novel alternative analysis approach for sparse Poisson inverse problems that 1) sidesteps the technical challenges present in previous work, 2) admits estimators that can readily be computed using off-the-shelf LASSO algorithms, and 3) hints at a general framework for broad classes of noise in sparse linear inverse problems. At the heart of this new approach lies a weighted LASSO estimator for which data-dependent weights are based on Poisson concentration inequalities. Unlike previous analyses of the weighted LASSO, the proposed analysis depends on conditions which can be checked or shown to hold in general settings with high probability. 2000 Math Subject Classification: 60E15, 62G05, 62G08, and 94A12.

High dimensional Poisson regression has become a standard framework for the analysis of massive counts datasets. In this work we estimate the intensity function of the Poisson regression model by using a dictionary approach, which generalizes the classical basis approach, combined with a Lasso or a group-Lasso procedure. Selection depends on penalty weights that need to be calibrated. Standard methodologies developed in the Gaussian framework can not be directly applied to Poisson models due to heteroscedasticity. Here we provide data-driven weights for the Lasso and the group-Lasso derived from concentration inequalities adapted to the Poisson case. We show that the associated Lasso and group-Lasso procedures satisfy fast and slow oracle inequalities. Simulations are used to assess the empirical performance of our procedure, and an original application to the analysis of Next Generation Sequencing data is provided.