Gersende Fort

Bio: Gersende Fort is an academic researcher from Institut de Mathématiques de Toulouse. The author has contributed to research in topics: Markov chain Monte Carlo & Markov chain. The author has an hindex of 27, co-authored 120 publications receiving 2856 citations. Previous affiliations of Gersende Fort include École Polytechnique & Télécom ParisTech.

Classification using partial least squares with penalized logistic regression

Abstract: Motivation: One important aspect of data-mining of microarray data is to discover the molecular variation among cancers. In microarray studies, the number n of samples is relatively small compared to the number p of genes per sample (usually in thousands). It is known that standard statistical methods in classification are efficient (i.e. in the present case, yield successful classifiers) particularly when n is (far) larger than p. This naturally calls for the use of a dimension reduction procedure together with the classification one. Results: In this paper, the question of classification in such a high-dimensional setting is addressed. We view the classification problem as a regression one with few observations and many predictor variables. We propose a new method combining partial least squares (PLS) and Ridge penalized logistic regression. We review the existing methods based on PLS and/or penalized likelihood techniques, outline their interest in some cases and theoretically explain their sometimes poor behavior. Our procedure is compared with these other classifiers. The predictive performance of the resulting classification rule is illustrated on three data sets: Leukemia, Colon and Prostate. Availability: Software that implements the procedures and data source on which this paper focuses are freely available at http://www-lmc.imag.fr/SMS/membres/Gersende_Fort,Sophie_Lambert.html Contact: sophie.lambert@imag.fr

Practical drift conditions for subgeometric rates of convergence

Abstract: We present a new drift condition which implies rates of convergence to the stationary distribution of the iterates of a \psi-irreducible aperiodic and positive recurrent transition kernel. This condition, extending a condition introduced by Jarner and Roberts [Ann. Appl. Probab. 12 (2002) 224-247] for polynomial convergence rates, turns out to be very convenient to prove subgeometric rates of convergence. Several applications are presented including nonlinear autoregressive models, stochastic unit root models and multidimensional random walk Hastings-Metropolis algorithms.

Convergence of the Monte Carlo expectation maximization for curved exponential families

Abstract: The Monte Carlo expectation maximization (MCEM) algorithm is a versatile tool for inference in incomplete data models, especially when used in combination with Markov chain Monte Carlo simulation methods. In this contribution, the almost-sure convergence of the MCEM algorithm is established. It is shown, using uniform versions of ergodic theorems for Markov chains, that MCEM converges under weak conditions on the simulation kernel. Practical illustrations are presented, using a hybrid random walk Metropolis Hastings sampler and an independence sampler. The rate of convergence is studied, showing the impact of the simulation schedule on the fluctuation of the parameter estimate at the convergence. A novel averaging procedure is then proposed to reduce the simulation variance and increase the rate of convergence.

Practical drift conditions for subgeometric rates of convergence

Abstract: We present a new drift condition which implies rates of convergence to the stationary distribution of the iterates of a \psi-irreducible aperiodic and positive recurrent transition kernel. This condition, extending a condition introduced by Jarner and Roberts [Ann. Appl. Probab. 12 (2002) 224-247] for polynomial convergence rates, turns out to be very convenient to prove subgeometric rates of convergence. Several applications are presented including nonlinear autoregressive models, stochastic unit root models and multidimensional random walk Hastings-Metropolis algorithms.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

Convergence of Probability Measures

Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of probability measures

Abstract: The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an