Abstract:
We introduce a new interpretation of two related notions - conditional utility and utility independence. Unlike the traditional interpretation, the new interpretation renders the notions the direct analogues of their probabilistic counterparts. To capture these notions formally, we appeal to the notion of utility distribution, introduced in previous paper. We show that utility distributions, which have a structure that is identical to that of probability distributions, can be viewed as a special case of an additive multiattribute utility functions, and show how this special case permits us to capture the novel senses of conditional utility and utility independence. Finally, we present the notion of utility networks, which do for utilities what Bayesian networks do for probabilities. Specifically, utility networks exploit the new interpretation of conditional utility and utility independence to compactly represent a utility distribution.

Abstract:
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.

Abstract:
Possibilistic conditional independence is investigated: we propose a definition of this notion similar to the one used in probability theory. The links between independence and non-interactivity are investigated, and properties of these relations are given. The influence of the conjunction used to define a conditional measure of possibility is also highlighted: we examine three types of conjunctions: Lukasiewicz - like T-norms, product-like T-norms and the minimum operator.

Abstract:
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}).

Abstract:
Independence screening is a powerful method for variable selection for `Big Data' when the number of variables is massive. Commonly used independence screening methods are based on marginal correlations or variations of it. In many applications, researchers often have some prior knowledge that a certain set of variables is related to the response. In such a situation, a natural assessment on the relative importance of the other predictors is the conditional contributions of the individual predictors in presence of the known set of variables. This results in conditional sure independence screening (CSIS). Conditioning helps for reducing the false positive and the false negative rates in the variable selection process. In this paper, we propose and study CSIS in the context of generalized linear models. For ultrahigh-dimensional statistical problems, we give conditions under which sure screening is possible and derive an upper bound on the number of selected variables. We also spell out the situation under which CSIS yields model selection consistency. Moreover, we provide two data-driven methods to select the thresholding parameter of conditional screening. The utility of the procedure is illustrated by simulation studies and analysis of two real data sets.

Abstract:
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.

Abstract:
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.

Abstract:
It is well-known that the notion of (strong) conditional independence (CI) is too restrictive to capture independencies that only hold in certain contexts. This kind of contextual independency, called context-strong independence (CSI), can be used to facilitate the acquisition, representation, and inference of probabilistic knowledge. In this paper, we suggest the use of contextual weak independence (CWI) in Bayesian networks. It should be emphasized that the notion of CWI is a more general form of contextual independence than CSI. Furthermore, if the contextual strong independence holds for all contexts, then the notion of CSI becomes strong CI. On the other hand, if the weak contextual independence holds for all contexts, then the notion of CWI becomes weak independence (WI) nwhich is a more general noncontextual independency than strong CI. More importantly, complete axiomatizations are studied for both the class of WI and the class of CI and WI together. Finally, the interesting property of WI being a necessary and sufficient condition for ensuring consistency in granular probabilistic networks is shown.

Abstract:
In broad applications, it is routinely of interest to assess whether there is evidence in the data to refute the assumption of conditional independence of $Y$ and $X$ conditionally on $Z$. Such tests are well developed in parametric models but are not straightforward in the nonparametric case. We propose a general Bayesian approach, which relies on an encompassing nonparametric Bayes model for the joint distribution of $Y$, $X$ and $Z$. The framework allows $Y$, $X$ and $Z$ to be random variables on arbitrary spaces, and can accommodate different dimensional vectors having a mixture of discrete and continuous measurement scales. Using conditional mutual information as a scalar summary of the strength of the conditional dependence relationship, we construct null and alternative hypotheses. We provide conditions under which the correct hypothesis will be consistently selected. Computational methods are developed, which can be incorporated within MCMC algorithms for the encompassing model. The methods are applied to variable selection and assessed through simulations and criminology applications.

Abstract:
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.