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an> The idea is that although the treatment and potential outcomes may be unconditionally associated (due to confounding), within levels of π , they are not associated. In other words, there is <span>no confounding within levels of π because controlling for π has made the treatment groups comparable. <span>

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Association flows along all unblocked paths. In causal graphs, causation flows along directed paths. Recall from Section 1.3.2 that not only is association not causation, but causation is a sub-category of association. Thatβs why association and causation both flow along directed paths.

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The minimal building blocks of DAGs consist of chains, forks, immoralities, two unconnected nodes, and two connected nodes.

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The main reason for moving from exchangeability (Assumption 2.1) to conditional exchangeability (Assumption 2.2) was that it seemed like a more realistic assumption

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Definition 3.4 (d-separation) Two (sets of) nodes π and π are d-separated by a set of nodes π if all of the paths between (any node in) π and (any node in) π are blocked by π Source: Pearl (1988), Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference

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We assumed π is discrete when we summed over its values, but we can simply replace the sum with an integral if π is continuous. Throughout this book, that will be the case, so we usually wonβt point it out To jest kluczowe rΓ³wnanie.

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sfying unconfoundedness, it can lead to a higher chance of violating positivity. As we increase the dimension of the covariates, we make the subgroups for any level π₯ of the covariates smaller. <span>The Positivity-Unconfoundedness Tradeoff This is related to the curse of dimensionality. As each subgroup gets smaller, there is a higher and higher chance that either the whole subgroup will have treatment or the whole subgroup will have control. For example, once the size of any subgroup has decreased to one, positivity is guaranteed to not hold. <span>

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Exchangeability means that the treatment groups are exchangeable in the sense that if they were swapped, the new treatment group would observe the same outcomes as the old treatment group