3. What Changes Under Intervention

Chapter 2 gave us this causal model:

General health affects both treatment and recovery.

So when treatment is merely observed, a patient's value of $T$ can also carry information about $G$.

An intervention changes that.

Under:

$$ do(T=1), $$

treatment is set independently of its ordinary causes.

This gives a more precise meaning to this notation informally introduced in Chapter 1: In the graph, the incoming arrow from general health to treatment is removed while everything else remains untouched:

Pearl calls the removal of the incoming arrows into the intervened variable graph surgery. This makes the difference between observation and intervention visible.

In:

$$ P(R\mid T=1), $$

we condition on patients whose treatment status arose through the ordinary hospital process.

In:

$$ P(R\mid do(T=1)), $$

treatment has been fixed, so general health no longer determines whether it is received.

But changing the graph does not yet give us the probability of recovery under the intervention.

Our records came from the original hospital process, where general health helped determine treatment. We do not have records from the modified process represented by:

$$ do(T=1). $$

Can the observational data we already have tell us what recovery would look like after that intervention?