In the research area of potential outcomes and individual treatment effect (ITE) estimation, a common assumption called ''strong ignorability'' is often made. Given a graphical model with the following variables: treatment $T=\{0,1\}$ (e.g. giving medication or not), covariates $X$ (e.g. patient history), and outcome $Y$ (e.g. health of a patient). The corresponding visualized graphical model would look like as follows:

$Y \leftarrow X \rightarrow T \rightarrow Y$

(where Y is the same here, see image below)

[![enter image description here](https://i.stack.imgur.com/ChbJs.jpg)](https://i.stack.imgur.com/ChbJs.jpg)

Then, strong ignorability is defined as :

$(Y\_0, Y\_1) \perp\!\!\!\perp T \mid X$

where $Y\_0 = Y(T=0)$ and $Y\_1 = Y(T=1)$.

My question is, if this assumption is made, then this means the outcome is independent of the treatment given $X$. But how can the outcome ever be independent of the treatment? Why do we even bother to solve the ITE problem if we start out with the assumption that the treatment does not really make a difference for the outcome?

Isn't the whole idea of ITE estimation, to determine the effect of a treatment on the outcome Y by estimating the difference between the two potential outcomes $Y(T=0)$ and $Y(T=1)$, one of which we observe as factual observation from our observational dataset?

What I am missing here and why is my understanding incorrect?

I guess it has something to do with the fact, that if we know $X$ (i.e. when X is given), then there is no uncertainty anymore about the treatment $T$ because knowing $X$ makes $T$ deterministic (as we can see from the graphical model above?)

Moreover, I think I do not understand the difference between the following four things:

$Y \perp\!\!\!\perp T \mid X$

$(Y\_0, Y\_1) ⊥ T \mid X$

$Y\_0 \perp\!\!\!\perp T \mid X$

$Y\_1 \perp\!\!\!\perp T \mid X$