So you know the Bayes rule. How does it relate to machine learning? It can be quite difficult to grasp how the puzzle pieces fit together - we know it took us a while. This article is an introduction we wish we had back then. While we have some grasp on the matter, we're not experts, so the following might contain inaccuracies or even outright errors.

We represent knowledge as integrity constraints in a formalization of probabilistic spatio-temporal knowledge bases. We start by defining the syntax and semantics of a formalization called PST knowledge bases. This definition generalizes an earlier version, called SPOT, which is a declarative framework for the representation and processing of probabilistic spatio-temporal data where probability is represented as an interval because the exact value is unknown. We augment the previous definition by adding a type of non-atomic formula that expresses integrity constraints. The result is a highly expressive formalism for knowledge representation dealing with probabilistic spatio-temporal data. We obtain complexity results both for checking the consistency of PST knowledge bases and for answering queries in PST knowledge bases, and also specify tractable cases. All the domains in the PST framework are finite, but we extend our results also to arbitrarily large finite domains.

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.

Regularized least-squares approaches have been successfully applied to linear system identification. Recent approaches use quadratic penalty terms on the unknown impulse response defined by stable spline kernels, which control model space complexity by leveraging regularity and bounded-input bounded-output stability. This paper extends linear system identification to a wide class of nonsmooth stable spline estimators, where regularization functionals and data misfits can be selected from a rich set of piecewise linear quadratic penalties. This class encompasses the 1-norm, huber, and vapnik, in addition to the least-squares penalty, and the approach allows linear inequality constraints on the unknown impulse response. We develop a customized interior point solver for the entire class of proposed formulations. By representing penalties through their conjugates, we allow a simple interface that enables the user to specify any piecewise linear quadratic penalty for misfit and regularizer, together with inequality constraints on the response. The solver is locally quadratically convergent, with O(n2(m+n)) arithmetic operations per iteration, for n impulse response coefficients and m output measurements. In the system identification context, where n << m, IPsolve is competitive with available alternatives, illustrated by a comparison with TFOCS and libSVM. The modeling framework is illustrated with a range of numerical experiments, featuring robust formulations for contaminated data, relaxation systems, and nonnegativity and unimodality constraints on the impulse response. Incorporating constraints yields significant improvements in system identification. The solver used to obtain the results is distributed via an open source code repository.

In previous discussions of Bayesian Inference we introduced Bayesian Statistics and considered how to infer a binomial proportion using the concept of conjugate priors. We discussed the fact that not all models can make use of conjugate priors and thus calculation of the posterior distribution would need to be approximated numerically. In this article we introduce the main family of algorithms, known collectively as Markov Chain Monte Carlo (MCMC), that allow us to approximate the posterior distribution as calculated by Bayes' Theorem. In particular, we consider the Metropolis Algorithm, which is easily stated and relatively straightforward to understand. It serves as a useful starting point when learning about MCMC before delving into more sophisticated algorithms such as Metropolis-Hastings, Gibbs Samplers and Hamiltonian Monte Carlo. Once we have described how MCMC works, we will carry it out using the open-source PyMC3 library, which takes care of many of the underlying implementation details, allowing us to concentrate on Bayesian modelling.

Recently, I began a series on exploratory data analysis; so far, I have written about computing descriptive statistics and creating box plots in R for a univariate data set with missing values. Today, I will continue this series by introducing the underlying concepts of kernel density estimation, a useful non-parametric technique for visualizing the underlying distribution of a continuous variable. In the second half of this blog post that will be published later here on AnalyticBridge, I will show how to construct kernel density estimates and plot them in R. I will also introduce rug plots and show how they can complement kernel density plots. Before defining kernel density estimation, let's define a kernel. A kernel is a special type of probability density function (PDF) with the added property that it must be even.

All the regression theory developed by statisticians over the last 200 years (related to the general linear model) is useless. Regression can be performed as accurately without statistical models, including the computation of confidence intervals (for estimates, predicted values or regression parameters). The non-statistical approach is also more robust than theory described in all statistics textbooks and taught in all statistical courses. It does not require Map-Reduce when data is really big, nor any matrix inversion, maximum likelihood estimation, or mathematical optimization (Newton algorithm). It is indeed incredibly simple, robust, easy to interpret, and easy to code (no statistical libraries required).

This practical introduction is geared towards scientists who wish to employ Bayesian networks for applied research using the BayesiaLab software platform. Through numerous examples, this book illustrates how implementing Bayesian networks involves concepts from many disciplines, including computer science, probability theory, information theory, machine learning, and statistics. Each chapter explores a real-world problem domain, exploring aspects of Bayesian networks and simultaneously introducing functions of BayesiaLab. The book can serve as a self-study guide for learners and as a reference manual for advanced practitioners.

An intelligent agent interacting with the real world will encounter individual people, courses, test results, drugs prescriptions, chairs, boxes, etc., and needs to reason about properties of these individuals and relations among them as well as cope with uncertainty. Uncertainty has been studied in probability theory and graphical models, and relations have been studied in logic, in particular in the predicate calculus and its extensions. This book examines the foundations of combining logic and probability into what are called relational probabilistic models. It introduces representations, inference, and learning techniques for probability, logic, and their combinations. The book focuses on two representations in detail: Markov logic networks, a relational extension of undirected graphical models and weighted first-order predicate calculus formula, and Problog, a probabilistic extension of logic programs that can also be viewed as a Turing-complete relational extension of Bayesian networks.

We propose new metrics on sets, ontologies, and functions that can be used in various stages of probabilistic modeling, including exploratory data analysis, learning, inference, and result interpretation. These new functions unify and generalize some of the popular metrics on sets and functions, such as the Jaccard and bag distances on sets and Marczewski-Steinhaus distance on functions. We then introduce information-theoretic metrics on directed acyclic graphs drawn independently according to a fixed probability distribution and show how they can be used to calculate similarity between class labels for the objects with hierarchical output spaces (e.g., protein function). Finally, we provide evidence that the proposed metrics are useful by clustering species based solely on functional annotations available for subsets of their genes. The functional trees resemble evolutionary trees obtained by the phylogenetic analysis of their genomes.