Notes of Causal Inference-1

• What Does Imply Causation?
• Identificability
• Assumption 1: Ignorability
• Assumption 1: Conditional exchangeability
• Assumption 2: Unconfoundedness
• Assumption 2: Positivity
• Assumption 3: No Interference
• Assumption 4: Consistency
• Summary

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What Does Imply Causation?

Consider following example, we cannot take $Y_i \mid _{T=1} - Y_i \mid _{T=0}$ as the causal effect because $Y_i\mid _{T=1}$ cannot represent the potential outcome of “if we take $T=1$”.

The main difference between causation and correlation is $Y_i\mid _{T=1}\neq Y_i\mid _{do(T=1)}$

Here we calculate the true causal effect, $Y_i\mid _{do(T=1)}-Y_i\mid _{do(T=0)}$, noted as $Y_i(1)-Y_i(0)$:

which is because we have confounding association $C$:

Identificability

Our goal is to calculate causal quantity with statistic quantity:

Assumption 1: Ignorability

The igorability can be showed as following: \(\mathbb{E}[Y(1)]-\mathbb{E}[Y(0)] = \mathbb{E}[Y(1)\mid T=1]-\mathbb{E}[Y(0)\mid T=0)]\)

which means that the potential outcome $Y(1)$ is regardless of what the $T$ value is in practice. In the following table, the blankets in $Y(1)$ and $Y(0)$ columns are the unobservable outcomes. Those values are independent of whether corresponding $T$ is taken.

The ignorability can be described as:

Well, in another perspective, the ingorability can be viewed as exchangeability:

which means that the outcomes is independent of what data you choose (swaping A and B doesn’t changes the expectations $y_0$ and $y_1$).

Assumption 1: Conditional exchangeability

However, we don’t know whether the data set satisfies exchangeability. Looking at the following example, we find that the distribution of “drunk/sober” in $T=1$ is different from it in $T=0$, where exchangeability is not satisfied.

So the conditional exchangeability is proposed that $(Y(1),Y(0))\perp\perp T\mid X$, where $X$ represents “drunk/sober”. So we have:

Assumption 2: Unconfoundedness

Unconfoundedness is an untestable assumption that $X$ is the only confoundedness and there is no unobservable “W”.

Assumption 2: Positivity

Positivity demands that the support set is consist of the set $X$.

And if we don’t have positivity, we have to predict the potential outcomes:

Assumption 3: No Interference

No interference means that the choice of $T$ in other samples doesn’t influence this outcome:

Assumption 4: Consistency

The same $T$ must correspond to the same outcome.

Summary