Related papers: Statistical Independence and the Brockwell Transfo…
The notion of a tensor product with projections or with inclusions is defined. It is shown that the definition of stochastic independence relies on such a structure and that independence can be defined in an arbitrary category with a tensor…
We develope the framework of transitional conditional independence. For this we introduce transition probability spaces and transitional random variables. These constructions will generalize, strengthen and unify previous notions of…
Structural independence is the (conditional) independence that arises from the structure rather than the precise numerical values of a distribution. We develop this concept and relate it to $d$-separation and structural causal models.…
We examine the statistical number of states, from which statistical entropy can be derived, and we show that it is an explicit function of the metric and thus observer dependent. We find a constraint on a transformation of the metric that…
Stochastic independence has a complex status in probability theory. It is not part of the definition of a probability measure, but it is nonetheless an essential property for the mathematical development of this theory. Bayesian decision…
Copulas are mathematical objects that fully capture the dependence structure among random variables and hence, offer a great flexibility in building multivariate stochastic models. In statistics, a copula is used as a general way of…
Given well-shuffled data, can we determine whether the data items are statistically (in)dependent? Formally, we consider the problem of testing whether a set of exchangeable random variables are independent. We will show that this is…
Dependence is undoubtedly a central concept in statistics. Though, it proves difficult to locate in the literature a formal definition which goes beyond the self-evident 'dependence = non-independence'. This absence has allowed the term…
(To appear in The American Statistician.) Distance covariance (Sz\'ekely, Rizzo, and Bakirov, 2007) is a fascinating recent notion, which is popular as a test for dependence of any type between random variables $X$ and $Y$. This approach…
We propose a new nonparametric test for the supposition of independence between two continuous random variables. The test is based on the size of the longest increasing subsequence of a random permutation. We identified the independence…
Bell's theorem is often said to imply that quantum mechanics violates local causality, and that local causality cannot be restored with a hidden-variables theory. This however is only correct if the hidden-variables theory fulfils an…
We develop a new framework of uncertainty variables to model uncertainty. An uncertainty variable is characterized by an uncertainty set, in which its realization is bound to lie, while the conditional uncertainty is characterized by a set…
High dimensional random dynamical systems are ubiquitous, including -- but not limited to -- cyber-physical systems, daily return on different stocks of S&P 1500 and velocity profile of interacting particle systems around McKeanVlasov…
In this paper we study different concepts of independence for convex sets of probabilities. There will be two basic ideas for independence. The first is irrelevance. Two variables are independent when a change on the knowledge about one…
We show that one can perform causal inference in a natural way for continuous-time scenarios using tools from stochastic analysis. This provides new alternatives to the positivity condition for inverse probability weighting. The probability…
Over the last decade, a series of applied mathematics papers have explored a type of inverse problem--called by a variety of names including "inverse sensitivity", "pushforward based inference", "consistent Bayesian inference", or…
Recently, Forr\'e (arXiv:2104.11547, 2021) introduced transitional conditional independence, a notion of conditional independence that provides a unified framework for both random and non-stochastic variables. The original paper establishes…
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…
In this paper we characterise the global stability, global boundedness and recurrence of solutions of a scalar nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable autonomous…
In this article we provide a substantial discussion on the statistical concept of conditional independence, which is not routinely mentioned in most elementary statistics and mathematical statistics textbooks. Under the assumption of…