arXiv subjects

Elisabeth Gassiat

Publications and source records attributed to Elisabeth Gassiat.

35 records · Page 2Linked to original sources

arXiv2016

Pattern Coding Meets Censoring: (almost) Adaptive Coding on Countable Alphabets

Adaptive coding faces the following problem: given a collection of source classes such that each class in the collection has non-trivial minimax redundancy rate, can we design a single code which is asymptotically minimax over each class in the collection? In particular, adaptive coding makes sense when there is no universal code on the union of classes in the collection. In this paper, we deal with classes of sources over an infinite alphabet, that are characterized by a dominating envelope. We provide asymptotic equivalents for the redundancy of envelope classes enjoying a regular variation property. We finally construct a computationally efficient online prefix code, which interleaves the encoding of the so-called pattern of the message and the encoding of the dictionary of discovered symbols. This code is shown to be adaptive, within a $\log\log n$ factor, over the collection of regularly varying envelope classes. The code is both simpler and less redundant than previously described contenders. In contrast with previous attempts, it also covers the full range of slowly varying envelope classes.

cs.IT↗

arXiv2016

Improving heritability estimation by a variable selection approach in sparse high dimensional linear mixed models

Motivated by applications in neuroanatomy, we propose a novel methodology for estimating the heritability which corresponds to the proportion of phenotypic variance which can be explained by genetic factors. Estimating this quantity for neuroanatomical features is a fundamental challenge in psychiatric disease research. Since the phenotypic variations may only be due to a small fraction of the available genetic information, we propose an estimator of the heritability that can be used in high dimensional sparse linear mixed models. Our method consists of three steps. Firstly, a variable selection stage is performed in order to recover the support of the genetic effects -- also called causal variants -- that is to find the genetic effects which really explain the phenotypic variations. Secondly, we propose a maximum likelihood strategy for estimating the heritability which only takes into account the causal genetic effects found in the first step. Thirdly, we compute the standard error and the 95% confidence interval associated to our heritability estimator thanks to a nonparametric bootsrap approach. Our main contribution consists in providing an estimation of the heritability with standard errors substantially smaller than methods without variable selection when the genetic effects are very sparse. Since the real genetic architecture is in general unknown in practice, we also propose an empirical criterion which allows the user to decide whether it is relevant to apply a variable selection based approach or not. We illustrate the performance of our methodology on synthetic and real neuroanatomic data coming from the Imagen project. We also show that our approach has a very low computational burden and is very efficient from a statistical point of view.

math.ST↗

arXiv2015

Consistent estimation of the filtering and marginal smoothing distributions in nonparametric hidden Markov models

In this paper, we consider the filtering and smoothing recursions in nonparametric finite state space hidden Markov models (HMMs) when the parameters of the model are unknown and replaced by estimators. We provide an explicit and time uniform control of the filtering and smoothing errors in total variation norm as a function of the parameter estimation errors. We prove that the risk for the filtering and smoothing errors may be uniformly upper bounded by the risk of the estimators. It has been proved very recently that statistical inference for finite state space nonparametric HMMs is possible. We study how the recent spectral methods developed in the parametric setting may be extended to the nonparametric framework and we give explicit upper bounds for the L2-risk of the nonparametric spectral estimators. When the observation space is compact, this provides explicit rates for the filtering and smoothing errors in total variation norm. The performance of the spectral method is assessed with simulated data for both the estimation of the (nonparametric) conditional distribution of the observations and the estimation of the marginal smoothing distributions.

math.ST↗

arXiv2015

Heritability estimation in high dimensional linear mixed models

Motivated by applications in genetic fields, we propose to estimate the heritability in high dimensional sparse linear mixed models. The heritability determines how the variance is shared between the different random components of a linear mixed model. The main novelty of our approach is to consider that the random effects can be sparse, that is may contain null components, but we do not know neither their proportion nor their positions. The estimator that we consider is strongly inspired by the one proposed by Pirinen et al. (2013), and is based on a maximum likelihood approach. We also study the theoretical properties of our estimator, namely we establish that our estimator of the heritability is $\sqrt{n}$-consistent when both the number of observations $n$ and the number of random effects $N$ tend to infinity under mild assumptions. We also prove that our estimator of the heritability satisfies a central limit theorem which gives as a byproduct a confidence interval for the heritability. Some Monte-Carlo experiments are also conducted in order to show the finite sample performances of our estimator.

math.ST↗

arXiv2015

About Adaptive Coding on Countable Alphabets: Max-Stable Envelope Classes

In this paper, we study the problem of lossless universal source coding for stationary memoryless sources on countably infinite alphabets. This task is generally not achievable without restricting the class of sources over which universality is desired. Building on our prior work, we propose natural families of sources characterized by a common dominating envelope. We particularly emphasize the notion of adaptivity, which is the ability to perform as well as an oracle knowing the envelope, without actually knowing it. This is closely related to the notion of hierarchical universal source coding, but with the important difference that families of envelope classes are not discretely indexed and not necessarily nested. Our contribution is to extend the classes of envelopes over which adaptive universal source coding is possible, namely by including max-stable (heavy-tailed) envelopes which are excellent models in many applications, such as natural language modeling. We derive a minimax lower bound on the redundancy of any code on such envelope classes, including an oracle that knows the envelope. We then propose a constructive code that does not use knowledge of the envelope. The code is computationally efficient and is structured to use an {E}xpanding {T}hreshold for {A}uto-{C}ensoring, and we therefore dub it the \textsc{ETAC}-code. We prove that the \textsc{ETAC}-code achieves the lower bound on the minimax redundancy within a factor logarithmic in the sequence length, and can be therefore qualified as a near-adaptive code over families of heavy-tailed envelopes. For finite and light-tailed envelopes the penalty is even less, and the same code follows closely previous results that explicitly made the light-tailed assumption. Our technical results are founded on methods from regular variation theory and concentration of measure.

cs.IT↗

arXiv2014

About the posterior distribution in hidden Markov models with unknown number of states

We consider finite state space stationary hidden Markov models (HMMs) in the situation where the number of hidden states is unknown. We provide a frequentist asymptotic evaluation of Bayesian analysis methods. Our main result gives posterior concentration rates for the marginal densities, that is for the density of a fixed number of consecutive observations. Using conditions on the prior, we are then able to define a consistent Bayesian estimator of the number of hidden states. It is known that the likelihood ratio test statistic for overfitted HMMs has a nonstandard behaviour and is unbounded. Our conditions on the prior may be seen as a way to penalize parameters to avoid this phenomenon. Inference of parameters is a much more difficult task than inference of marginal densities, we still provide a precise description of the situation when the observations are i.i.d. and we allow for $2$ possible hidden states.

math.ST↗

arXiv2014

About adaptive coding on countable alphabets

This paper sheds light on universal coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate. We prove that the auto-censuring AC code introduced by Bontemps (2011) is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy % of small source classes by Opper and Haussler (1997) and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on non-asymptotic bounds for maxima of samples from discrete distributions with finite and non-decreasing hazard rate.

math.ST↗

arXiv2013

Finite state space non parametric Hidden Markov Models are in general identifiable

In this paper, we prove that finite state space non parametric hidden Markov models are identifiable as soon as the transition matrix of the latent Markov chain has full rank and the emission probability distributions are linearly independent. We then propose several non parametric likelihood based estimation methods, which we apply to models used in applications. We finally show on examples that the use of non parametric modeling and estimation may improve the classification performances.

stat.ME↗

arXiv2013

Non parametric finite translation mixtures with dependent regime

In this paper we consider non parametric finite translation mixtures. We prove that all the parameters of the model are identifiable as soon as the matrix that defines the joint distribution of two consecutive latent variables is non singular and the translation parameters are distinct. Under this assumption, we provide a consistent estimator of the number of populations, of the translation parameters and of the distribution of two consecutive latent variables, which we prove to be asymptotically normally distributed under mild dependency assumptions. We propose a non parametric estimator of the unknown translated density. In case the latent variables form a Markov chain (Hidden Markov models), we prove an oracle inequality leading to the fact that this estimator is minimax adaptive over regularity classes of densities.

math.ST↗

arXiv2012

Joint estimation of intersecting context tree models

We study a problem of model selection for data produced by two different context tree sources. Motivated by linguistic questions, we consider the case where the probabilistic context trees corresponding to the two sources are finite and share many of their contexts. In order to understand the differences between the two sources, it is important to identify which contexts and which transition probabilities are specific to each source. We consider a class of probabilistic context tree models with three types of contexts: those which appear in one, the other, or both sources. We use a BIC penalized maximum likelihood procedure that jointly estimates the two sources. We propose a new algorithm which efficiently computes the estimated context trees. We prove that the procedure is strongly consistent. We also present a simulation study showing the practical advantage of our procedure over a procedure that works separately on each dataset.

math.ST↗

arXiv2012

The local geometry of finite mixtures

We establish that for q>=1, the class of convex combinations of q translates of a smooth probability density has local doubling dimension proportional to q. The key difficulty in the proof is to control the local geometric structure of mixture classes. Our local geometry theorem yields a bound on the (bracketing) metric entropy of a class of normalized densities, from which a local entropy bound is deduced by a general slicing procedure.

math.ST↗

arXiv2012

Consistent order estimation and minimal penalties

Consider an i.i.d. sequence of random variables whose distribution f* lies in one of a nested family of models M_q, q>=1. The smallest index q* such that M_{q*} contains f* is called the model order. We establish strong consistency of the penalized likelihood order estimator in a general setting with penalties of order η(q) log log n, where η(q) is a dimensional quantity. Moreover, such penalties are shown to be minimal. In contrast to previous work, an a priori upper bound on the model order is not assumed. The results rely on a sharp characterization of the pathwise fluctuations of the generalized likelihood ratio statistic under entropy assumptions on the model classes. Our results are applied to the geometrically complex problem of location mixture order estimation, which is widely used but poorly understood.

math.ST↗

arXiv2008

Model-Based Clustering using multi-allelic loci data with loci selection

We propose a Model-Based Clustering (MBC) method combined with loci selection using multi-allelic loci genetic data. The loci selection problem is regarded as a model selection problem and models in competition are compared with the Bayesian Information Criterion (BIC). The resulting procedure selects the subset of clustering loci, the number of clusters, estimates the proportion of each cluster and the allelic frequencies within each cluster. We prove that the selected model converges in probability to the true model under a single realistic assumption as the size of the sample tends to infinity. The proposed method named MixMoGenD (Mixture Model using Genetic Data) was implemented using c++ programming language. Numerical experiments on simulated data sets was conducted to highlight the interest of the proposed loci selection procedure.

stat.AP↗

arXiv2008

Semiparametric regression estimation using noisy nonlinear non invertible functions of the observations

We investigate a semiparametric regression model where one gets noisy non linear non invertible functions of the observations. We focus on the application to bearings-only tracking. We first investigate the least squares estimator and prove its consistency and asymptotic normality under mild assumptions. We study the semiparametric likelihood process and prove local asymptotic normality of the model. This allows to define the efficient Fisher information as a lower bound for the asymptotic variance of regular estimators, and to prove that the parametric likelihood estimator is regular and asymptotically efficient. Simulations are presented to illustrate our results.

math.ST↗

arXiv2008

Forgetting of the initial distribution for non-ergodic Hidden Markov Chains

In this paper, the forgetting of the initial distribution for a non-ergodic Hidden Markov Models (HMM) is studied. A new set of conditions is proposed to establish the forgetting property of the filter, which significantly extends all the existing results. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using generic models of non-ergodic HMM and extend all the results known so far.

math.PR↗

arXiv2008

Coding on countably infinite alphabets

This paper describes universal lossless coding strategies for compressing sources on countably infinite alphabets. Classes of memoryless sources defined by an envelope condition on the marginal distribution provide benchmarks for coding techniques originating from the theory of universal coding over finite alphabets. We prove general upper-bounds on minimax regret and lower-bounds on minimax redundancy for such source classes. The general upper bounds emphasize the role of the Normalized Maximum Likelihood codes with respect to minimax regret in the infinite alphabet context. Lower bounds are derived by tailoring sharp bounds on the redundancy of Krichevsky-Trofimov coders for sources over finite alphabets. Up to logarithmic (resp. constant) factors the bounds are matching for source classes defined by algebraically declining (resp. exponentially vanishing) envelopes. Effective and (almost) adaptive coding techniques are described for the collection of source classes defined by algebraically vanishing envelopes. Those results extend ourknowledge concerning universal coding to contexts where the key tools from parametric inference

math.ST↗

arXiv2005

Parameter estimation in pair hidden Markov models

This paper deals with parameter estimation in pair hidden Markov models (pair-HMMs). We first provide a rigorous formalism for these models and discuss possible definitions of likelihoods. The model being biologically motivated, some restrictions with respect to the full parameter space naturally occur. Existence of two different Information divergence rates is established and divergence property (namely positivity at values different from the true one) is shown under additional assumptions. This yields consistency for the parameter in parametrization schemes for which the divergence property holds. Simulations illustrate different cases which are not covered by our results.

math.ST↗