Notes: Elements of Causal Inference (Peters, Janzing, Schölkopf)

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This seems a conceptually novel way of looking at equivalence of partial pooling in causal models. Might be exact equivalence though.

Though it is worth noting that the independence seems to be a bigger term than in statistics. Independence of cause and mechanism - mechanism being $p(t|a)$ and cause being $p(a).$ If we start changing the cause distribution then the t → a causal model does not keep the mechanism intact.

The generative process or mechanism generating the effect from the cause is indpeendent of the generative process or mechanism that is generating the cause.

Three differnet visions of independence.

1. Mechanistic - the light, the position of a chair and angle you are watching the chair are indpendent and there exists the assumption of generic viewpoint. You can change the light, position of the chair or perspective without being surprised.

2. Information theoretic independence - learning about the structure of the conditional does not in any way inform you about the structure of the prior.

3. Independence of noises - where is noise generated from with regards to the generative process. Not sure how this generalises last - it was mentioned that this can model or homogeneous processes as well. To break independence we can look at mechanisms or subprocesses $f_k$ such that the process is selected randomly when rv $X_t=k$. When we for example know that $X_{t+1}=X_t+1$ then processes $f_k$ at $\tau =t$ or $\tau =t+1$ are not independent.

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Problem 3.5.

from plots import hist_demo
 
hist_demo()

Problem 3.6

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Problem 3.8 Test ABC4

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Problem 4.15 (ANMs)

Problem 4.16 (Maximum likelihood)

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Problem 5.1 (Independence of mechanisms)

Helpers: convolution and deconvolution