Mixture modeling is a general technique for making any simple model more expressive through weighted combination. This generality and simplicity in part explains the success of the Expectation Maximization (EM) algorithm, in which updates are easy to derive for a wide class of mixture models. However, the likelihood of a mixture model is non-convex, so EM has no known global convergence guarantees. Recently, method of moments approaches offer global guarantees for some mixture models, but they do not extend easily to the range of mixture models that exist. In this work, we present Polymom, an unifying framework based on method of moments in which estimation procedures are easily derivable, just as in EM. Polymom is applicable when the moments of a single mixture component are polynomials of the parameters. Our key observation is that the moments of the mixture model are a mixture of these polynomials, which allows us to cast estimation as a Generalized Moment Problem. We solve its relaxations using semidefinite optimization, and then extract parameters using ideas from computer algebra. This framework allows us to draw insights and apply tools from convex optimization, computer algebra and the theory of moments to study problems in statistical estimation.

Machine learning is becoming a buzzword, everybody talks aboit it and few seem to be interested in the math underneath (I find statements like "I wanted to know more but all sources were too statistical/mathematical and I wanted more practical stuff"). Let me tell you something: You can't really use Machine Learning if you don't know the statistical/mathematical basis. I am really upset when I see a Youtube video of some guy in T-Shirt probably working at a large organization ranting about Machine Learning and Data Science, telling programmers that maths is easy to grasp. Everybody knows how to press a button or, if you force me, almost everybody knows how to fix something in their Windows control panel, but that does not mean we can trust them when talking about building a secure payment system, Everybody can use Mahout or the like but that does not mean he knows jack about what he is doing using Naive Bayes to predict the class from thre variables (x, y, z) where z x 2 and x belongs to the range [-1,1]. Machine Learning is just a fancy word for the statistical/mathematical tools lying underneath, whose objective is to extract something that we may loosely call knowledge (or something that we understand) from data (or something chaotic that we do not understand), so that computers may take action based on the inferred knowledge.

Here we discuss general applications of statistical models, whether they arise from data science, operations research, engineering, machine learning or statistics. We do not discuss specific algorithms such as decision trees, logistic regression, Bayesian modeling, Markov models, data reduction or feature selection. Instead, I discuss frameworks - each one using its own types of techniques and algorithms - to solve real life problems. Most of the entries below are found in Wikipedia, and I have used a few definitions or extracts from the relevant Wikipedia articles, in addition to personal contributions. Spatial dependency is the co-variation of properties within geographic space: characteristics at proximal locations appear to be correlated, either positively or negatively. Methods for time series analyses may be divided into two classes: frequency-domain methods and time-domain methods.

We investigate the choice of tuning parameters for a Bayesian multi-level group lasso model developed for the joint analysis of neuroimaging and genetic data. The regression model we consider relates multivariate phenotypes consisting of brain summary measures (volumetric and cortical thickness values) to single nucleotide polymorphism (SNPs) data and imposes penalization at two nested levels, the first corresponding to genes and the second corresponding to SNPs. Associated with each level in the penalty is a tuning parameter which corresponds to a hyperparameter in the hierarchical Bayesian formulation. Following previous work on Bayesian lassos we consider the estimation of tuning parameters through either hierarchical Bayes based on hyperpriors and Gibbs sampling or through empirical Bayes based on maximizing the marginal likelihood using a Monte Carlo EM algorithm. For the specific model under consideration we find that these approaches can lead to severe overshrinkage of the regression parameter estimates in the high-dimensional setting or when the genetic effects are weak. We demonstrate these problems through simulation examples and study an approximation to the marginal likelihood which sheds light on the cause of this problem. We then suggest an alternative approach based on the widely applicable information criterion (WAIC), an asymptotic approximation to leave-one-out cross-validation that can be computed conveniently within an MCMC framework.

K-means clustering is one of the simplest clustering algorithms one can use to find natural groupings of an unlabeled data set. Another way of stating this is that k-means clustering is an unsupervised learning algorithm. "learning the structure of X without being given Y". K-means clustering finds "k" different means (surprise surprise) which represent the centers of k clusters and assigns each data point to one of these clusters. The cluster it is assigned to is the one where the distance (usually Euclidean) from the point to the mean is smallest.

This article discusses the quadratization of Markov Logic Networks, which enables efficient approximate MAP computation by means of maximum flows. The procedure relies on a pseudo-Boolean representation of the model, and allows handling models of any order. The employed pseudo-Boolean representation can be used to identify problems that are guaranteed to be solvable in low polynomial-time. Results on common benchmark problems show that the proposed approach finds optimal assignments for most variables in excellent computational time and approximate solutions that match the quality of ILP-based solvers.

This thesis focuses on gaining linguistic insights into textual discussions on a word level. It was of special interest to distinguish messages that constructively contribute to a discussion from those that are detrimental to them. Thereby, we wanted to determine whether "I"- and "You"-messages are indicators for either of the two discussion styles. These messages are nowadays often used in guidelines for successful communication. Although their effects have been successfully evaluated multiple times, a large-scale analysis has never been conducted. Thus, we used Wikipedia Articles for Deletion (short: AfD) discussions together with the records of blocked users and developed a fully automated creation of an annotated data set. In this data set, messages were labelled either constructive or disruptive. We applied binary classifiers to the data to determine characteristic words for both discussion styles. Thereby, we also investigated whether function words like pronouns and conjunctions play an important role in distinguishing the two. We found that "You"-messages were a strong indicator for disruptive messages which matches their attributed effects on communication. However, we found "I"-messages to be indicative for disruptive messages as well which is contrary to their attributed effects. The importance of function words could neither be confirmed nor refuted. Other characteristic words for either communication style were not found. Yet, the results suggest that a different model might represent disruptive and constructive messages in textual discussions better.

The present work proposes hybridization of Expectation-Maximization (EM) and K-Means techniques as an attempt to speed-up the clustering process. Though both K-Means and EM techniques look into different areas, K-means can be viewed as an approximate way to obtain maximum likelihood estimates for the means. Along with the proposed algorithm for hybridization, the present work also experiments with the Standard EM algorithm. Six different datasets are used for the experiments of which three are synthetic datasets. Clustering fitness and Sum of Squared Errors (SSE) are computed for measuring the clustering performance. In all the experiments it is observed that the proposed algorithm for hybridization of EM and K-Means techniques is consistently taking less execution time with acceptable Clustering Fitness value and less SSE than the standard EM algorithm. It is also observed that the proposed algorithm is producing better clustering results than the Cluster package of Purdue University.

We introduce Markov substitute processes, a new model at the crossroad of statistics and formal grammars, and prove its main property : Markov substitute processes with a given support form an exponential family.

We present a method for performing Hamiltonian Monte Carlo that largely eliminates sample rejection for typical hyperparameters. In situations that would normally lead to rejection, instead a longer trajectory is computed until a new state is reached that can be accepted. This is achieved using Markov chain transitions that satisfy the fixed point equation, but do not satisfy detailed balance. The resulting algorithm significantly suppresses the random walk behavior and wasted function evaluations that are typically the consequence of update rejection. We demonstrate a greater than factor of two improvement in mixing time on three test problems. We release the source code as Python and MATLAB packages.