1. From Observation To Intervention
What If I Sleep Longer?
You have been keeping track of your sleep and mood.
After a short period of time, you notice already: on days after you sleep longer, you are usually in a better mood.
The pattern becomes convincing enough that you start saying: "I feel better because I sleep longer."
You may even decide to act on it. If more sleep produces the better mood, going to bed earlier should help.
But most of the long nights in your record were weekends and holidays. You did not have to get up for work, and the day was less hurried.
So what would actually happen if you deliberately started sleeping longer?
Your records tell you what mood was like after nights on which longer sleep occurred.
Let $S=1$ mean that you slept longer than usual, and let $M=1$ mean that you were in a good mood that day. $S=0$ is your usual sleep baseline.
Your records suggest:
$$ P(M=1\mid S=1)>P(M=1\mid S=0). $$
In other words, good mood was more common after nights on which you slept longer than after nights on which you slept as usual.
But what you want to know now is what mood would be like if you made longer sleep occur. The question looks similar, but asks something quite different.
Pearl introduced notation to keep this intervention question apart from the observational question the records answer¹.
The observational comparison tells us what happened on nights when longer sleep occurred.
But those nights differed from the other nights in more than sleep duration. Those circumstances could affect both how long you slept and how you felt the next day.
Observing $S=1$ can carry information about those circumstances.
Now imagine arranging things so that you sleep longer.
We represent that intervention as:
$$ P(M\mid do(S=1)). $$
The expression $do(S=1)$ represents intervening so that longer sleep occurs, rather than selecting nights from the record on which it happened anyway.
Pearl's notation gives us a way to keep observation and intervention apart as the inquiry becomes more formal.
That is important when probability alone leaves us with a problem.
As with logic and probability, this essay is not an introduction to Pearl's theory of causality. The aim is to examine how causal assumptions and graphical representations can constrain questions about intervention, and what gives their application to a particular problem convincing force. We will go only as far into the theory as we need to for that purpose.
- cf. Judea Pearl, Causality: Models, Reasoning, and Inference, 2nd ed., Cambridge University Press, 2009.