I would like to ask a few questions on Judea Pearls approach to causal inference.

In judea pearls book we want to estimate:


now to get to the estimand we can rewrite this in terms of our initial causal model e.g.


So we have re-written this 'do' expression into one using conditional probabilities from our original data we had defined. What i don't understand is that if we write Probability of Y given X, doesn't the fact that we have now started using given X mean we are implying association? So we are saying that we can write an causal equation in terms of associations/ observed data?

My second point of confusion is how we estimate from data . Now we have an estimand e.g. like above we can calculate the causal effect. But when we say we 'calculate it using the observed data' how are we doing this exactly e.g. computing the probabilities - is the estimand a 'recipe' on an individual level? Some information on HOW it is estimated with a working example would be very useful.

  • 1
    $\begingroup$ Question 1: Yes, that's the purpose of the identification result - to express a counterfactual term as an observed data term. Question 2: This is not a causal inference question and Pearl's approach to causality (or any other approach) doesn't prescribe how to estimate observed data terms. This is a statistical question for which a deep theory exists. A simple estimator is a "plug in" estimator, in which you first estimate P(Y|X,Z) and P(Z) then second plug those into the expression. $\endgroup$
    – Ben
    Commented Jul 11, 2022 at 20:34

1 Answer 1


Good question! It was a while since I read Pearl, so I had to go back to my notes. I'm not sure if my explanation will be satisfactory to you, but I'll try.

First of all, it's very important to understand what P(Y|X) means. It could mean that X increases the propensity for ("cause") Y directly. This is the case where e.g. X is "speeding in a residential area" and Y is "pedestrian killed".

However, P(Y|X) could also mean that something else increases the propensity of both X and Y. In my country, we could have Y being "melanoma" and X being "high vacation spending". This is not because high vacation spending directly cause melanoma, it's because people who fly to sunny destinations every year both expose themselves to sun and spend a lot of money on vacations.

The way to interpret P(Y|X) is not that X causes Y, but rather that X is evidence of a situation in which Y is more likely.

With that out of the way, we can move on to your actual question. In the equation you wrote down, Z is that "situation in which Y is more likely", the one that X is evidence for.

More concretely, Z could be "vacations in the sun".

To find out if high travel costs truly cause melanoma directly, we have to control for "vacations in the sun". The experimental way to do this is to round up a bunch of random people, force half of them to spend a lot on their vacations (without changing anything else about their vacation plans) and then seeing if those people develop melanoma more often (then it would be just from spending.)

That experiment is what Judea would write down as P(Y|do(X)). Ensure people spend, see how likely melanoma is then.

We can't do that experiment, obviously.

But we can simulate the experiment from observational data, if our causal model is complete enough! That's a really neat trick.

In the sum

$$\sum_Z P(Y|X,Z) P(Z)$$

Two things are happening: first, we are looking at the probability of Y given that we are informed about both X and Z. In the example, this means that someone asks us to predict a person's risk of melanoma with information on their vacation spending and how often they vacation in the sun. We intuitively know the outcome here: since spending doesn't cause melanoma, we can think of this as predicting melanoma given just the information about vacations in the sun. (Then we multiply by the probability of spending vacations in the sun to get the proper joint probability.)

A more interesting example could be where e.g. Y is bad health, and X is low income. We know that low income causes bad health, we know that bad health causes low income, and we know that a lot of things can cause both bad health and low income.

Importantly, P(Y|X) means "if we observe someone with low income, what's their probability of bad health?" Note that this says nothing about low income causing bad health. It's saying that low income is evidence of a situation and personal history that can also lead to bad health.

There are many such things: war, drought, etc, etc.

The meaning of P(Y|do(X)) in that context is "if we force people to have bad income, what happens to their risk of bad health?"

This eliminates all other variables (war, drought, etc) from the equation and makes it all about the effect of income on health.

Observationally, we look instead at the sum:

$$P(health|income, war) P(war) + ...$$

Meaning "what would we predict about someone's health given that we know they have low income and have suffered from war? And by the way, how many people suffer from war, more generally?"

When we do the same for all the hidden factors, we have accounted for them separately and they no longer influence the probability of bad health any more than they do in the baseline population. Thus any effect we see is purely from low income.

  • $\begingroup$ Pearl's causal inference calculus acts as if $n=\infty$. It doesn't deal with data uncertainties. So it's problematic to some statisticians. Contrast that with Bayesian causal reasoning, e.g. this. $\endgroup$ Commented Jul 11, 2022 at 20:58
  • $\begingroup$ what about the estimation part e.g. how can we approximate the statisical estimand using propensity scores? $\endgroup$
    – Maths12
    Commented May 25, 2023 at 14:25

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