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Consider $X, Y$ and suppose you have some i.i.d. observations of $Y$ and $X + \text{do}(Y)$ with observation noise $\epsilon_0$ and $\epsilon_1$ (Gaussian). So the observations of the latter should not influence out estimate of $Y$.

Is there some nice way of expressing this when doing the analysis (I've made this problem very simple, but consider doing regression or bayesian updates).

So you have some log-likelihood like this (for example)

$$ J = \frac{(\hat{Z}_0 - Y)^2}{\epsilon_0^2} + \frac{(\hat{Z}_1 - \text{do}(Y) - X)^2}{\epsilon_1^2} $$

Are there any tricks to using the DAG matrix to effectively do a stop gradient?

This post seems related? Mathematical notation for suppressing differentiation

Another way of asking this question is suppose you have a standard algorithm for dealing with the non-causal situation here (consider a filter or online mean) is there a simple way besides recursively solving the system, to change that solution into a causal one?

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  • $\begingroup$ What are you trying to do with the log-likelihood? As in, what does the model you have in mind currently do, and what should it be able to do after making it "causal"? $\endgroup$
    – Scriddie
    Jun 6, 2023 at 12:48

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