Conditional independence and natural conditional functions

AbstractThe concept of conditional independence (CI) within the framework of natural conditional functions (NCFs) is studied. An NCF is a function asribing natural numbers to possible states of the world; it is the central concept of Spohn's theory of deterministic epistemology. Basic properties of CI within this framework are recalled, and further results analogous to the results concerning probabilistic CI are proved. Firstly, the intersection of two CI-models is shown to be a CI-model. Using this, it is proved that CI-models for NCFs have no finite complete axiomatic characterization (by means of a simple deductive system describing relationships among CI-statements). The last part is devoted to the marginal problem for NCFs. It is shown that (pairwise) consonancy is equivalent to consistency iff the running intersection property holds

Nonparametric tests for conditional independence using conditional distributions

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 acknowledgedThe 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

Nonparametric Identification and Estimation of Transformation Models

This paper derives sufficient conditions for nonparametric transformation models to be identified and develops estimators of the identified components. Our nonparametric identification result is global, and is derived under conditions that are substantially weaker than full independence. In particular, we show that a completeness assumption combined with conditional independence with respect to one of the regressors suffices for the model to be identified. The identification result is also constructive in the sense that it yields explicit expressions of the functions of interest. We show how natural estimators can be developed from these expressions, and analyze their theoretical properties. Importantly, it is demonstrated that the proposed estimator of the unknown transformation function converges at the parametric rate.nonparametric identification; transformation models; kernel estimation

Credal Networks under Epistemic Irrelevance

A credal network under epistemic irrelevance is a generalised type of Bayesian network that relaxes its two main building blocks. On the one hand, the local probabilities are allowed to be partially specified. On the other hand, the assessments of independence do not have to hold exactly. Conceptually, these two features turn credal networks under epistemic irrelevance into a powerful alternative to Bayesian networks, offering a more flexible approach to graph-based multivariate uncertainty modelling. However, in practice, they have long been perceived as very hard to work with, both theoretically and computationally. The aim of this paper is to demonstrate that this perception is no longer justified. We provide a general introduction to credal networks under epistemic irrelevance, give an overview of the state of the art, and present several new theoretical results. Most importantly, we explain how these results can be combined to allow for the design of recursive inference methods. We provide numerous concrete examples of how this can be achieved, and use these to demonstrate that computing with credal networks under epistemic irrelevance is most definitely feasible, and in some cases even highly efficient. We also discuss several philosophical aspects, including the lack of symmetry, how to deal with probability zero, the interpretation of lower expectations, the axiomatic status of graphoid properties, and the difference between updating and conditioning

Independence and Product Systems

Starting from elementary considerations about independence and Markov processes in classical probability we arrive at the new concept of conditional monotone independence (or operator-valued monotone independence). With the help of product systems of Hilbert modules we show that monotone conditional independence arises naturally in dilation theory.Comment: To appear in Proceedings of the ``First Sino-German Meeting on Stochastic Analysis'', Beijing, 200

On a Nonparametric Notion of Residual and its Applications

Let $(X, \mathbf{Z})$ be a continuous random vector in $\mathbb{R} \times \mathbb{R}^d$, $d \ge 1$. In this paper, we define the notion of a nonparametric residual of $X$ on $\mathbf{Z}$ that is always independent of the predictor $\mathbf{Z}$. We study its properties and show that the proposed notion of residual matches with the usual residual (error) in a multivariate normal regression model. Given a random vector $(X, Y, \mathbf{Z})$ in $\mathbb{R} \times \mathbb{R} \times \mathbb{R}^d$, we use this notion of residual to show that the conditional independence between $X$ and $Y$, given $\mathbf{Z}$, is equivalent to the mutual independence of the residuals (of $X$ on $\mathbf{Z}$ and $Y$ on $\mathbf{Z}$) and $\mathbf{Z}$. This result is used to develop a test for conditional independence. We propose a bootstrap scheme to approximate the critical value of this test. We compare the proposed test, which is easily implementable, with some of the existing procedures through a simulation study.Comment: 19 pages, 2 figure