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Let X be a smooth hypersurface of degree d in P^n over an algebraically closed field of characteristic p. We show that X must be separably rationally connected and must contain a free line if either p is at least d or if p is at least d-1…

This paper characterizes the values of partial regression coefficients, defined as projection coefficients onto the space spanned by explanatory variables, for random variables generated by linear structural equation models using graphical…

This paper deals with the problem of estimating the probability that one event was a cause of another in a given scenario. Using structural-semantical definitions of the probabilities of necessary or sufficient causation (or both), we show…

The aim in many sciences is to understand the mechanisms that underlie the observed distribution of variables, starting from a set of initial hypotheses. Causal discovery allows us to infer mechanisms as sets of cause and effect…

In this paper, the notion of conditionally bi-free independence for pairs of algebras is introduced. The notion of conditional $(\ell, r)$-cumulants are introduced and it is demonstrated that conditionally bi-free independence is equivalent…

This paper is concerned with test of the conditional independence. We first establish an equivalence between the conditional independence and the mutual independence. Based on the equivalence, we propose an index to measure the conditional…

This paper is concerned with graphical criteria that can be used to solve the problem of identifying casual effects from nonexperimental data in a causal Bayesian network structure, i.e., a directed acyclic graph that represents causal…

Analyzing the geometry of correlation sets constrained by general causal structures is of paramount importance for foundational and quantum technology research. Addressing this task is generally challenging, prompting the development of…

Identifying the effect of a treatment from observational data typically requires assuming a fully specified causal diagram. However, such diagrams are rarely known in practice, especially in complex or high-dimensional settings. To overcome…

We suggest an enhancement to structural coding through the use of (a) causally bound codes, (b) basic constructs of graph theory and (c) statistics. As is the norm with structural coding, the codes are collected into categories. The…

We consider the problem of learning causal networks with interventions, when each intervention is limited in size under Pearl's Structural Equation Model with independent errors (SEM-IE). The objective is to minimize the number of…

The main approach to defining equivalence among acyclic directed causal graphical models is based on the conditional independence relationships in the distributions that the causal models can generate, in terms of the Markov equivalence.…

Representation learners that disentangle factors of variation have already proven to be important in addressing various real world concerns such as fairness and interpretability. Initially consisting of unsupervised models with independence…

Disaggregated evaluation across subgroups is critical for assessing the fairness of machine learning models, but its uncritical use can mislead practitioners. We show that equal performance across subgroups is an unreliable measure of…

In this paper we consider the separability problem for bipartite quantum states arising from graphs. Earlier it was proved that the degree criterion is the graph-theoretic counterpart of the familiar positive partial transpose criterion for…

Causal models defined in terms of a collection of equations, as defined by Pearl, are axiomatized here. Axiomatizations are provided for three successively more general classes of causal models: (1) the class of recursive theories (those…

Understanding causal relationships between variables is fundamental across scientific disciplines. Most causal discovery algorithms rely on two key assumptions: (i) all variables are observed, and (ii) the underlying causal graph is…

The notion of causal effect is fundamental across many scientific disciplines. Traditionally, quantitative researchers have studied causal effects at the level of variables; for example, how a certain drug dose (W) causally affects a…

In this dissertation we develop a new formal graphical framework for causal reasoning. Starting with a review of monoidal categories and their associated graphical languages, we then revisit probability theory from a categorical perspective…

Deep learning has revolutionized the field of artificial intelligence. Based on the statistical correlations uncovered by deep learning-based methods, computer vision has contributed to tremendous growth in areas like autonomous driving and…