Identification Algorithms

backdoor_adjustment_set(g, X, Y)

Find a valid backdoor adjustment set for estimating the causal effect of X on Y.

Delegates to CausalInference.find_min_backdoor_adjustment. Returns nothing if no valid set exists (false from CausalInference).

Arguments

• g: Directed acyclic graph
• X: Treatment node
• Y: Outcome node

Returns

• Set{Int} of adjustment nodes (possibly empty if no backdoors), or nothing

is_backdoor_adjustable(g, X, Y)

Return true if a backdoor adjustment set exists (including the empty set when there are no backdoor paths).

frontdoor_adjustment_set(g, X, Y, M)

Check if M is a valid frontdoor adjustment set for estimating the causal effect of X on Y.

The frontdoor criterion states that a set M is a valid frontdoor adjustment set if:

1. M blocks all directed paths from X to Y
2. There are no backdoor paths from X to M
3. All backdoor paths from M to Y are blocked by X

Uses reachability (BFS), not path enumeration — safe on dense DAGs.

Arguments

• g: Directed acyclic graph
• X: Treatment node
• Y: Outcome node
• M: Potential mediator set

Returns

• true if M is a valid frontdoor adjustment set, false otherwise

find_frontdoor_mediators(g, X, Y)

Find single-node frontdoor mediators between X and Y.

Candidates are nodes on some directed path X → Y (nodes_on_directed_paths), checked with frontdoor_adjustment_set. Avoids enumerating all simple paths.

find_instruments(g, X, Y)

Find instrumental variables for estimating the causal effect of X on Y.

An instrumental variable Z must satisfy:

1. Z has a causal effect on X
2. Z affects Y only through X (exclusion restriction)
3. Z is independent of confounders of X and Y (independence)

Arguments

• g: Directed acyclic graph
• X: Treatment node
• Y: Outcome node

Returns

• Vector of potential instrumental variables

Examples

using CausalDynamics, Graphs

# IV example
g = DiGraph(4)
add_edge!(g, 1, 2)  # Z → X
add_edge!(g, 2, 3)  # X → Y
add_edge!(g, 4, 2)  # U → X
add_edge!(g, 4, 3)  # U → Y

# Z is a valid instrument
instruments = find_instruments(g, 2, 3)  # [1]

References

• Angrist, J. D., & Pischke, J. S. (2009). Mostly Harmless Econometrics

is_valid_instrument(g, Z, X, Y)

Check if Z is a valid instrumental variable for X → Y.

An instrument Z must satisfy three conditions:

1. Relevance: Z has a causal effect on X (directed path Z → X)
2. Exclusion restriction: Z affects Y only through X (all paths Z → Y go through X)
3. Independence: Z is independent of confounders (no backdoor paths Z → Y)

Arguments

• g::AbstractGraph: Directed acyclic graph
• Z::Int: Potential instrument node
• X::Int: Treatment node
• Y::Int: Outcome node

Returns

• Bool: true if Z is a valid instrument, false otherwise

Examples

using CausalDynamics, Graphs

# Valid instrument
g = DiGraph(4)
add_edge!(g, 1, 2)  # Z → X
add_edge!(g, 2, 3)  # X → Y
add_edge!(g, 4, 2)  # U → X
add_edge!(g, 4, 3)  # U → Y

is_valid_instrument(g, 1, 2, 3)  # true

# Invalid: Z has direct path to Y
g2 = DiGraph(3)
add_edge!(g2, 1, 2)  # Z → X
add_edge!(g2, 1, 3)  # Z → Y (violates exclusion)
add_edge!(g2, 2, 3)  # X → Y

is_valid_instrument(g2, 1, 2, 3)  # false

Notes

• The independence condition (3) is checked conservatively: if any backdoor paths exist, returns false
• In practice, independence may hold even with backdoor paths if they are blocked

References

• Angrist, J. D., & Pischke, J. S. (2009). Mostly Harmless Econometrics, Chapter 4

find_all_adjustment_sets(g, X, Y)

List valid backdoor adjustment sets via CausalInference.list_backdoor_adjustment.

is_valid_adjustment_set(g, X, Y, Z)

Check if Z is a valid backdoor adjustment set via CausalInference.

minimal_adjustment_set(g, X, Y)

Minimal backdoor adjustment set (same as backdoor_adjustment_set).