Bayes and maximum likelihood for L1L^1-Wasserstein deconvolution of Laplace mixtures

We consider the problem of recovering a distribution function on the real line from observations additively contaminated with errors following the standard Laplace distribution. Assuming that the latent distribution is completely unknown leads to a nonparametric deconvolution problem. We begin by studying the rates of convergence relative to the $L^2$-norm and the Hellinger metric for the direct problem of estimating the sampling density, which is a mixture of Laplace densities with a possibly unbounded set of locations: the rate of convergence for the Bayes' density estimator corresponding to a Dirichlet process prior over the space of all mixing distributions on the real line matches, up to a logarithmic factor, with the $n^{-3/8}\log^{1/8}n$ rate for the maximum likelihood estimator. Then, appealing to an inversion inequality translating the $L^2$-norm and the Hellinger distance between general kernel mixtures, with a kernel density having polynomially decaying Fourier transform, into any $L^p$-Wasserstein distance, $p\geq1$, between the corresponding mixing distributions, provided their Laplace transforms are finite in some neighborhood of zero, we derive the rates of convergence in the $L^1$-Wasserstein metric for the Bayes' and maximum likelihood estimators of the mixing distribution. Merging in the $L^1$-Wasserstein distance between Bayes and maximum likelihood follows as a by-product, along with an assessment on the stochastic order of the discrepancy between the two estimation procedures

Fast performance estimation of block codes

Importance sampling is used in this paper to address the classical yet important problem of performance estimation of block codes. Simulation distributions that comprise discreteand continuous-mixture probability densities are motivated and used for this application. These mixtures are employed in concert with the so-called g-method, which is a conditional importance sampling technique that more effectively exploits knowledge of underlying input distributions. For performance estimation, the emphasis is on bit by bit maximum a-posteriori probability decoding, but message passing algorithms for certain codes have also been investigated. Considered here are single parity check codes, multidimensional product codes, and briefly, low-density parity-check codes. Several error rate results are presented for these various codes, together with performances of the simulation techniques

Network Tomography: Identifiability and Fourier Domain Estimation

The statistical problem for network tomography is to infer the distribution of $\mathbf{X}$, with mutually independent components, from a measurement model $\mathbf{Y}=A\mathbf{X}$, where $A$ is a given binary matrix representing the routing topology of a network under consideration. The challenge is that the dimension of $\mathbf{X}$ is much larger than that of $\mathbf{Y}$ and thus the problem is often called ill-posed. This paper studies some statistical aspects of network tomography. We first address the identifiability issue and prove that the $\mathbf{X}$ distribution is identifiable up to a shift parameter under mild conditions. We then use a mixture model of characteristic functions to derive a fast algorithm for estimating the distribution of $\mathbf{X}$ based on the General method of Moments. Through extensive model simulation and real Internet trace driven simulation, the proposed approach is shown to be favorable comparing to previous methods using simple discretization for inferring link delays in a heterogeneous network.Comment: 21 page

Approximate Profile Maximum Likelihood

We propose an efficient algorithm for approximate computation of the profile maximum likelihood (PML), a variant of maximum likelihood maximizing the probability of observing a sufficient statistic rather than the empirical sample. The PML has appealing theoretical properties, but is difficult to compute exactly. Inspired by observations gleaned from exactly solvable cases, we look for an approximate PML solution, which, intuitively, clumps comparably frequent symbols into one symbol. This amounts to lower-bounding a certain matrix permanent by summing over a subgroup of the symmetric group rather than the whole group during the computation. We extensively experiment with the approximate solution, and find the empirical performance of our approach is competitive and sometimes significantly better than state-of-the-art performance for various estimation problems

A Method of Moments for Mixture Models and Hidden Markov Models

Mixture models are a fundamental tool in applied statistics and machine learning for treating data taken from multiple subpopulations. The current practice for estimating the parameters of such models relies on local search heuristics (e.g., the EM algorithm) which are prone to failure, and existing consistent methods are unfavorable due to their high computational and sample complexity which typically scale exponentially with the number of mixture components. This work develops an efficient method of moments approach to parameter estimation for a broad class of high-dimensional mixture models with many components, including multi-view mixtures of Gaussians (such as mixtures of axis-aligned Gaussians) and hidden Markov models. The new method leads to rigorous unsupervised learning results for mixture models that were not achieved by previous works; and, because of its simplicity, it offers a viable alternative to EM for practical deployment