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

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

The Poisson Bracket for Poisson Forms in Multisymplectic Field Theory

We present a general definition of the Poisson bracket between differential forms on the extended multiphase space appearing in the geometric formulation of first order classical field theories and, more generally, on exact multisymplectic manifolds. It is well defined for a certain class of differential forms that we propose to call Poisson forms and turns the space of Poisson forms into a Lie superalgebra.Comment: 40 pages LaTe

A GEOGRAPHY OF ILLICIT CROPS (COCA LEAF) AND ARMED CONFLICT IN COLOMBIA

Colombia is currently the world´s largest producer of coca leaf and the principal producer of opium poppies in the Americas; the plants are the basic raw materials used to produce cocaine and heroin. This document will analyse the current relationship between these crops and illegal armed groups in Colombia, using the hypothesis that the geographical intensification of the conflict is the principal cause of expanding illicit crop production. This relationship was analysed using a theoretic model, in which an interaction between illegal armed activity and strategic territorial control lead to cocaine production. Spatial analysis techniques were then applied, especially spatial association indicators; and a clear spatial dynamic was observed, related to the two aspects mentioned above. Non parametric exercises were also carried out using matching estimators, to determine the effect illegal armed groups have on coca crops, and also to analyse the efficiency of aerial eradication policies. The results suggest that a large percentage of coca production in Colombia is due to the effects of illegal armed activity. We therefore conclude that the expansion of illegal crop growing is a consequence of the expanding conflict. In contrast, coca crops can only be used to explain a small part of the armed conflict in Colombia. In addition, we found that crop eradication via aerial spraying has not been an efficient tool in the fight against coca production in the country.Illicit crops