26 research outputs found
Some distance bounds of branching processes and their diffusion limits
We compute exact values respectively bounds of "distances" - in the sense of
(transforms of) power divergences and relative entropy - between two
discrete-time Galton-Watson branching processes with immigration GWI for which
the offspring as well as the immigration is arbitrarily Poisson-distributed
(leading to arbitrary type of criticality). Implications for asymptotic
distinguishability behaviour in terms of contiguity and entire separation of
the involved GWI are given, too. Furthermore, we determine the corresponding
limit quantities for the context in which the two GWI converge to Feller-type
branching diffusion processes, as the time-lags between observations tend to
zero. Some applications to (static random environment like) Bayesian decision
making and Neyman-Pearson testing are presented as well.Comment: 45 page
R\'enyi Divergence and Kullback-Leibler Divergence
R\'enyi divergence is related to R\'enyi entropy much like Kullback-Leibler
divergence is related to Shannon's entropy, and comes up in many settings. It
was introduced by R\'enyi as a measure of information that satisfies almost the
same axioms as Kullback-Leibler divergence, and depends on a parameter that is
called its order. In particular, the R\'enyi divergence of order 1 equals the
Kullback-Leibler divergence.
We review and extend the most important properties of R\'enyi divergence and
Kullback-Leibler divergence, including convexity, continuity, limits of
-algebras and the relation of the special order 0 to the Gaussian
dichotomy and contiguity. We also show how to generalize the Pythagorean
inequality to orders different from 1, and we extend the known equivalence
between channel capacity and minimax redundancy to continuous channel inputs
(for all orders) and present several other minimax results.Comment: To appear in IEEE Transactions on Information Theor
Convergence rates of posterior distributions for noniid observations
We consider the asymptotic behavior of posterior distributions and Bayes
estimators based on observations which are required to be neither independent
nor identically distributed. We give general results on the rate of convergence
of the posterior measure relative to distances derived from a testing
criterion. We then specialize our results to independent, nonidentically
distributed observations, Markov processes, stationary Gaussian time series and
the white noise model. We apply our general results to several examples of
infinite-dimensional statistical models including nonparametric regression with
normal errors, binary regression, Poisson regression, an interval censoring
model, Whittle estimation of the spectral density of a time series and a
nonlinear autoregressive model.Comment: Published at http://dx.doi.org/10.1214/009053606000001172 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
On an information-type inequality for the Hellinger process
Let (Ω,픉, 픽) be a filtered space with two probability measures P and P' on (Ω,픉). Let X be a d-dimensional locally square-integrable semimartingale relative to P and P' with the canonical decomposition X = X0 + M + A and X = X0 + M' + A' respectively. We give a lower bound for the Hellinger process h(1⁄2; P, P') of order 1/2 between P and P' in terms of A, A' and the quadratic characteristic of M and M'. This result implies simple sufficient conditions for the entire separation of measures in a linear regression model with martingale errors
Robust Procedures for Estimating and Testing in the Framework of Divergence Measures
The scope of the contributions to this book will be to present new and original research papers based on MPHIE, MHD, and MDPDE, as well as test statistics based on these estimators from a theoretical and applied point of view in different statistical problems with special emphasis on robustness. Manuscripts given solutions to different statistical problems as model selection criteria based on divergence measures or in statistics for high-dimensional data with divergence measures as loss function are considered. Reviews making emphasis in the most recent state-of-the art in relation to the solution of statistical problems base on divergence measures are also presented
Community detection and stochastic block models: recent developments
The stochastic block model (SBM) is a random graph model with planted
clusters. It is widely employed as a canonical model to study clustering and
community detection, and provides generally a fertile ground to study the
statistical and computational tradeoffs that arise in network and data
sciences.
This note surveys the recent developments that establish the fundamental
limits for community detection in the SBM, both with respect to
information-theoretic and computational thresholds, and for various recovery
requirements such as exact, partial and weak recovery (a.k.a., detection). The
main results discussed are the phase transitions for exact recovery at the
Chernoff-Hellinger threshold, the phase transition for weak recovery at the
Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial
recovery, the learning of the SBM parameters and the gap between
information-theoretic and computational thresholds.
The note also covers some of the algorithms developed in the quest of
achieving the limits, in particular two-round algorithms via graph-splitting,
semi-definite programming, linearized belief propagation, classical and
nonbacktracking spectral methods. A few open problems are also discussed
CLADAG 2021 BOOK OF ABSTRACTS AND SHORT PAPERS
The book collects the short papers presented at the 13th Scientific Meeting of the Classification and Data Analysis Group (CLADAG) of the Italian Statistical Society (SIS). The meeting has been organized by the Department of Statistics, Computer Science and Applications of the University of Florence, under the auspices of the Italian Statistical Society and the International Federation of Classification Societies (IFCS). CLADAG is a member of the IFCS, a federation of national, regional, and linguistically-based classification societies. It is a non-profit, non-political scientific organization, whose aims are to further classification research