Probabilistic Easy Variational Causal Effect
Usef Faghihi Amir Saki
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
$Z$.