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Graphical Representation of Causal Effects

A causal graph consists the following elements:

  • Nodes in causal graphs represent random variables.
  • Edges in causal graphs represent direct causal effects.
    • A trail is a sequence of edges that does not meet the same node twice.
    • A path is a trail that all the edges are pointing in the same direction.
  • Conventionally, time flows from left to right in causal graphs.
  • Causal graphs follow the properties of Bayesian networks (acyclic, markov factorization).

Markov Factorization

The joint distribution of a set of random variables can be factorized into a product of conditional distributions of each variable given its parents in the graph.

\[ P(X_1, \dots, X_n) = \prod_{i=1}^n P(X_i \mid \text{Pa}(X_i)) \]

where \(\text{Pa}(X_i)\) is the set of parents of \(X_i\).

Markov factorization is equivalent to the following local independence property:

\[ X_i \perp \text{ND}(X_i) \mid \text{Pa}(X_i) \]

where \(\text{ND}(X_i)\) is the set of non-descendants of \(X_i\).

A causal graph contains the following building blocks: fork, chain, collider.

D-seperation and Independence

Conditional independence relations are encoded in the graph structure.

Blocked Trails and D-seperation

Trails in causal graphs are regarded as blocked or open according to the following rules:

  1. If there are no nodes being observed, a path is blocked if and only if there exists one or more colliders on the path.
  2. If the trail contains a non-collider that is observed, the trail is blocked.
  3. A collider which is observed does not block the trail.
  4. A collider which any of its descendants is observed does not block the trail.

A trail is blocked if it satisfies any of the following conditions:

  1. There exists a collider on the trail that itself or all of its descendants are not observed.
  2. There exists a non-collider on the trail that is observed.

If all possible trails between two nodes \(X, Y\) are blocked \(Z\), the two nodes are d-separated. D-separated nodes are conditionally independent given the observed nodes, i.e., \(X \perp Y \mid Z\).

Faithfulness assumption

Faithfulness assumptions states that the observed conditional independence relations are consistent with the graph structure. That is: observed conditional independence \(X \perp Y \mid Z\) implies that \(X\) and \(Y\) are d-separated given \(Z\) in the graph.

Flows of Association and Causation

If two nodes are not d-separated, there exists at least one open trail between the nodes.

  • Association flows along open trails.
  • Causation flows along open paths.

There may be multiple trails and paths between two nodes, each trail provides a flow of association or causation. The overall observed association is affected by all the flows of association and causation. Existence of undesired flows of association may lead to lack of exchangeability and biased estimation of causal effects. If there are no flow of association between two nodes, the association is causal.

There two types of flows of association:

  1. Common cause: two nodes are associated because they are both affected by a common cause.

  2. Condition on common effect: two nodes are associated because they are both affecting an observed common effect.

Blocking a trail can be regarded as blocking the flow of association or causation on the graph.

Hidden Properties

Causal graphs only encode conditional independence relations and flows of causation or association. Some properties or assumptions cannot be represent in causal graphs.

  • Positivity: Positivity in causal graphs can be represented that all of the edges in the graph are not deterministic.
  • Consistency: Consistency is implicitly assumed in causal graphs.

In a causal graph, the treatment node should lead to well-defined treatment.\

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