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General ADMG ID Algorithm

This vignette introduces the general ADMG identification tools in CausalGraphs.jl. identify() is the estimator-routing function: it checks the effect classes for which estimate_causal() has estimators. ID_algorithm() is the more general symbolic Pearl-Shpitser ID check for ADMG total-effect queries.

How the ID Algorithm Works

The ID algorithm answers one question:

Can P(Y | do(A)) be written using only the observed joint distribution P(V)?

Here V is the set of observed vertices in the ADMG, A is the intervention, and Y is the outcome. Directed edges represent causal relations among observed variables. Bidirected edges represent latent common causes. Identification means that, despite those latent common causes, the intervention distribution has a unique expression in terms of the observed law.

The recursive algorithm repeatedly simplifies the query:

  1. Restrict to ancestors of the outcome. If some observed variables are not ancestors of Y, they cannot affect Y under the current intervention query. The algorithm sums them out and continues in the ancestral subgraph.
  2. Add irrelevant interventions. After removing incoming arrows into A, some variables may no longer be ancestors of Y. Those variables can be treated as fixed as well, because changing them cannot change the target distribution.
  3. Remove the intervention nodes and inspect districts. In the graph induced by V \ A, districts are the bidirected-connected components. Multiple districts mean the post-intervention law factorizes into smaller pieces, so ID recursively identifies each district and multiplies the results.
  4. Handle a single remaining district. If the remaining district is already a district in the original graph, the needed factor can be read from the observed joint distribution using a topological product of kernels. If it is contained in a larger district, the algorithm recurses inside that larger district with the corresponding kernel as the new source law.
  5. Report a hedge when no reduction is possible. If the graph is one large district and the intervention cannot separate the outcome from hidden confounding, ID fails. The returned hedge is a pair of district structures witnessing non-identification.

The output is not an estimate. It is a symbolic functional built from sums, products, joint probabilities, and conditional kernels. Estimation is a second step: for finite-support variables, estimate_id() evaluates that functional by plugging in empirical probabilities.

Front-Door as ID

The front-door graph has hidden treatment-outcome confounding, represented by A <-> Y, but the effect is still identified through the mediator M.

using CausalGraphs

g_fd = make_graph(
    vertices = [:A, :M, :Y],
    di_edges = [(:A, :M), (:M, :Y)],
    bi_edges = [(:A, :Y)],
)

draw_graph(g_fd; direction="LR")
%3 A A M M A->M Y Y A->Y M->Y

For estimation, this graph is p-fixable, so identify() routes to the NPS-TMLE estimator.

identify(g_fd, :A, :Y).strategy
:p_fixable

The same graph is also identified by the general ID algorithm:

id_fd = ID_algorithm(g_fd, :A, :Y)
(identified = id_fd.identified,
 expression = string(id_fd.expression))
(identified = true, expression = "sum_{M} P(M | A) * sum_{A} P(A) * P(Y | A, M)")

The printed expression is intentionally compact. In the front-door graph it is the familiar formula: average the mediator distribution under the intervention, then average the mediator-outcome regression over the observed treatment distribution.

Finite-Support Plug-In Estimation

For discrete variables, estimate_id() can evaluate the symbolic ID functional automatically by enumerating the observed support and plugging in empirical conditional probabilities.

using DataFrames, Random
Random.seed!(24)

n = 800
A = Float64.(rand(n) .< 0.5)
M = Float64.(rand(n) .< (0.2 .+ 0.6 .* A))
Y = Float64.(rand(n) .< (0.1 .+ 0.2 .* A .+ 0.4 .* M))
df = DataFrame(A=A, M=M, Y=Y)

id_est = estimate_id(a=[1, 0], data=df, graph=g_fd, treatment=:A, outcome=:Y)
r = x -> round(x, sigdigits=4)
(ACE = r(id_est[:IDPlugin].ACE),
 lower_ci = r(id_est[:IDPlugin].lower_ci),
 upper_ci = r(id_est[:IDPlugin].upper_ci),
 SE = r(id_est[:IDPlugin].standard_error),
 total_probability_a1 = r(id_est[:IDPlugin].total_probability_a1),
 total_probability_a0 = r(id_est[:IDPlugin].total_probability_a0))
(ACE = 0.2429, lower_ci = 0.1895, upper_ci = 0.2962, SE = 0.0272, total_probability_a1 = 1.0, total_probability_a0 = 1.0)

The same plug-in route is used by estimate_causal() when identify() returns :id_algorithm. The estimator is deliberately labeled :IDPlugin: it is an automatic finite-support estimator with a finite-support EIF standard error, not a general TMLE for arbitrary continuous ID functionals.

Non-Identification

The bow graph has both A -> Y and hidden confounding A <-> Y. The observed association cannot separate the directed effect from the latent common cause.

g_bow = make_graph(
    vertices = [:A, :Y],
    di_edges = [(:A, :Y)],
    bi_edges = [(:A, :Y)],
)

draw_graph(g_bow; direction="LR")
%3 A A Y Y A->Y A->Y 
id_bow = ID_algorithm(g_bow, :A, :Y)
(identified = id_bow.identified,
 hedge = id_bow.hedge)
(identified = false, hedge = (S = [:Y], F = [:A, :Y], treatment = [:A], outcome = [:Y]))

When ID fails, the returned hedge is a graph-theoretic witness of non-identification. In practice, this is more useful than a bare false because it points to the district structure responsible for the failure.

Fixing Sequences

Fixing is the graph operation that turns random vertices into intervention-like or conditioned-on vertices by removing incoming directed edges and bidirected edges involving the fixed node. A vertex is fixable when it is not in the same district as one of its proper descendants.

g_bd = make_graph(
    vertices = [:X, :A, :Y],
    di_edges = [(:X, :A), (:X, :Y), (:A, :Y)],
)

fixing_sequence(g_bd, [:A])
(fixable = true, fixing_order = [:A], graph = ADMG([:X, :A, :Y], [(:X, :Y), (:A, :Y)], Tuple{Symbol, Symbol}[], Dict{Symbol, Vector{Symbol}}(), Dict{Symbol, Bool}(:A => 1, :X => 0, :Y => 0)))

For the backdoor graph, fixing A succeeds immediately because A is not bidirected-connected to its descendant Y.

Nested-Fixability Diagnostics

Nested-fixability uses fixing operations inside induced subgraphs. It is more general than ordinary backdoor adjustment and p-fixability, and it underlies the ANIPW/NIPW estimator route in this package.

g_hiv = make_graph(
    vertices = [:ViralLoad, :Income, :Exercise, :T, :Toxicity, :Adherence, :CD4],
    di_edges = [
        (:ViralLoad, :T), (:ViralLoad, :CD4), (:Income, :T), (:Exercise, :CD4),
        (:T, :Toxicity), (:T, :CD4), (:Toxicity, :Adherence), (:Adherence, :CD4),
    ],
    bi_edges = [
        (:Income, :Toxicity), (:Exercise, :Income), (:Exercise, :T),
        (:ViralLoad, :Adherence), (:ViralLoad, :CD4),
    ],
)

nested = nested_fixability(g_hiv, :T, :CD4)
(a_fixable = is_fix(g_hiv, :T),
 p_fixable = is_p_fix(g_hiv, :T),
 nested_identified = nested.identified,
 ystar = nested.ystar,
 nested_order = nested.n_order)
(a_fixable = false, p_fixable = false, nested_identified = true, ystar = [:CD4, :ViralLoad, :Exercise, :Adherence, :Toxicity], nested_order = [:ViralLoad, :Income, :T, :Exercise, :Toxicity, :Adherence, :CD4])

The nested diagnostics show which outcome ancestors remain after fixing treatment and which districts must be reachable by fixing vertices outside the district.

Practical Boundary

Use the functions at different levels of generality:

  • is_fix() and is_p_fix() are simple graph checks.
  • nested_fixability() explains the nested/fixing route used by the ANIPW estimator.
  • ID_algorithm() answers the more general symbolic ADMG identification question and returns a hedge when the effect is not identified.
  • identify() combines these checks for package routing.
  • estimate_id() estimates symbolic ID functionals by finite-support plug-in when the variables are discrete.
  • estimate_causal() estimates a-fixable, p-fixable, nested-fixable, and discrete ID-algorithm functionals.

General ID can produce complicated functionals. Those functionals are useful identification results. For continuous or high-dimensional variables, they still require specialized estimation machinery beyond finite support enumeration.