ignorable assignment mechanism in causal studies In the causal studies, there is so-called ignorable assignment mechanism. For instance,

The vast majority of causal studies assume certain versions of an
ignorable assignment mechanism, where the treatment as assignment is
independent of the potential outcomes conditional on some observed
variables.

How to understand this assumption?
 A: $\newcommand{\pperp}{\perp\kern-5pt\perp}$
A good reference for this is

Morgan, Stephen L., and Christopher Winship. Counterfactuals and causal inference. Cambridge University Press, 2015.

In the potential outcome framework, if we consider a binary treatment variable $D$ (e.g. placebo or real medicine), each individual is assigned two so-called potential outcomes $Y^0$ and $Y^1$ (e.g. cured or not cured), one outcome $Y^0$ that we would get for this individual if it was not treated (placebo) and one outcome $Y^1$ for this individual if it was treated (real medicine). I.e., if $Y$ is the outcome random variable, then
$$
\begin{align}
Y=Y^0 \quad &\mbox{if}\quad D=0\\
Y=Y^1 \quad &\mbox{if}\quad D=1.
\end{align}
$$
Clearly, the assignment of treatment should be independent of the potential outcome:
$$
(Y^0, Y^1) \pperp D.
$$
E.g., imagine that the real medicine is mainly administered to really sick people while the placebo is mainly given to healthier individuals that would recover anyway. This would skew the results. Note, that this independence is always given if you have proper randomization in the assignment of the treatment. Now, this independence is almost what is called ignorability, but not quite. Read on.
Unfortunately, very often we don't have randomized data. Then the challenge is to still obtain proper treatment effects from the nonrandomized data. This can often be done by e.g. the back door criterion. In a nutshell, you find a set $S$ of additional variables such that, when conditioned on, you will obtain the above-mentioned independence, i.e.:
$$
\tag{*}\label{ignor}
(Y^0, Y^1) \pperp D \;|\; S.
$$
This reads: "The potential outcomes $Y^0, Y^1$ are independent of the treatment $D$ given the covariates in the set $S$". E.g., in the example above, imagine that $S$ would consist of only one binary variable, namely the variable that the individual is either very sick or only mildly sick. That means that, if I consider only the mildly sick ones, there would be no dependence between $D$ and $(Y^0, Y^1)$ and equally for the very sick ones.
And this is the full definition of ignorability: if you find a set of covariates $S$ such that $\eqref{ignor}$ is satisfied, then the treatment assignment satisfies ignorability.
