Network monitoring and diagnosis are key to improving network performance. The difficulties of performance monitoring lie in today's fast growing Internet, accompanied by increasingly heterogeneous and unregulated structures. Moreover, these tasks become even harder since one cannot rely on the collaboration of individual routers and servers to directly measure network traffic. Even though the aggregatory nature of possible network measurements gives rise to inverse problems, existing methods for solving inverse problems are usually computationally intractable or statistically inefficient.

The Expectation Maximization (EM) algorithm is a powerful computational technique for locating maxima of functions...

Most problems in frequentist statistics involve optimization of a function such as a likelihood or a sum of squares. EM algorithms are among the most effective algorithms for maximum likelihood estimation because they consistently drive the likelihood uphill by maximizing a simple surrogate function for the loglikelihood. Iterative optimization of a surrogate function as exemplified by an EM algorithm does not necessarily require missing data. Indeed, every EM algorithm is a special case of the more general class of MM optimization algorithms, which typically exploit convexity rather than missing data in majorizing or minorizing an objective function. In our opinion, MM algorithms deserve to part of the standard toolkit of professional statisticians. The current article explains the principle behind MM algorithms, suggests some methods for constructing them, and discusses some of their attractive features. We include numerous examples throughout the article to illustrate the concepts described. In addition to surveying previous work on MM algorithms, this article introduces some new material on constrained optimization and standard error estimation. Key words and phrases: constrained optimization, EM algorithm, majorization, minorization, Newton-Raphson 1 1

The EM algorithm and its extensions are popular tools for modal estimation but are often criticised for their slow convergence. We propose a new method that can often make EM much faster. The intuitive idea is to use a 'covariance adjustment ' to correct the analysis of the M step, capitalising on extra information captured in the imputed complete data. The way we accomplish this is by parameter expansion; we expand the complete-data model while preserving the observed-data model and use the expanded complete-data model to generate EM. This parameter-expanded EM, PX-EM, algorithm shares the simplicity and stability of ordinary EM, but has a faster rate of convergence since its M step performs a more efficient analysis. The PX-EM algorithm is illustrated for the multivariate t distribution, a random effects model, factor analysis, probit regression and a Poisson imaging model.

Abstract. Technological innovations have revolutionized the process of scientific research and knowledge discovery. The availability of massive data and challenges from frontiers of research and development have reshaped statistical thinking, data analysis and theoretical studies. The challenges of high-dimensionality arise in diverse fields of sciences and the humanities, ranging from computational biology and health studies to financial engineering and risk management. In all of these fields, variable selection and feature extraction are crucial for knowledge discovery. We first give a comprehensive overview of statistical challenges with high dimensionality in these diverse disciplines. We then approach the problem of variable selection and feature extraction using a unified framework: penalized likelihood methods. Issues relevant to the choice of penalty functions are addressed. We demonstrate that for a host of statistical problems, as long as the dimensionality is not excessively large, we can estimate the model parameters as well as if the best model is known in advance. The persistence property in risk minimization is also addressed. The applicability of such a theory and method to diverse statistical problems is demonstrated. Other related problems with high-dimensionality are also discussed.

The Bradley–Terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among the many studies of the model in the past 75 years, numerous authors have generalized it in several directions, sometimes providing iterative algorithms for obtaining maximum likelihood estimates for the generalizations. Building on a theory of algorithms known by the initials MM, for minorization–maximization, this paper presents a powerful technique for producing iterative maximum likelihood estimation algorithms for a wide class of generalizations of the Bradley–Terry model. While algorithms for problems of this type have tended to be custom-built in the literature, the techniques in this paper enable their mass production. Simple conditions are stated that guarantee that each algorithm described will produce a sequence that converges to the unique maximum likelihood estimator. Several of the algorithms and convergence results herein are new. 1. Introduction. In

Variable selection is fundamental to high-dimensional statistical modeling. Many variable selection techniques may be implemented by maximum penalized likelihood using various penalty functions. Optimizing the penalized likelihood function is often challenging because it may be nondifferentiable and/or nonconcave. This article proposes a new class of algorithms for finding a maximizer of the penalized likelihood for a broad class of penalty functions. These algorithms operate by perturbing the penalty function slightly to render it differentiable, then optimizing this differentiable function using a minorize–maximize (MM) algorithm. MM algorithms are useful extensions of the well-known class of EM algorithms, a fact that allows us to analyze the local and global convergence of the proposed algorithm using some of the techniques employed for EM algorithms. In particular, we prove that when our MM algorithms converge, they must converge to a desirable point; we also discuss conditions under which this convergence may be guaranteed. We exploit the Newton–Raphson-like aspect of these algorithms

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... This article studies this problem asymptotically in the setting of gaussian mixtures under the theoretical framework of Xu and Jordan (1996). It has been proved that the asymptotic convergence rate of the EM algorithm for gaussian mixtures locally around the true solution 2 is o.e 0:5" .2 //, where " > 0 is an arbitrarily small number , o.x/ means that it is a higher-order in#nitesimal as x ! 0, and e.2 / is a measure of the average overlap of gaussians in the mixture. In other words, the large sample local convergence rate for the EM algorithm tends to be asymptotically superlinear when e.2 / tends to zero.

In this contribution, we propose a generic online (also sometimes called adaptive or recursive) version of the Expectation-Maximisation (EM) algorithm applicable to latent variable models of independent observations. Compared to the algorithm of Titterington (1984), this approach is more directly connected to the usual EM algorithm and does not rely on integration with respect to the complete data distribution. The resulting algorithm is usually simpler and is shown to achieve convergence to the stationary points of the Kullback-Leibler divergence between the marginal distribution of the observation and the model distribution at the optimal rate, i.e., that of the maximum likelihood estimator. In addition, the proposed approach is also suitable for conditional (or regression) models, as illustrated in the case of the mixture of linear regressions model. Keywords: Latent data models, Expectation-Maximisation, adaptive algorithms, online estimation, stochastic approximation, Polyak-Ruppert averaging, mixture of regressions. 1