Bayesian Principal Stratification: Modeling Treatment Effects

So there is some, I think like Avi Feller and some other people have done some Bayesian principles graphification models in Stan.

And the idea is that you would characterize people that there's a latent variable.

Well, it's observed for the people who got the mailer and not observed for the others.

And it's whether you would have, well, I can't remember how it works, but like the people who, the latent variable is, If you receive the stage one treatment, would you do the stage two treatment? And for the people who did receive the stage one treatment, you know that.

But the people who didn't, you don't know that.

But you can form a model.

And you can do inference on that.

You'd want to use pre-treatment predictors.

You'd want to use characteristics of the people in the study, their age and where they live and whatever spending patterns they have in the past.

And then you should be able to do that model.

And that will, then you can then, from that, you can estimate the effect of the, you can compare the people who would or would not do the stage two treatment, and you can estimate what would happen if someone who didn't do it were to do it.

But basically you have to model the whole process.

I haven't done this myself, but I know people have done it.

It seems like a very natural Bayesian problem.

And I think that standard solutions such as instrumental variables would correspond to special cases of this model, assuming various things are zero or with flat priors and various...

Usually standard procedures usually correspond to some mix of some flat priors and some parameters and priors with spikes at zero on others.

Okay.

Yeah.

And so Richard, I'm sure you want to say something on that.

That sounds reasonable to me.

I recognize this as the per protocol or intent to treat kind of problem.

That's a big literature.

What Andrew says sounds like it's very sensible there.

You try to do the best you can, but you've got to be careful about what you're adding because now you have...

It's this downstream from treatment problem.

But people have written a lot about this.

But this is the per protocol intent to treat sort of issue.

If you want to figure out mechanism, how the treatment's really working, it's hard, even if the treatment's randomized.

But there are ways to do it, as Andrew has discussed.

And that's the way I approach it.

Yeah.

Awesome.

Yeah, I love these problems.

I think these kind of hard causal inference problems are among the most interesting ones because also that's where you get to do the most customized models.

And these are the fun ones where you have to, these are kind of Lego bricks that you want to add together.

So I love working on these kind of models.