A Bayesian Nonparametric model for microbiome data analysis We develop a statistical model to analyse microbiome profiling data based on sequencing of genetic fingerprints in 16S ribosomal RNA. The analysis allows us to quantify the uncertainty in  ecological ordination and clustering methods commonly applied in microbiome research. In addition, it can be extended into a…

Predicting the Present with Bayesian Structural Time Series This article describes a system for short term forecasting based on an ensemble prediction that averages over different combinations of predictors. The system combines a structural time series model for the target series with regression component capturing the contributions of contemporaneous search query data. A spike-and-slab prior…

Bayesian Factor Models in High Dimensions Sparse Bayesian factor models are routinely implemented for parsimonious dependence modeling and dimensionality reduction in high-dimensional applications. We provide theoretical understanding of such Bayesian procedures in terms of posterior convergence rates in inferring high-dimensional covariance matrices where the dimension can be larger than the sample size. We will also…

Nonparametric Bayesian inference for discretely sampled diffusions We consider the nonlinear statistical inverse problem ofmaking inference on the unknown parameters of a diffusion processdescribing the solution of a stochastic differential equation. Theobservation regime is such that the process is sampled at discretetime points that are a fixed distance apart, and we investigate theasymptotic regime when…

Robust Approximate Bayesian Inference The likelihood function is the basis of both frequentist and Bayesian methods. However, the stability of likelihood-based procedures requires strict adherence to the model assumptions: mild deviations from the model can lead to misleading inferential results. A possible Bayesian solution to robustness is to use a of robust pseudo likelihood, such…

The information complexity of sequential resource allocation I will talk about sequential resource allocation, under the so-called stochastic multi-armed bandit model. In this model, an agent interacts with a set of (unknown) probability distributions, called 'arms' (in reference to 'one-armed bandits', another name for slot machines in a casino). When the agent draws an arm,…

An inferential theory of clustering for functional data Recently, it has been shown that Morse theory can be exploited to de- velop a sound inferential background for clustering: one can rigorously define both population and empirical clusters by means of the gradient flows asso- ciated to the population density p and the estimated density pˆ.…

Bayesian inference for intra-tumor heterogeneity in mutations and copy number variation Tissue samples from the same tumor are heterogeneous. They consist of different subclones that can be characterized by differences in DNA nucleotide sequences and copy numbers on multiple loci. Inference on tumor heterogeneity thus involves the identification of the subclonal copy number and single…

Relative exchangeability Symmetry arguments lie at the heart of classical considerations in inductive inference and statistics.  In statistics, de Finetti's notion of exchangeability is the most prominent symmetry assumption, laying the foundation for Bayesian inference.  In practice, many statistical and scientific problems exhibit only partial symmetry determined by some underlying structure in a population.  As…

Exact simulation of the Wright-Fisher diffusion The Wright-Fisher family of diffusion processes is a class of evolutionary models widely used in population genetics, with applications also in finance and Bayesian statistics. Simulation and inference from these diffusions is therefore of widespread interest. However, simulating a Wright-Fisher diffusion is difficult because there is no known closed-form…

Consistency of posterior model probabilities in high-dimensional model selection In recent years there has been an increasing interest in developing Bayesian formulations that remain effective in high-dimensional and non-standard problems. We focus on high-dimensional model selection problems where the number of parameters may grow with the sample size, and review results characterizing the situations under…

Determinantal point process models and statistical inference In this talk, I will demonstrate that Determinantal point processes (DPPs) provide useful models for the description of repulsive spatial point processes. Such data are usually modeled by Gibbs point processes, where the likelihood and moment expressions are intractable and simulations are time consuming. I will recall the…

The Poisson gamma belief network A key issue in deep learning is to define an appropriate network structure, including both the depth of the network and the width of each hidden layer, which may be naturally addressed with completely random measures. We propose the Poisson gamma belief network (PGBN), which factorizes each of its layers…

Flexible behavioral capture-recapture modelling We develop some new strategies for building and fitting new flexible classes of para- metric capture-recapture models for closed populations which can be used to address a better understanding of behavioural patterns. We first rely on previous approaches based on a conditional probability parameterization and review how to regard a large…

Deep factorisation of stable processes The Lamperti-Kiu transformation for real-valued self-similar Markov processes (rssMp) states that, associated to each rssMp via a space-time transformation, is a Markov additive process (MAP). In the case that the rssMp is taken to be an α-stable process with α∈(0,2), the characteristics of the matrix exponent of the semi-group of…

Maximum likelihood estimation for linear Gaussian covariance Models We study parameter estimation in linear Gaussian covariance models, which are p-dimensional Gaussian models with linear constraints on the covariance matrix. Maximum likelihood estimation for this class of models leads to a non-convex optimization problem which typically has many local optima. We prove that the log-likelihood function…

Information geometry of the Gaussian distribution in view of stochastic optimization: first and second order geometry We study the optimization of a continuous function by its stochastic relaxation, i.e., the optimization of the expected value of the function itself with respect to a density in a statistical model. In the first part of the talk…

Sequential decisions, time dependence, and central limit theorems We prove a central limit theorem for the sum of functions of (1+m)-dimensional vectors from a time non-homogeneous Markov chain and we show several examples in which this central limit theorem can be used to easily establish the asymptotic normality of the optimal total reward of finite…

Bayesian nonparametric testing for embedded hypotheses with application to shape constrains If Bayesian nonparametric methods have received a great interest in the literature, only a few is known for testing nonparametric hypotheses, and especially the asymptotic properties of such tests. The problem of testing between two nonparametric hypotheses is known to be difficult, but the…

Spatial gamma processes in disease mapping In this talk we will present Bayesian models based on Markov random fields of gamma type to model the relative risk in disease mapping data. The spatial gamma processes allow for different spatial dependence among neighbours. We describe the properties of ths processes and use them as prior distributions…