We address the problem of estimating the distribution of presumed i.i.d. observations within the framework of Bayesian statistics. We propose a new posterior distribution that shares some similarities with the classical Bayesian one. In particular, when the statistical model is exact, we show that this new posterior distribution concentrates its mass around the target distribution, just as the classical Bayes posterior would do. However, unlike the Bayes posterior, we prove that these concentration properties remain stable when the equidistribution assumption is violated or when the data are i.i.d. with a distribution that does not belong to our model but only lies close enough to it. The results we obtain are non-asymptotic and involve explicit numerical constants.[-]

Les processus de Hawkes forment une classe des processus ponctuels pour lesquels l'intensité s'écrit comme :

$\lambda(t)= \int_{0}^{t^-} h(t-s)dN_s +\nu$

où $N$ représente le processus de Hawkes, et $\nu > 0$. Les processus de Hawkes multivariés ont une intensité similaire sauf que des interractions entre les différentes composantes du processus de Hawkes sont autorisées. Les paramètres de ce modèle sont donc les fonctions d'interractions $h_{k,\ell}, k, \ell \le M$ et les constantes $\nu_\ell, \ell \le M$. Dans ce travail nous étudions une approche bayésienne nonparamétrique pour estimer les fonctions $h_{k,\ell}$ et les constantes $\nu_\ell$. Nous présentons un théorème général caractérisant la vitesse de concentration de la loi a posteriori dans de tels modèles. L'intérêt de cette approche est qu'elle permet la caractérisation de la convergence en norme $L_1$ et demande assez peu d'hypothèses sur la forme de la loi a priori. Une caractérisation de la convergence en norme $L_2$ est aussi considérée. Nous étudierons un exemple de lois a priori adaptées à l'étude des interractions neuronales. Travail en collaboration avec S. Donnet et V. Rivoirard.[-]

This talk focuses on the estimation of the distribution of unobserved nodes in large random graphs from the observation of very few edges. These graphs naturally model tournaments involving a large number of players (the nodes) where the ability to win of each player is unknown. The players are only partially observed through discrete valued scores (edges) describing the results of contests between players. In this very sparse setting, we present the first nonasymptotic risk bounds for maximum likelihood estimators (MLE) of the unknown distribution of the nodes. The proof relies on the construction of a graphical model encoding conditional dependencies that is extremely efficient to study n-regular graphs obtained using a round-robin scheduling. This graphical model allows to prove geometric loss of memory properties and deduce the asymptotic behavior of the likelihood function. Following a classical construction in learning theory, the asymptotic likelihood is used to define a measure of performance for the MLE. Risk bounds for the MLE are finally obtained by subgaussian deviation results derived from concentration inequalities for Markov chains applied to our graphical model.[-]

This course will give a gentle introduction to SMC (Sequential Monte Carlo algorithms):
• motivation: state-space (hidden Markov) models, sequential analysis of such models; non-sequential problems that may be tackled using SMC.
• Formalism: Markov kernels, Feynman-Kac distributions.
• Monte Carlo tricks: importance sampling and resampling
• standard particle filters: bootstrap, guided, auxiliary
• maximum likelihood estimation of state-stace models
• Bayesian estimation of these models: PMCMC, SMC$^2$.[-]

Data mining methods based on finite mixture models are quite common in many areas of applied science, such as marketing, to segment data and to identify subgroups with specific features. Recent work shows that these methods are also useful in micro econometrics to analyze the behavior of workers in labor markets. Since these data are typically available as time series with discrete states, clustering kernels based on Markov chains with group-specific transition matrices are applied to capture both persistence in the individual time series as well as cross-sectional unobserved heterogeneity. Markov chains clustering has been applied to data from the Austrian labor market, (a) to understanding the effect of labor market entry conditions on long-run career developments for male workers (Frühwirth-Schnatter et al., 2012), (b) to study mothers' long-run career patterns after first birth (Frühwirth-Schnatter et al., 2016), and (c) to study the effects of a plant closure on future career developments for male worker (Frühwirth-Schnatter et al., 2018). To capture non- stationary effects for the later study, time-inhomogeneous Markov chains based on time-varying group specific transition matrices are introduced as clustering kernels. For all applications, a mixture-of-experts formulation helps to understand which workers are likely to belong to a particular group. Finally, it will be shown that Markov chain clustering is also useful in a business application in marketing and helps to identify loyal consumers within a customer relationship management (CRM) program.[-]

In many health studies, interest often lies in assessing health effects on a large set of outcomes or specific outcome subtypes, which may be sparsely observed, even in big data settings. For example, while the overall prevalence of birth defects is not low, the vast heterogeneity in types of congenital malformations leads to challenges in estimation for sparse groups. However, lumping small groups together to facilitate estimation is often controversial and may have limited scientific support.
There is a very rich literature proposing Bayesian approaches for clustering starting with a prior probability distribution on partitions. Most approaches assume exchangeability, leading to simple representations in terms of Exchangeable Partition Probability Functions (EPPF). Gibbs-type priors encompass a broad class of such cases, including Dirichlet and Pitman-Yor processes. Even though there have been some proposals to relax the exchangeability assumption, allowing covariate-dependence and partial exchangeability, limited consideration has been given on how to include concrete prior knowledge on the partition. We wish to cluster birth defects into groups to facilitate estimation, and we have prior knowledge of an initial clustering provided by experts. As a general approach for including such prior knowledge, we propose a Centered Partition (CP) process that modifies the EPPF to favor partitions close to an initial one. Some properties of the CP prior are described, a general algorithm for posterior computation is developed, and we illustrate the methodology through simulation examples and an application to the motivating epidemiology study of birth defects.[-]

With the growing capabilities of Geographic Information Systems (GIS) and user-friendly software, statisticians today routinely encounter geographically referenced data containing observations from a large number of spatial locations and time points. Over the last decade, hierarchical spatiotemporal process models have become widely deployed statistical tools for researchers to better understand the complex nature of spatial and temporal variability. However, fitting hierarchical spatiotemporal models often involves expensive matrix computations with complexity increasing in cubic order for the number of spatial locations and temporal points. This renders such models unfeasible for large data sets. I will present a focused review of two methods for constructing well-defined highly scalable spatiotemporal stochastic processes. Both these processes can be used as ``priors" for spatiotemporal random fields. The first approach constructs a low-rank process operating on a lower-dimensional subspace. The second approach constructs a Nearest-Neighbor Gaussian Process (NNGP) that ensures sparse precision matrices for its finite realizations. Both processes can be exploited as a scalable prior embedded within a rich hierarchical modeling framework to deliver full Bayesian inference. These approaches can be described as model-based solutions for big spatiotemporal datasets. The models ensure that the algorithmic complexity has n floating point operations (flops), where n is the number of spatial locations (per iteration). We compare these methods and provide some insight into their methodological underpinnings.[-]