Statistical and Linear Independence of Binary Random Variables

Linear cryptanalysis makes use of statistical models that consider linear approximations over practical and ideal block ciphers as binary random variables. Recently, more complex models have been proposed that take also into account the statistical behavior of correlations of linear approximations over the key space of the cipher and over the randomness of the ideal cipher. The goal of this ongoing work is to investigate independence properties of linear approximations and their relationships. In this third revised version we show that the assumptions of Proposition~1 of the previous version are contradictory and hence renders that result useless. In particular, we prove that linear and statistical independence of binary random variables are equivalent properties in a vector space of variables if and only if all non-zero variables in this vector space are balanced, that is, correspond to components of a permutation. This study is motivated by finding reasonable wrong-key hypotheses for linear cryptanalysis and its generalizations which will also be discussed

Binary Independent Component Analysis with OR Mixtures

Independent component analysis (ICA) is a computational method for separating a multivariate signal into subcomponents assuming the mutual statistical independence of the non-Gaussian source signals. The classical Independent Components Analysis (ICA) framework usually assumes linear combinations of independent sources over the field of realvalued numbers R. In this paper, we investigate binary ICA for OR mixtures (bICA), which can find applications in many domains including medical diagnosis, multi-cluster assignment, Internet tomography and network resource management. We prove that bICA is uniquely identifiable under the disjunctive generation model, and propose a deterministic iterative algorithm to determine the distribution of the latent random variables and the mixing matrix. The inverse problem concerning inferring the values of latent variables are also considered along with noisy measurements. We conduct an extensive simulation study to verify the effectiveness of the propose algorithm and present examples of real-world applications where bICA can be applied.Comment: Manuscript submitted to IEEE Transactions on Signal Processin

Multivariate Bernoulli distribution

In this paper, we consider the multivariate Bernoulli distribution as a model to estimate the structure of graphs with binary nodes. This distribution is discussed in the framework of the exponential family, and its statistical properties regarding independence of the nodes are demonstrated. Importantly the model can estimate not only the main effects and pairwise interactions among the nodes but also is capable of modeling higher order interactions, allowing for the existence of complex clique effects. We compare the multivariate Bernoulli model with existing graphical inference models - the Ising model and the multivariate Gaussian model, where only the pairwise interactions are considered. On the other hand, the multivariate Bernoulli distribution has an interesting property in that independence and uncorrelatedness of the component random variables are equivalent. Both the marginal and conditional distributions of a subset of variables in the multivariate Bernoulli distribution still follow the multivariate Bernoulli distribution. Furthermore, the multivariate Bernoulli logistic model is developed under generalized linear model theory by utilizing the canonical link function in order to include covariate information on the nodes, edges and cliques. We also consider variable selection techniques such as LASSO in the logistic model to impose sparsity structure on the graph. Finally, we discuss extending the smoothing spline ANOVA approach to the multivariate Bernoulli logistic model to enable estimation of non-linear effects of the predictor variables.Comment: Published in at http://dx.doi.org/10.3150/12-BEJSP10 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Binary Models for Marginal Independence

Log-linear models are a classical tool for the analysis of contingency tables. In particular, the subclass of graphical log-linear models provides a general framework for modelling conditional independences. However, with the exception of special structures, marginal independence hypotheses cannot be accommodated by these traditional models. Focusing on binary variables, we present a model class that provides a framework for modelling marginal independences in contingency tables. The approach taken is graphical and draws on analogies to multivariate Gaussian models for marginal independence. For the graphical model representation we use bi-directed graphs, which are in the tradition of path diagrams. We show how the models can be parameterized in a simple fashion, and how maximum likelihood estimation can be performed using a version of the Iterated Conditional Fitting algorithm. Finally we consider combining these models with symmetry restrictions

Hierarchical Models for Independence Structures of Networks

We introduce a new family of network models, called hierarchical network models, that allow us to represent in an explicit manner the stochastic dependence among the dyads (random ties) of the network. In particular, each member of this family can be associated with a graphical model defining conditional independence clauses among the dyads of the network, called the dependency graph. Every network model with dyadic independence assumption can be generalized to construct members of this new family. Using this new framework, we generalize the Erd\"os-R\'enyi and beta-models to create hierarchical Erd\"os-R\'enyi and beta-models. We describe various methods for parameter estimation as well as simulation studies for models with sparse dependency graphs.Comment: 19 pages, 7 figure

Algebraic Aspects of Conditional Independence and Graphical Models

This chapter of the forthcoming Handbook of Graphical Models contains an overview of basic theorems and techniques from algebraic geometry and how they can be applied to the study of conditional independence and graphical models. It also introduces binomial ideals and some ideas from real algebraic geometry. When random variables are discrete or Gaussian, tools from computational algebraic geometry can be used to understand implications between conditional independence statements. This is accomplished by computing primary decompositions of conditional independence ideals. As examples the chapter presents in detail the graphical model of a four cycle and the intersection axiom, a certain implication of conditional independence statements. Another important problem in the area is to determine all constraints on a graphical model, for example, equations determined by trek separation. The full set of equality constraints can be determined by computing the model's vanishing ideal. The chapter illustrates these techniques and ideas with examples from the literature and provides references for further reading.Comment: 20 pages, 1 figur

Markov Properties of Discrete Determinantal Point Processes

Determinantal point processes (DPPs) are probabilistic models for repulsion. When used to represent the occurrence of random subsets of a finite base set, DPPs allow to model global negative associations in a mathematically elegant and direct way. Discrete DPPs have become popular and computationally tractable models for solving several machine learning tasks that require the selection of diverse objects, and have been successfully applied in numerous real-life problems. Despite their popularity, the statistical properties of such models have not been adequately explored. In this note, we derive the Markov properties of discrete DPPs and show how they can be expressed using graphical models.Comment: 9 pages, 1 figur

Exchangeable Variable Models

A sequence of random variables is exchangeable if its joint distribution is invariant under variable permutations. We introduce exchangeable variable models (EVMs) as a novel class of probabilistic models whose basic building blocks are partially exchangeable sequences, a generalization of exchangeable sequences. We prove that a family of tractable EVMs is optimal under zero-one loss for a large class of functions, including parity and threshold functions, and strictly subsumes existing tractable independence-based model families. Extensive experiments show that EVMs outperform state of the art classifiers such as SVMs and probabilistic models which are solely based on independence assumptions.Comment: ICML 201