Lg updates causal effects

[yuchen-zhu]

Hey there! Here comes the reviews:

For ATE:

• I think ATE is in general just the difference in expected causal consequence between two actions? And the binary case is just a (perhaps useful) special case. So I would say the definition of ATE should be:

Given two random variables X and Y, and suppose X is intervened at two different place, say x_1 and x_2. The Average Treatment Effect(ATE) from x_1 to x_2 is defined as

ATE(x_1, x_2) = E[Y|do(X=x_2)] - E[Y|do(x_1)].

Often, X is a binary random variable, e.g. treatment or no treatment in medical trials, and in cases when this is clear, we often just write ATE to mean ATE = E[Y|do(X=1)] - E[Y|do(X=0)].

• The intuition you then gave is great! Do you think it's better to put it outside of the bounding box of the definition or do you think it's better to leave it there?

• Wait, usually when we say ATE, are we actually assuming that X is binary? In this case, it is possibly better to just state that assumption upfront, and say why this assumption is commonly used, before defining it. And then say how to extend to categorical or continuous.

• typo: '...... higher socio-economic status, that in turn* might .....'

For CATE & ITE:

• Comments about ATE carries over I think, but otherwise I think it's great!

[yuchen-zhu]

For the selection bias page:

• Note that there is a wikipedia article for selection bias, could be a good idea to link to that.

• Feels like selection bias is a well-known phenomenon that is well studied in statistics (given the wikipedia page), so it feels like perhaps a contribution we can make is to illustrate it with some graphical models like in the fusion paper? For example, we can say when both the treatment and the outcome affect the sampling, that introduces a collider at the samples. When we observe those samples, we unblock the path and introduce a spurious path from X to Y, affecting our causal estimates (hence linking it back to causal inference).

• What do you think about this structure for the selection page then: -- Keep the first two paragraphs which essentially say what sampling bias is and give intuitions. -- Given some graphical examples (let's say two?) of sampling bias; remark that there is a wikipedia page on it and link to that. -- Make the third paragraph under the heading 'ways to deal with sampling bias for causal inference'

I'm happy to share this load if you want!