IPW computation

[winston-zillow]

Briefly read the codes around IPW, in particular this line:

# weight_matrix = 1.0 / probabilities
weight_matrix = probabilities.rdiv(1.0)

My understanding of IPW is that outcomes are weighted by: $w_i = \frac{a_i}{e_i} + \frac{(1 - a_i)}{(1 - e_i)}$. do I miss something or this is a bug?

[ehudkr]

Hi Winston, not a bug 🙃 🐞

The equation you presented is limited to binary treatment scenarios, while causallib's implementation scales to arbitrary number of treatment values. Reformulating your equation in the binary case, treated observations ($A_i=1$) are weighted with $\frac{1}{\Pr[A=1|X]}$ and control observations ($A_i=0$) are weighted with $\frac{1}{1-\Pr[A=1|X]} = \frac{1}{\Pr[A=0|X]}$. The generalization is therefore quite straightforward if we observe that each observation $i$ is weighted by the inverse probability of being assigned to their observed treatment group: $w_i=\frac{1}{\Pr[A=a_i|X=x_i]}$. Causallib's implementation tries to make the most generalized calculation by first considering for each $i$ the probabilities for every possible treatment (hence the probabilities matrix, which is similar to the output of a predict_proba()), and only slicing for the required values according to the treatment assignment at the very end.

Thanks for bringing this up, and sorry for causing any confusion. I hope I was able to clear some of it up. Please feel free if there's anything else.

[ehudkr]

To justify this further. There are other small benefits to this implementation that I have yet to seize. For example, generalizing the average treatment effect on the treated/controls (ATT/ATC) to any arbitrary treatment level v can be simply done by multiplying the inverse probability matrix (weight_matrix) by the corresponding probability column: weight_matrix *= probabilities[v]. Then the column corresponding to v is all ones ($p_{i,v}$ times its inverse) and all the rest of the columns are the odds with respect to v.

I hope this gives better context to the design decisions.

[winston-zillow]

I see. You turn the binary one-class label to a multi-class label first. That works. Thanks.