# Probabistic formulation of (Rubin's?) Causal Inference Model: Do I have the right picture in mind?

Over the course of my life (not that long!), I've probably gone over a dozen time over section 3.2.1 "The Conditional Independence Assumption" of "Mostly Harmless Econometrics" which lays down --- what I understand to be --- Rubin's Causal Inference model.

Each time I read the section, I form a mental representation of the model which I convince myself makes perfect sense. But then, invariantly, I ended up going back to reading the section because something just doesn't make sense further down the line. So I open section 3.2.1 again and promise myself that, this time, I will get this straight, just to restart the cycle a couple of months later.

This is one of those times, except I am tired of failing over and over and I am really decided to make this crystal-clear.

I realized one thing that was causing this endless cycle was that I never quite got a clear picture of the model from a basic probabilistic standpoint. More precisely, I was never quite clear on what in the model was a random variable, what wasn't, what sample space these random variables related to, etc.

As often, when one needs some "informal" presentation to be laid down in a more systematic probabilistic way, Wasserman's "All of Statistics" does it. So I consulted chapter 16, and I feel like --- hope --- that the stars are starting to align. Still:

1. I really want to make sure I get it this time, and
2. I felt like doing so requires me to go one level lower than Wasserman and start with a sample space rather than right away with random variables, so
3. I decided to make a picture of my understanding of the model from a sample space perspective.

What I want to ask is whether my picture below is an accurate representation of the causal model that Wasserman lays down in Chapter 16. I am not claiming it is enlighting in any way, or that it would help anyone other than myself understand what the model is. I am just asking whether it is formally accurate as a representation of Rubin's Causal Model (or rather of an example thereof, as the values of random variables and many other aspects of the picture below are obviously completely arbitrary).

The picture is below.

In Wasserman's notation (beginning of Chapter 16):

• "$$X$$ is a binary treatment variable where $$X = 1$$ means 'treated' and $$X = 0$$ means 'not treated'",
• "$$Y$$ [is] some outcome variable such as presence or absence of disease".
• "$$C_0$$ is the outcome if the subject is not treated ($$X = 0$$) and $$C_1$$ is the outcome if the subject is treated ($$X = 1$$)", and
• the so-called "consistency relationship" requires that $$Y = C_0$$ if $$X=0$$ and $$Y= C_1$$ if $$X=1$$.

In my picture below,

1. I view the sample outcomes $$a,b,c,d,e,$$ and $$f$$ as potential "subjects" to the experiment (understood in a broad sense), and
2. I view the sample space $$S$$ as being made of the six sample outcomes $$a,b,c,d,e,$$ and $$f$$ only.

• Pearl's $X \to Y$ is so much simpler, in my opinion. Mar 12, 2020 at 19:21
• @Adrian Keister: Understanding Pearl's framework is the next item on my bucket list :) (and the next chapter in Wasserman!). I am a little old school in that way, trained as an economist in an age were moving econometrics classes from Wooldridge to Mostly Harmless already seemed like a revolution, and people were not ready for the next move from Mostly Harmless to Pearl (are they now? Is it even where econmetrics education should go? Not in a position to judge). So I figured before trying to understand something I was not taught, I would first try to really understand what I was taught.
– FZS
Mar 12, 2020 at 19:30