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Let $X$ and $Z$ be random vectors, and $Y=g(X,Z)$. In this paper, on the one hand, for the case that $X$ and $Z$ are continuous, by using the ideas from the total variation and the flux of $g$, we develop a point of view in causal inference capable of dealing with a broad domain of causal problems. Indeed, we focus on a function, called Probabilistic Easy Variational Causal Effect (PEACE), which can measure the direct causal effect of $X$ on $Y$ with respect to continuously and interventionally changing the values of $X$ while keeping the value of $Z$ constant. PEACE is a function of $d\ge 0$, which is a degree managing the strengths of probability density values $f(x|z)$. On the other hand, we generalize the above idea for the discrete case and show its compatibility with the continuous case. Further, we investigate some properties of PEACE using measure theoretical concepts. Furthermore, we provide some identifiability criteria and several examples showing the generic capability of PEACE. We note that PEACE can deal with the causal problems for which micro-level or just macro-level changes in the value of the input variables are important. Finally, PEACE is stable under small changes in $\partial g_{in}/\partial x$ and the joint distribution of $X$ and $Z$, where $g_{in}$ is obtained from $g$ by removing all functional relationships defining $X$ and $Z$., Comment: 45 pages, 9 Figures

In this paper, we introduce a new causal framework capable of dealing with probabilistic and non-probabilistic problems. Indeed, we provide a direct causal effect formula called Probabilistic vAriational Causal Effect (PACE) and its variations satisfying some ideas and postulates. Our formula of causal effect uses the idea of the total variation of a function integrated with probability theory. The probabilistic part is the natural availability of changing an exposure values given some variables. These variables interfere with the effect of the exposure on a given outcome. PACE has a parameter $d$ determining the degree of considering the natural availability of changing the exposure values. The lower values of $d$ refer to the scenarios for which rare cases are important. In contrast, with the higher values of $d$, our framework deals with the problems that are in nature probabilistic. Hence, instead of a single value for causal effect, we provide a causal effect vector by discretizing $d$. Further, we introduce the positive and negative PACE to measure the positive and the negative causal changes in the outcome while changing the exposure values. Furthermore, we provide an identifiability criterion for PACE to deal with observational studies. We also address the problem of computing counterfactuals in causal reasoning. We compare our framework to the Pearl, the mutual information, the conditional mutual information, and the Janzing et al. frameworks by investigating several examples., Comment: 59 pages, 7 figures

In this paper, we introduce a fundamental framework to create a bridge between Probability Theory and Fuzzy Logic. Indeed, our theory formulates a random experiment of selecting crisp elements with the criterion of having a certain fuzzy attribute. To do so, we associate some specific crisp random variables to the random experiment. Then, several formulas are presented, which make it easier to compute different conditional probabilities and expected values of these random variables. Also, we provide measure theoretical basis for our probabilistic fuzzy logic framework. Note that in our theory, the probability density functions of continuous distributions which come from the aforementioned random variables include the Dirac delta function as a term. Further, we introduce an application of our theory in Causal Inference., Comment: 60 pages, 7 figures, 1 table, 2 links to Github

Group equivariant non-expansive operators have been recently proposed as basic components in topological data analysis and deep learning. In this paper we study some geometric properties of the spaces of group equivariant operators and show how a space $\mathcal{F}$ of group equivariant non-expansive operators can be endowed with the structure of a Riemannian manifold, so making available the use of gradient descent methods for the minimization of cost functions on $\mathcal{F}$. As an application of this approach, we also describe a procedure to select a finite set of representative group equivariant non-expansive operators in the considered manifold., Comment: 21 pages, 1 figure. The introduction has been extended and a section on the group's action on the space of GENEOs has been added. Some minor fixes are made. The text has been simplified and made clearer