Probabilistic Graphical Models

Probabilistic Graphical Model

= a joint probability distribution that uses a graph structure to encode conditional independence assumptions.

A Graph (or Network) is a set of entities (called Nodes or Vertices) connected through Edges.

A node has N degree of the node = N outgoing and incoming edges
Two nodes directly connected by an edge are said to be Adjacent
Parent = the node at the “entrance” to the edge, Child = the one at the “exit” of the edge
path = sequences of nodes connected by edges

= a type of DAG that represents probabilistic relationships between variables (although there is nothing inherently Bayesian about such models)

each node is a random variable
each edge is a directed connection indicating a probabilistic dependency (not just correlation or causation, but conditional probability)
each node is conditionally independent of all its predecessors given its parents, i.e. showing Markov property (Markov Chain#^b013c4)

Confound type	DAG structure (schematic)	Description	Adjustment implications / Take care...	Example
The Fork	X ← Z → Y	Z is a common cause of X and Y. This creates a spurious association between X and Y if not controlled.	Must adjust for Z to block confounding.	Age (Z) affects both exercise (X) and health outcomes (Y).
The Pipe	X → Z → Y	Z is a mediator (on the causal pathway from X to Y).	Adjusting for Z blocks part of the causal effect → post-treatment bias. Don’t condition if interested in total effect.	Smoking (X) → blood pressure (Z) → heart disease (Y).
The Collider	X → Z ← Y	Z is a collider (caused by both X and Y). By default, X and Y are independent. Conditioning on Z opens a spurious path → collider bias.	Do not adjust for Z.	Genetics (X) and environment (Y) both affect admission to college (Z). Conditioning on Z creates a false association.

= A graphical method for identifying a set of variables to adjust for, in order to estimate the causal effect of X on Y

Idea: Block all “back-door paths” (paths that go into X through an arrow pointing backward) between treatment and outcome.
Formal definition: A set of variables Z satisfies the back-door criterion relative to (X, Y) if:
1. No node in Z is a descendant of X
2. Z blocks every path between X and Y that has an arrow into X
Adjustment formula:

P (Y ∣ d o (X)) = \sum_{z} P (Y ∣ X, Z = z) P (Z = z)

Intuition: Control for confounders that create spurious associations.
Example: Age is a confounder of Exercise (X) and Health (Y). Adjusting for Age satisfies the back-door criterion.

= A graphical method that can identify causal effects even with unmeasured confounders, if there exists a mediator we can observe.

Idea: Use a mediator M that transmits the effect of X on Y.
Conditions (Pearl):
1. M intercepts all directed paths from X to Y
2. There are no unblocked back-door paths from X to M
3. All back-door paths from M to Y are blocked by X
Adjustment formula:

P (Y ∣ d o (X)) = \sum_{m} P (M ∣ X) \sum_{x^{'}} P (Y ∣ M, X = x^{'}) P (X = x^{'})

Intuition: Even if X and Y are confounded, we can identify the causal effect through the mediator M.
Example: X = smoking, M = tar deposits in lungs, Y = lung cancer.
- There may be unmeasured confounders (e.g., genetic risk) affecting smoking and lung cancer, but tar (M) perfectly mediates the causal pathway.