Probabilistic Easy Variational Causal Effect

Abstract

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 $Z$.

Figures (9)

• Figure 1: (A) A $3$-cell centered at the point $(x,y,z)$ with the outward unit normal vector $\widehat{N}$. (B) The faces of the cell parallel to the $xy$ plane. (C) Two identical $3$-cells, one of which is on top of the other one. The outward unit normal vectors of these two cubes on their common face are in opposite directions.
• Figure 2: A compact region $\Omega\subseteq\mathbb{R}^3$, whose boundary $\mathscr{S}$ is smooth. The region $\Omega$ could be estimated by the union of the cells belonging to an admissible collection of cells for $\Omega$.
• Figure 3: Admissible collections could be similarly defined in any dimension. In this figure, for a better visualization, we consider an admissible collection $\Gamma=\{C_1,\ldots, C_6\}$ of $2$-cells for a 2-dimensional shape. Note that the subfigures (A), (B), and (C) shows $\lVert\Gamma\rVert$, $\mathscr{S}_{\Gamma}$, and $\mathscr{S}_{\lVert\Gamma\rVert}$, respectively. However, each of the red sides is considered twice as it is the common side of two adjacent squares.
• Figure 4: Let $g:[a,b]\to\mathbb{R}$ be continuosly differentiable. Define $\widetilde{g'}:D\to\mathbb{R}$ sending each $x\in D$ to $-|g'(x)|/g'(x)$, where $D=\{x\in(a,b):g'(x)\neq 0\}$. Also, assume that $r_0,\ldots, r_k$ are all zeros of $g'$ (in this figure, we have that $k=2$). For a given small enough $\varepsilon>0$, this figure shows the graph of a smooth extension $\varphi_{\varepsilon}:(a,b)\to\mathbb{R}$ of a restriction of $\widetilde{g'}$, which satisfies the following properties: 1) the support of $\varphi_{\varepsilon}$ is compact, 2) $\lVert\varphi_{\varepsilon}\rVert_{\infty}\le 1$, and 3) the integral of $\varphi_{\varepsilon}$ is an approximation of the integral of $\widetilde{g'}$. Let us assume that $r_{-1}=a+\varepsilon$ and $r_{k+1}=b-\varepsilon$. The thick segments on the $x$ axis have the length $2\varepsilon$. Also, $r_{-1}$, $r_0$, $r_1$, $r_2$, and $r_3$ are the midpoints of the five inner thick segments. The black segments on the graph come from the graph of $\widetilde{g'}$.
• Figure 5: The set of the vertices of each of the small cubes and their faces are a cube-like and a face-like, respectively.

Theorems & Definitions (49)

• Theorem 3.1
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Proposition 4.3
• proof
• Theorem 4.4
• proof
• Remark 4.5