Research Group "Stochastic Algorithms and Nonparametric Statistics"

Dominique Bontemps (Université Paul Sabatier, Toulouse)

Bayesian posterior consistency and contraction rates in the shape invariant model

Abstract:In this work, we consider the so-called Shape Invariant Model which stands for the estimation of a function f0 submitted to a random translation of law g0 in a white noise model. We are interested in such a model when the law of the deformations is unknown. We aim to recover the law of the process P(f0,g0) as well as f0 and g0. In this perspective, we adopt a Bayesian point of view and find prior on f and g such that the posterior distribution concentrates around P(f0,g0) at a polynomial rate when n goes to infinity. We obtain a logarithmic posterior contraction rate for the shape f0 and the distribution g0. We also derive logarithmic lower bounds for the estimation of f0 and g0 in a frequentist paradigm.

Prof. Denis Belomestny (Universität Essen)

On one inverse problem of nancial mathematics with error in the operator

Abstract: In this talk we consider a calibration problem for the so called Markov Functional Models (MFM) from the statistical point of view. It is shown that at each step of the calibration procedure one has to solve a nonlinear inverse problem with the operator estimated in the previous step. We propose a regularisation method and derive the optimal convergence rates.

Stéphane Gaïffas (École Polytechnique Palaiseau, France)

Link prediction in graphs with time-evolving features

Abstract:

Estimation of the transition density of a Markov chain

Abstract:

Victor-Emmanuel Brunel (ENSAE/ParisTech, France)

Adaptive estimation of convex polytopes

Abstract: We consider a sample of $n$ i.i.d. random variables, uniformly distributed in some unknown polytope $P$ in $\R^d$. We propose a maximum likelihood estimator whose accuracy, defined as the expectation of the Lebesgue measure of its symmetric difference with the true polytope, is at most of the order $r(\ln n)/n$, when $r$, the number of vertices of $P$, is known. Using a concentration inequality for this estimator, we develop a procedure in order to estimate $P$ adaptively with respect to $r$, based on a model selection method. The adaptive estimator achieves the same rate as for the case of known $r$.

Marc Hoffmann (Université Paris Dauphine)

Statistical estimation of a growth-fragmentation model observed on a genealogical tree

Abstract: We model the growth of a cell population by a piecewise deterministic Markov branching tree. Each cell splits into two osprings at a division rate B(x) that depends on its size x. The size of each cell grows exponentially in time, at a rate that varies for each individual. We show that the mean empirical measure of the model satises a growth-fragmentation type equation if structured in both size and growth rate as state variables. We construct a nonparametric estimator of the division rate B(x) based on the observation of the population over dierent sampling schemes of size n on the genealogical tree. Our estimator nearly achieves the rate n��s=(2s+1) in squared-loss error asymptotically. When the growth rate is assumed to be identical for every cell, we retrieve the classical growth-fragmentation model and our estimator improves on the rate n��s=(2s+3) obtained in a related framework through indirect observation schemes. Our method is consistently tested numerically and implemented on Escherichia coli data.

Jia Li (Duke University, Durham, USA)

Inference on volatility functional dependence

Abstract: We develop inference theory for models involving possibly nonlinear transforms of the elements of the spot covariance matrix of a multivariate continuous-time process observed at high frequency. The framework can be used to study the relationship among the elements of the latent spot covariance matrix and processes defined on the basis of it such as systematic and idiosyncratic variances, factor betas and correlations. The estimation is based on matching model-implied moment conditions under the occupation measure induced by the spot covariance process. We prove consistency and asymptotic mixed normality of our estimator of the (random) coefficients in the volatility model and further develop model specification tests. We apply our inference methods to study variance and correlation risks in nine sector portfolios comprising the S\&P 500 index. We document sector-specific variance risks in addition to that of the market and time-varying heterogeneous correlation risk among the market-neutral components of the sector portfolio returns. (Joint with V. Todorov and G. Tauchen)

Professor Matt Wand (University of Technology, Sydney, Australia)

Variational Approximations in Statistics I

Abstract:Variational approximations facilitate approximate inference for the parameters in complex statistical models and provide fast, deterministic alternatives to Monte Carlo methods. However, much of the contemporary literature on variational approximations is in Computer Science rather than Statistics, and uses terminology, notation, and examples from the former field. In this series of lectures we explain variational approximation in statistical terms. In particular, we illustrate the ideas of variational approximation using examples that are familiar to statisticians.

Professor Matt Wand (University of Technology, Sydney, Australia)

Variational Approximations in Statistics II

Abstract: Variational approximations facilitate approximate inference for the parameters in complex statistical models and provide fast, deterministic alternatives to Monte Carlo methods. However, much of the contemporary literature on variational approximations is in Computer Science rather than Statistics, and uses terminology, notation, and examples from the former field. In this series of lectures we explain variational approximation in statistical terms. In particular, we illustrate the ideas of variational approximation using examples that are familiar to statisticians.

Nicolai Meinshausen (ETH Zürich)

Challenges for high-dimensional inference

Abstract: High-dimensional inference has progressed rapidly in the last years. I will give a brief introduction and show examples from biology, neuroscience and physics, while also mentioning theoretical underpinnings. Some challenges of inference for large data sets will be discussed, including computational feasibility and inhomogeneity. I will highlight some more recent developments about the feasibility of constructing confidence intervals for regression coefficients when the number of variables exceeds the number of observations.

Prof. Thomas Kneib

Structured additive distributional regression

Abstract:In this talk, we present a generic Bayesian framework for inference in distributional regression models in which each parameter of a potentially complex response distribution and not only the mean is related to a structured additive predictor. The latter is composed additively of a variety of different functional effect types such as nonlinear effects, spatial effects, random coefficients, interaction surfaces or other (possibly non-standard) basis function representations. To enforce specific properties of the functional effects such as smoothness, informative multivariate Gaussian priors are assigned to the basis function coefficients. Inference is then based on efficient Markov chain Monte Carlo simulation techniques where a generic procedure makes use of distribution-specific iteratively weighted least squares approximations to the full conditionals. We provide detailed guidance on practical aspects of model choice including selecting an appropriate response distribution and predictor specification. The importance and flexibility of Bayesian structured additive distributional regression to estimate all parameters as functions of explanatory variables and therefore to obtain more realistic models, is exemplified in several applications with complex response distributions. We will also introduce extensions to multivariate distributional regression

Dr. Axel Bücher (Ruhr-Universität Bochum)

When uniform weak convergence fails: empirical processes for dependence functions via epi- and hypographs

Abstract:For copulas whose partial derivatives are not continuous everywhere on the interior of the unit cube, the empirical copula process does not converge weakly with respect to the supremum distance. This makes it hard to verify asymptotic properties of inference procedures for such copulas. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epi- and hypoconvergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies Lp convergence. Even in cases where uniform convergence fails, weak convergence with respect to the new metric is established for empirical copula and tail dependence processes. No additional assumptions are needed for tail dependence functions, and for copulas, the assumptions reduce to existence and continuity of the partial derivatives almost everywhere on the unit cube. The results are applied to obtain asymptotic properties of minimum distance estimators, goodness-of-t tests and resampling procedures.

Nicolai Meinshausen (ETH Zürich)

Challenges for high-dimensional inference

Abstract: High-dimensional inference has progressed rapidly in the last years. I will give a brief introduction and show examples from biology, neuroscience and physics, while also mentioning theoretical underpinnings. Some challenges of inference for large data sets will be discussed, including computational feasibility and inhomogeneity. I will highlight some more recent developments about the feasibility of constructing confidence intervals for regression coefficients when the number of variables exceeds the number of observations.

Bo Markussen (University of Copenhagen)

Functional data, operator calculus, and statistical computation

Abstract: Functional data consists of observations of continuous curves, e.g. human growth curves or chemical spectrograms measured by gas chromatography, say. In practice the underlying functions are observed at nitely many sample points giving a multivariate observation. Suppose that we have observed n curves at the same p sample points. Functional data analysis, however, is dierent from classical multi- variate statistics since the p-dimensional observations of each of the n samples originate from underlying continuous curves. Special problems related to functional data includes data representation, registration, and regularization. See the monograph by Ramsay and Silverman (2005) for an introduction. Data repre- sentation is often done by a functional basis for the underlying curves, e.g. using splines or wavelets, where the coecients are estimated from the multivariate observations. This method retains the data in a nitely dimensional framework, and the needed statistical computations are often done using sparse matrices. In this talk we advocate a dierent computational approach. The idea is to embed the discrete observation of each curve into the originating function space, and then to do the computations on genuine functions and not on vectors of basis coecients. This approach implies that the sparse matrix computations are replaced by operator calculus. In Markussen (Bernoulli, 2013) this approach was implemented for constant coecient dierential operators, where there exists semi-explicit formulae for the associated Greens func- tions, and in Rakt and Markussen (CSDA, 2014) the methodology was generalized to image data. In this talk I will discuss recent developments using a much more versatile class of integral operators, including a discussion of the associated numerical computations. The methods will be exemplied in the context of functional regression, where the discretely observed curves are used as covariates for a univariate (or low dimensional) response.

Bharath Sriperumbudur (University of Cambridge)

RKHS-induced innite dimensional exponential families: Density estimation and beyond

Abstract: In the rst part of the talk, we consider the problem of estimating densities in an innite dimensional exponential family indexed by functions in a reproducing kernel Hilbert space. Since standard techniques like maximum likelihood estimation (MLE) or pseudo MLE (based on the method of sieves) do not yield practically useful estimators, we propose an estimator based on the score matching method introduced by Hyvarinen, which involves solving a simple linear system. We show that the proposed estimator is consistent, and provide convergence rates under smoothness assumptions. We also empirically demonstrate that the proposed method outperforms the standard non-parametric kernel density estimator. The second part of the talk deals with certain mean element and covariance operator associated with the above considered exponential family. For an appropriate choice of kernel, we show that these objects uniquely characterize the probability measure and then provide an ecient approach for non-parametric two-sample testing and independence testing. Based on various joint works with Kenji Fukumizu (Institute of Statistical Mathematics, Tokyo) and Arthur Gretton (Gatsby Unit, University College London).