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## A scoring function for learning Bayesian networks based on mutual information and conditional independence tests

Campos Ib????ez, Luis Miguel
Fonte: MIT Press Publicador: MIT Press
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
56.32%
We propose a new scoring function for learning Bayesian networks from data using score+search algorithms. This is based on the concept of mutual information and exploits some well-known properties of this measure in a novel way. Essentially, a statistical independence test based on the chi-square distribution, associated with the mutual information measure, together with a property of additive decomposition of this measure, are combined in order to measure the degree of interaction between each variable and its parent variables in the network. The result is a non-Bayesian scoring function called MIT (mutual information tests) which belongs to the family of scores based on information theory. The MIT score also represents a penalization of the Kullback-Leibler divergence between the joint probability distributions associated with a candidate network and with the available data set. Detailed results of a complete experimental evaluation of the proposed scoring function and its comparison with the well-known K2, BDeu and BIC/MDL scores are also presented.

## A nonparametric copula based test for conditional independence with applications to granger causality

Bouezmarni, Taoufik; Rombouts, Jeroen V. K.; Taamouti, Abderrahim
Tipo: Trabalho em Andamento Formato: application/pdf
Relevância na Pesquisa
66.41%
This paper proposes a new nonparametric test for conditional independence, which is based on the comparison of Bernstein copula densities using the Hellinger distance. The test is easy to implement because it does not involve a weighting function in the test statistic, and it can be applied in general settings since there is no restriction on the dimension of the data. In fact, to apply the test, only a bandwidth is needed for the nonparametric copula. We prove that the test statistic is asymptotically pivotal under the null hypothesis, establish local power properties, and motivate the validity of the bootstrap technique that we use in finite sample settings. A simulation study illustrates the good size and power properties of the test. We illustrate the empirical relevance of our test by focusing on Granger causality using financial time series data to test for nonlinear leverage versus volatility feedback effects and to test for causality between stock returns and trading volume. In a third application, we investigate Granger causality between macroeconomic variables

## Nonparametric tests for conditional independence using conditional distributions

Bouezmarni, Taoufik; Taamouti, Abderrahim
Tipo: info:eu-repo/semantics/draft; info:eu-repo/semantics/workingPaper Formato: text/plain; application/pdf
Relevância na Pesquisa
66.56%
The concept of causality is naturally defined in terms of conditional distribution, however almost all the empirical works focus on causality in mean. This paper aim to propose a nonparametric statistic to test the conditional independence and Granger non-causality between two variables conditionally on another one. The test statistic is based on the comparison of conditional distribution functions using an L2 metric. We use Nadaraya-Watson method to estimate the conditional distribution functions. We establish the asymptotic size and power properties of the test statistic and we motivate the validity of the local bootstrap. Further, we ran a simulation experiment to investigate the finite sample properties of the test and we illustrate its practical relevance by examining the Granger non-causality between S&P 500 Index returns and VIX volatility index. Contrary to the conventional t-test, which is based on a linear mean-regression model, we find that VIX index predicts excess returns both at short and long horizons.; Financial support from the Natural Sciences and Engineering Research Council of Canada and from the Spanish Ministry of Education through grants SEJ 2007-63098 are also acknowledged

## Minimum Distance Estimation in Categorical Conditional Independence Models

Kahle, David John
Tipo: Thesis; Text Formato: 185 p.; application/pdf
Português
Relevância na Pesquisa
66.73%
One of the oldest and most fundamental problems in statistics is the analysis of cross-classified data called contingency tables. Analyzing contingency tables is typically a question of association - do the variables represented in the table exhibit special dependencies or lack thereof? The statistical models which best capture these experimental notions of dependence are the categorical conditional independence models; however, until recent discoveries concerning the strongly algebraic nature of the conditional independence models surfaced, the models were widely overlooked due to their unwieldy implicit description. Apart from the inferential question above, this thesis asks the more basic question - suppose such an experimental model of association is known, how can one incorporate this information into the estimation of the joint distribution of the table? In the traditional parametric setting several estimation paradigms have been developed over the past century; however, traditional results are not applicable to arbitrary categorical conditional independence models due to their implicit nature. After laying out the framework for conditional independence and algebraic statistical models, we consider three aspects of estimation in the models using the minimum Euclidean (L2E)...

## Technical Note : assessing predictive capacity and conditional independence of landslide predisposing factors for shallow landslide susceptibility models

Pereira, Susana da Silva; Zêzere, José Luís Gonçalves Moreira da Silva; Bateira, Carlos
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.41%
The aim of this study is to identify the landslide predisposing factors' combination using a bivariate statistical model that best predicts landslide susceptibility. The best model is one that has simultaneously good performance in terms of suitability and predictive power and has been developed using variables that are conditionally independent. The study area is the Santa Marta de Penaguião council (70 km2) located in the Northern Portugal. In order to identify the best combination of landslide predisposing factors, all possible combinations using up to seven predisposing factors were performed, which resulted in 120 predictions that were assessed with a landside inventory containing 767 shallow translational slides. The best landslide susceptibility model was selected according to the model degree of fitness and on the basis of a conditional independence criterion. The best model was developed with only three landslide predisposing factors (slope angle, inverse wetness index, and land use) and was compared with a model developed using all seven landslide predisposing factors. Results showed that it is possible to produce a reliable landslide susceptibility model using fewer landslide predisposing factors, which contributes towards higher conditional independence.

## Conditional independence relations among biological markers may improve clinical decision as in the case of triple negative breast cancers

Stefanini, Federico M; Coradini, Danila; Biganzoli, Elia
Fonte: BioMed Central Publicador: BioMed Central
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.59%
The associations existing among different biomarkers are important in clinical settings because they contribute to the characterisation of specific pathways related to the natural history of the disease, genetic and environmental determinants. Despite the availability of binary/linear (or at least monotonic) correlation indices, the full exploitation of molecular information depends on the knowledge of direct/indirect conditional independence (and eventually causal) relationships among biomarkers, and with target variables in the population of interest. In other words, that depends on inferences which are performed on the joint multivariate distribution of markers and target variables. Graphical models, such as Bayesian Networks, are well suited to this purpose. Therefore, we reconsidered a previously published case study on classical biomarkers in breast cancer, namely estrogen receptor (ER), progesterone receptor (PR), a proliferative index (Ki67/MIB-1) and to protein HER2/neu (NEU) and p53, to infer conditional independence relations existing in the joint distribution by inferring (learning) the structure of graphs entailing those relations of independence. We also examined the conditional distribution of a special molecular phenotype...

## Gaussian conditional independence relations have no finite complete characterization

Sullivant, Seth
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.52%
We show that there can be no finite list of conditional independence relations which can be used to deduce all conditional independence implications among Gaussian random variables. To do this, we construct, for each $n> 3$ a family of $n$ conditional independence statements on $n$ random variables which together imply that $X_1 \ind X_2$, and such that no subset have this same implication. The proof relies on binomial primary decomposition.; Comment: 6 pages

## A Scalable Conditional Independence Test for Nonlinear, Non-Gaussian Data

Ramsey, Joseph D.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
46.59%
Many relations of scientific interest are nonlinear, and even in linear systems distributions are often non-Gaussian, for example in fMRI BOLD data. A class of search procedures for causal relations in high dimensional data relies on sample derived conditional independence decisions. The most common applications rely on Gaussian tests that can be systematically erroneous in nonlinear non-Gaussian cases. Recent work (Gretton et al. (2009), Tillman et al. (2009), Zhang et al. (2011)) has proposed conditional independence tests using Reproducing Kernel Hilbert Spaces (RKHS). Among these, perhaps the most efficient has been KCI (Kernel Conditional Independence, Zhang et al. (2011)), with computational requirements that grow effectively at least as O(N3), placing it out of range of large sample size analysis, and restricting its applicability to high dimensional data sets. We propose a class of O(N2) tests using conditional correlation independence (CCI) that require a few seconds on a standard workstation for tests that require tens of minutes to hours for the KCI method, depending on degree of parallelization, with similar accuracy. For accuracy on difficult nonlinear, non-Gaussian data sets, we also compare a recent test due to Harris & Drton (2012)...

## Robustness and Conditional Independence Ideals

Rauh, Johannes; Ay, Nihat
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.41%
We study notions of robustness of Markov kernels and probability distribution of a system that is described by $n$ input random variables and one output random variable. Markov kernels can be expanded in a series of potentials that allow to describe the system's behaviour after knockouts. Robustness imposes structural constraints on these potentials. Robustness of probability distributions is defined via conditional independence statements. These statements can be studied algebraically. The corresponding conditional independence ideals are related to binary edge ideals. The set of robust probability distributions lies on an algebraic variety. We compute a Gr\"obner basis of this ideal and study the irreducible decomposition of the variety. These algebraic results allow to parametrize the set of all robust probability distributions.; Comment: 16 pages

## Extended Conditional Independence and Applications in Causal Inference

Constantinou, Panayiota; Dawid, A. Philip
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.59%
The goal of this paper is to integrate the notions of stochastic conditional independence and variation conditional independence under a more general notion of extended conditional independence. We show that under appropriate assumptions the calculus that applies for the two cases separately (axioms of a separoid) still applies for the extended case. These results provide a rigorous basis for a wide range of statistical concepts, including ancillarity and sufficiency, and, in particular, the Decision Theoretic framework for statistical causality, which uses the language and calculus of conditional independence in order to express causal properties and make causal inferences.

## Conditional Independence and Markov Properties in Possibility Theory

Vejnarova, Jirina
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.41%
Conditional independence and Markov properties are powerful tools allowing expression of multidimensional probability distributions by means of low-dimensional ones. As multidimensional possibilistic models have been studied for several years, the demand for analogous tools in possibility theory seems to be quite natural. This paper is intended to be a promotion of de Cooman's measure-theoretic approcah to possibility theory, as this approach allows us to find analogies to many important results obtained in probabilistic framework. First, we recall semi-graphoid properties of conditional possibilistic independence, parameterized by a continuous t-norm, and find sufficient conditions for a class of Archimedean t-norms to have the graphoid property. Then we introduce Markov properties and factorization of possibility distrubtions (again parameterized by a continuous t-norm) and find the relationships between them. These results are accompanied by a number of conterexamples, which show that the assumptions of specific theorems are substantial.; Comment: Appears in Proceedings of the Sixteenth Conference on Uncertainty in Artificial Intelligence (UAI2000)

## Kernel-based Conditional Independence Test and Application in Causal Discovery

Zhang, Kun; Peters, Jonas; Janzing, Dominik; Schoelkopf, Bernhard
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.59%
Conditional independence testing is an important problem, especially in Bayesian network learning and causal discovery. Due to the curse of dimensionality, testing for conditional independence of continuous variables is particularly challenging. We propose a Kernel-based Conditional Independence test (KCI-test), by constructing an appropriate test statistic and deriving its asymptotic distribution under the null hypothesis of conditional independence. The proposed method is computationally efficient and easy to implement. Experimental results show that it outperforms other methods, especially when the conditioning set is large or the sample size is not very large, in which case other methods encounter difficulties.

## Bayesian test of significance for conditional independence: The multinomial model

Andrade, Pablo de Morais; Stern, Julio Michael; Pereira, Carlos Alberto de Bragança
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.41%
Conditional independence tests (CI tests) have received special attention lately in Machine Learning and Computational Intelligence related literature as an important indicator of the relationship among the variables used by their models. In the field of Probabilistic Graphical Models (PGM)--which includes Bayesian Networks (BN) models--CI tests are especially important for the task of learning the PGM structure from data. In this paper, we propose the Full Bayesian Significance Test (FBST) for tests of conditional independence for discrete datasets. FBST is a powerful Bayesian test for precise hypothesis, as an alternative to frequentist's significance tests (characterized by the calculation of the \emph{p-value}).; Comment: 24 pages, 33 figures

## Smoothness of Gaussian conditional independence models

Drton, Mathias; Xiao, Han
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.52%
Conditional independence in a multivariate normal (or Gaussian) distribution is characterized by the vanishing of subdeterminants of the distribution's covariance matrix. Gaussian conditional independence models thus correspond to algebraic subsets of the cone of positive definite matrices. For statistical inference in such models it is important to know whether or not the model contains singularities. We study this issue in models involving up to four random variables. In particular, we give examples of conditional independence relations which, despite being probabilistically representable, yield models that non-trivially decompose into a finite union of several smooth submodels.

## Conditional Independence in Uncertainty Theories

Shenoy, Prakash P.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.6%
This paper introduces the notions of independence and conditional independence in valuation-based systems (VBS). VBS is an axiomatic framework capable of representing many different uncertainty calculi. We define independence and conditional independence in terms of factorization of the joint valuation. The definitions of independence and conditional independence in VBS generalize the corresponding definitions in probability theory. Our definitions apply not only to probability theory, but also to Dempster-Shafer's belief-function theory, Spohn's epistemic-belief theory, and Zadeh's possibility theory. In fact, they apply to any uncertainty calculi that fit in the framework of valuation-based systems.; Comment: Appears in Proceedings of the Eighth Conference on Uncertainty in Artificial Intelligence (UAI1992)

## A Graph-Based Inference Method for Conditional Independence

Shachter, Ross D.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.58%
The graphoid axioms for conditional independence, originally described by Dawid [1979], are fundamental to probabilistic reasoning [Pearl, 19881. Such axioms provide a mechanism for manipulating conditional independence assertions without resorting to their numerical definition. This paper explores a representation for independence statements using multiple undirected graphs and some simple graphical transformations. The independence statements derivable in this system are equivalent to those obtainable by the graphoid axioms. Therefore, this is a purely graphical proof technique for conditional independence.; Comment: Appears in Proceedings of the Seventh Conference on Uncertainty in Artificial Intelligence (UAI1991)

## Conditional independence relations and log-linear models for random permutations

Csiszár, V.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.41%
We propose a new class of models for random permutations, which we call log-linear models, by the analogy with log-linear models used in the analysis of contingency tables. As a special case, we study the family of all Luce-decomposable distributions, and the family of those random permutations, for which the distribution of both the permutation and its inverse is Luce-decomposable. We show that these latter models can be described by conditional independence relations. We calculate the number of free parameters in these models, and describe an iterative algorithm for maximum likelihood estimation, which enables us to test if a set of data satisfies the conditional independence relations or not.; Comment: 25 pages

## Conditions Under Which Conditional Independence and Scoring Methods Lead to Identical Selection of Bayesian Network Models

Cowell, Robert G.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.41%
It is often stated in papers tackling the task of inferring Bayesian network structures from data that there are these two distinct approaches: (i) Apply conditional independence tests when testing for the presence or otherwise of edges; (ii) Search the model space using a scoring metric. Here I argue that for complete data and a given node ordering this division is a myth, by showing that cross entropy methods for checking conditional independence are mathematically identical to methods based upon discriminating between models by their overall goodness-of-fit logarithmic scores.; Comment: Appears in Proceedings of the Seventeenth Conference on Uncertainty in Artificial Intelligence (UAI2001)

## Nonparametric testing of conditional independence by means of the partial copula

Bergsma, Wicher
Tipo: Artigo de Revista Científica
We propose a new method to test conditional independence of two real random variables $Y$ and $Z$ conditionally on an arbitrary third random variable $X$. %with $F_{.|.}$ representing conditional distribution functions, The partial copula is introduced, defined as the joint distribution of $U=F_{Y|X}(Y|X)$ and $V=F_{Z|X}(Z|X)$. We call this transformation of $(Y,Z)$ into $(U,V)$ the partial copula transform. It is easy to show that if $Y$ and $Z$ are continuous for any given value of $X$, then $Y\ind Z|X$ implies $U\ind V$. Conditional independence can then be tested by (i) applying the partial copula transform to the data points and (ii) applying a test of ordinary independence to the transformed data. In practice, $F_{Y|X}$ and $F_{Z|X}$ will need to be estimated, which can be done by, e.g., standard kernel methods. We show that under easily satisfied conditions, and for a very large class of test statistics for independence which includes the covariance, Kendall's tau, and Hoeffding's test statistic, the effect of this estimation vanishes asymptotically. Thus, for large samples, the estimation can be ignored and we have a simple method which can be used to apply a wide range of tests of independence, including ones with consistency for arbitrary alternatives...
In this paper, the maximal nonlinear conditional correlation of two random vectors $X$ and $Y$ given another random vector $Z$, denoted by $\rho_1(X,Y|Z)$, is defined as a measure of conditional association, which satisfies certain desirable properties. When $Z$ is continuous, a test for testing the conditional independence of $X$ and $Y$ given $Z$ is constructed based on the estimator of a weighted average of the form $\sum_{k=1}^{n_Z}f_Z(z_k)\rho^2_1(X,Y|Z=z_k)$, where $f_Z$ is the probability density function of $Z$ and the $z_k$'s are some points in the range of $Z$. Under some conditions, it is shown that the test statistic is asymptotically normal under conditional independence, and the test is consistent.; Comment: Published in at http://dx.doi.org/10.1214/09-AOS770 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)