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…

Dependent species sampling models We consider a novel Bayesian nonparametric model for density estimation with an underlying spatial structure. The model is built on a class of species sampling models, which are discrete random probability measures that can be represented as a mixture of random support points and random weights. Specifically, we construct a collection…

Generalized partially additive Bayesian spectral analysis regression models In this talk, we present a Bayesian method for generalized partially additive regression using a spectral analysis of Gaussian process priors  for the  regression function. The smoothing prior distribution for the spectral coefficients incorporates hyper parameters that control the smoothness of the function and the tradeoff between…

False discovery rate smoothing Many approaches for multiple testing begin with the assumption that all tests in a given study should be combined into a global false-discovery-rate analysis.  But this may be inappropriate for many of today's large-scale screening problems, where test statistics have a natural spatial lattice structure (voxels in the brain, distance along…

SMC Samplers for Applications in High Dimensions Sequential Monte Carlo (SMC) methods are nowadays routinely applied in a variety of complex applications: hidden Markov models, dynamical systems, target tracking, control problems, just to name a few. Whereas SMC methods have been greatly refined in the last decades and are now much better understood, they are…

Convergence rates of posterior distributions in nonparametric inverse problems Since the seminal works of Ghosal, Ghosh and van der Vaart (2000) and Shen and Wasserman (2001), posterior contraction has attracted much attention, resulting in the rich literature on this subject. However, these results are not suitable to deal with trully ill-posed inverse problems, where one…

A Flexible Bayesian Approach to Monotone Missing Data in Longitudinal Studies with Informative Missingness with Application to An Acute Schizophrenia Clinical Trial We develop a Bayesian nonparametric model for a longitudinal response in the presence of nonignorable missing data. Our general approach is to first specify a {em working model} that flexibly models the missingness…

Behaviour of the posterior distribution in HMM models when the number of states is misspecified In this paper we study the asymptotic behaviour of the posterior distribution for parametric HMM models with finite number of components. We concentrate in particular on the case where the number of states of the hidden Markov chain in the…

On the ancestral process of long-range seed bank models It has been observed that, in some bacterial species, spores may remain dormant for a long time, to wake up much later, even up to "order of population size" generations later. When they wake up, they can still participate in the population's reproduction. This incredibly relaxed…

The scaling limit of the minimum spanning tree of the complete graph Consider the complete graph on n vertices with independent and identically distributed edge-weights having some absolutely continuous distribution. The minimum spanning tree (MST) is simply the spanning subtree of smallest weight.  It is straightforward to construct the MST using one of several natural…

Sequential Quasi Monte Carlo We develop a new class of algorithms, SQMC (Sequential Quasi Monte Carlo), as a variant of SMC (Sequential Monte Carlo) based on low-discrepancy points. The complexity of SQMC is O(N log N) where N is the number of simulations at each iteration, and its error rate is smaller than the Monte…

The algebraic method in experimental designs This seminar is part of the 3rd Carlo Alberto Stochastics Workshop

Chain event graphs and the geometry of causation This seminar is part of the 3rd Carlo Alberto Stochastics Workshop

Testability and consistency Bayesian consistency theorems come in (at least) three distinct types, e.g. Doob's prior-almost-sure consistency on Polish spaces, Schwartz's Hellinger consistency with KL-priors and the `tailfree' weak consistency of Dirichlet posteriors. In this talk we ask the question how these notions of convergence are related and argue that one characterises them most conveniently…